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Applying Graph Theory in
Ecological Research
MARK R. T. DALE
University of Northern British Columbia

www.cambridge.org
Information on this title: www.cambridge.org/9781107089310
DOI: 10.1017/9781316105450
C
Cambridge University Press 2017
First published 2017
Printed in the United Kingdom by TJ International Ltd. Padstow Cornwall
A catalogue record for this publication is available from the British Library
ISBN 978-1-107-08931-0 Hardback
ISBN 978-1-107-46097-3 Paperback

Contents
Preface page x
1 Graphs as Structure in the Ecological Context 1
Introduction 1
1.1 Graphs as Structure 3
1.2 Graphs and Ecological Relationships 10
1.3 Graphs and Locations: Spatial and Temporal 11
1.4 Networks and Dynamics 17
1.5 Graphs and Data 18
1.6 Ecological Hypotheses and Graph Theory 27
1.7 Statistical Tests and Hypothesis Evaluation 29
1.8 Concluding Comments 35
2 Shapes of Graphs: Trees to Triangles 37
Introduction 37
2.1 Acyclic Graphs 37
2.2 Digraphs and Directed Acyclic Graphs 41
2.3 Weighted Directed Trees 45
2.4 Lattice Graphs 46
2.5 Triangles 49
2.6 Smaller Than Triangles: Singletons, Isolated Pairs and Whiskers 50
2.7 How It Looks 51
3 Species Interaction Networks 54
Introduction 54
3.1 Objects 57
3.2 Properties 60
3.3 Generative Models 68
3.4 Comparisons 72

4 Trophic Networks: Structure, Partitioning and Dynamics 79
Introduction 79
4.1 Trophic Networks and Derived Graphs 82
4.2 Trophic Network Characteristics 86
5 Species Associations, Communities and Graphs of Social Structure 105
Introduction 105
5.1 Graphs of Social Structure 107
5.2 Cluster Detection in Graphs and Networks 113
5.3 Transitivity and Reciprocity 121
5.4 Balance 122
5.5 Change 124
5.6 Key Nodes; Key Edges 126
6 Competition: Hierarchies and Reversals 128
Introduction 128
6.1 Concepts for Competition Interaction Graphs 130
6.2 Measuring Competitive Outcomes 135
6.3 Choosing Edges and Finding Hierarchies 137
6.4 Example: Arabidopsis thaliana Ecotypes 141
7 Mutualism, Parasitism and Bipartite Graphs 147
Introduction 147
7.1 Internal Structure of Bipartite Graphs 147
7.2 Applications of Bipartite Graphs 158
8 Temporal or Time-Only Graphs 164
Introduction 164
8.1 Properties of Temporal Graphs 170
8.2 Techniques for Temporal Graphs: Testing Significance 180
8.3 Applications of Techniques 183
8.4 Conclusions and Advice 185
9 Spatial Graphs 191
Introduction 191
9.1 Properties of Spatial Graphs 193

9.2 Techniques for Spatial Graphs: Testing Significance and Other
Assessments 201
9.3 Choice and Applications of Techniques 202
10 Spatio-temporal Graphs 222
Introduction 222
10.1 Characteristics 226
10.2 Two Spatio-temporal Properties 228
10.3 Examples of Ecological Applications 232
11 Graph Structure and System Function: Graphlet Methods 252
Introduction 252
11.1 Graphs for Structure and Dynamics in Ecological Systems 257
11.2 Graph Characteristics and Methods Based on Graphlets 259
12 Synthesis and Future Directions 271
Introduction 271
12.1 Comparisons and Matching 271
12.2 What Next? 276
Glossary 286
References 297
Index 328
Appendix 333
Colour plates are found between pages 212 and 213.

Preface
Applications of graph theory have been proliferating throughout ecology over the past
several decades, whether explicitly realized or implicit in the approaches used, and not
only in the cases which fall clearly into the popular category of networks. The reasons
for this increased interest are as diverse as the areas of research. A basic impetus is
that graphs and graph theory are about structure and provide the methods to analyze
structure as abstracted from almost any ecological (or other) system. The second rea-
son is the great popularity of network studies and network theory, originally applied to
social relationships, communications (including the Internet as a prime example), trans-
portation and the spread of disease. It is an obvious step to take network concepts and
models from these sources and see how well they apply to ecological systems. Such
network studies are obvious sources of inspiration for investigations of ecological inter-
actions of all kinds (such as predation, competition, mutualism and facilitation) using
the methods developed for those other systems. A third reinforcement for graph theory
applications arises from the growing sophistication of ecologists in analyzing spatial
data or time-ordered data or the complexities of spatio-temporal data; and, once again,
methods based on graph theory provide the right mix of simplicity of concept but flex-
ibility in application to provide valuable insights that would otherwise be impossible.
Putting together interaction networks and spatio-temporal data brings a researcher to
the challenges and rewards of studying the interplay of form and function (or “pattern
and process” or “structure and dynamics”) in ecological systems in which both form
and function change through time by reciprocal influences and effects.
The book is organized in an order that reflects this range of sources. First is an intro-
duction to thinking with graphs based on the theme of graphs and structure (Chapters
1 and 2). There are then several chapters on ecological interaction networks, first in
general (Chapter 3), followed by more specific topics: predation (Chapter 4), social
structure (Chapter 5), competition (Chapter 6) and mutualism (Chapter 7). The next
three chapters are about locational graphs, in which the nodes have positions in one or
more dimensions: time only (Chapter 8), space only (Chapter 9) and spatio-temporal
(Chapter 10). Chapter 11 describes approaches to studying the dynamics of networks
in the context of the reciprocal effects of form and function, focussing on the fascinat-
ing and promising methods based on graphlets. The last chapter (Chapter 12) attempts
to draw together a number of the themes that emerged throughout the book and pro-
vide a synthesis of the common threads; it also takes on the risky task of making some
predictions about future directions and developments to be expected in this field.

The working title started out as “Smart Things Ecologists Can Do with Graph The-
ory”; and that is a good description of the intention. The book is not primarily an intro-
duction to graph theory developed for ecologists; it is intended to make researchers
aware of the wide range of possibilities for their own research projects, even when (or
especially when) they have yet to be fully tried out in ecological systems. A prime
example is the many forms of analysis based on graphlets that are recently developed
and applied in other biological systems (e.g. protein-protein interactions) but not yet in
ecology. The goal is to provide enough background that the researcher knows how and
where to start and where to find some examples that will provide inspiration and support.
The treatments of the various topics are very heterogeneous; some have a good range of
examples to be cited (e.g. food webs or trophic networks; mutualism), but others have
virtually none.
My own interest in graph theory as a useful approach to answering ecological ques-
tions related to structure started with my MSc research many years ago, and I owe a large
debt to my then-supervisor, Tony Yarranton, who suggested the area and encouraged my
exploration of the field. I owe thanks to John Moon, who helped me understand some
of the more formal aspects of graph theory and its application (look at his Topics on
Tournaments, if you have not already: a great example). In acknowledging people who
have helped with this book, I thank the following for reading chapters, sometimes as
they developed: Alex Aravind, Tan Bao, Conan Vietch, JC Cahill and Brendan Wilson.
I thank Marie-Josée Fortin, especially; she read all the chapters, and some more than
once! For data used in examples, there are many to be acknowledged, including Tan Bao
and JC Cahill for the Arabadopsis competition tournament material and Gord Thomas
for the rich data set on Saskatchewan weed communities. I thank NSERC Canada and
UNBC for their support over many years.
I greatly enjoyed writing this book, and discovering all the exciting material I had
not known was very rewarding. It is my hope that the readers will find the work equally
rewarding and that it will help create pathways to more that is useful, more that is new
and more that is surprising.

1 Graphs as Structure in the
Ecological Context
Introduction
Ecology is the study of organisms in the context of their environment, including both
abiotic effects and interactions among organisms. Ecologists, like other scientists, are
looking for patterns in these phenomena that can be used reliably to make predictions,
and those predictions can extend the findings to other organisms, to ecological systems
not yet studied or merely to similar groups of organisms in different places or at different
times. Those predictions may also refer to how a system’s form or structure determines
its function and dynamics and how function and dynamics constrain or modify structure
and form.
A long but not exhaustive list of the kinds of problems ecologists study might include
the following:
r the fate of individuals as determined by neighbours and environmental conditions
r the interactions of individuals in a social structure and their effects on population
dynamics
r the movement of individuals through their environment and their reactions to it
r the dynamics of populations and communities in fragmented habitats
r the flow of energy and the population and community effects of predation in trophic
networks
r the effects of competition, both intra- and inter-specific, on survival, growth and
reproduction
r the dynamics of species interactions, such as mutualism, commensalism and para-
sitism
r the determinants of species composition of multi-species communities in island sys-
tems
Almost all of these can be approached in a theoretical or abstracted way, or quite explic-
itly with locations in time or space, and almost all of these are studied in the context
of a system of some sort and usually in the context of that system’s structure. In fact,
explicit references to “structure” arise in almost every study of ecological systems, from
behaviour to trophic networks and from individuals to community interactions. The term
“structure” usually refers to how systems are put together or to the relationships among
units that determine how they work together. Structure, like pattern, suggests some
05:35:16 at

2 1 Graphs as Structure in the Ecological Context
Figure 1.1 A graph. The basic graph consists of nodes (•) and edges (──) joining pairs of nodes.
Nodes can have labels, weights or locations. Edges can have directions, signs, weights,
functional equations or locations.
predictability in the way a phenomenon is organized, even if the process that gives rise
to it has a random origin or stochastic component, such as the fates of individual organ-
isms. Even structures generated by fully random processes may have predictable char-
acteristics, as we will see in Chapter 3. Graph theory is the mathematics of the basics
of structure (objects and their connections), providing a rich technical vocabulary and
a formal treatment of the concepts and outcomes. Because of the importance of under-
standing and quantifying structure in all ecological systems, graph theory has important
contributions to make to a broad range of ecological studies, including trophic networks
(Kondoh et al. 2010), mutualisms (Bascompte Jordano 2014), epidemiology (Meyers
2007) and conservation ecology (Keitt et al. 1997), where the graphs depict functional
connections among organisms or physical connections among spatially structured pop-
ulations (Grant et al. 2007).
The graphs that are the focus of graph theory are deceptively simple mathematical
objects, each consisting of a set of points with a set of lines joining them in pairs. The
points are called nodes, represented by dots in a diagram (Figure 1.1), and the lines are
edges, represented by straight or curving lines in a diagram, although a range of terms
can be found in the literature (see Harary 1969; West 2001).
Graphs are about connections and the pattern of connections. In a diagram of the
most basic graph, the positions of the nodes on the page and the lengths and shapes of
the edges joining them have no meaning; they are placed for convenience and clarity. It
is the set of connections made by the edges that determines the graph’s topology. The
nodes usually represent components or units of organization, and the essence of the
graph lies in what is connected to what: really very simple! In this way, the graph is an
abstract description of structure or topology because the edges show the relationships
among organizational components that the nodes represent.
Graphs and graph theory lend themselves extremely well to applications in many
areas of science because there is a wealth of mathematical knowledge that has been
developed over the years from studying these simple components. Graph theory inves-
tigates all aspects of combinations of nodes with edges joining them; and “all” is no
exaggeration. What is continually impressive about graph theory is the way that it can go
from what seems simple and intuitive to very sophisticated (and, yes, difficult) results;
advances in recent decades have really changed the field, and it has important links (pun
intended) to many other branches of mathematics, such as algebra, number theory and
05:35:16 at

topology. An obvious example is the application of graph theory to understanding the
properties and vulnerabilities of information networks like the Internet.
A second reason for the great value of graph theory for ecologists is the flexibility
of the approach for meaningful applications to a range of ecological phenomena. This
is accomplished by including different characteristics in the graphs beyond the simple
nodes and edges. These include the following:
r node labels that identify the node as an individual and identifiable component of the
system, such as a species name; labels make a difference when counting the number
of different structures
r node weights that record qualitative or quantitative characteristics of the components,
such as relative abundance
r node locations: the nodes may have spatial or temporal locations, such as the time and
place of a single predation event; temporal location allows the possibility of nodes that
come into existence or cease to exist
and
r directions for the edges so that A to B is distinct from B to A
r signs for the edges, indicating positive or negative interactions between the nodes
r weights for the edges, or equations describing flow or function
r locations for the edges, spatial or temporal, dependent on the locations of their end-
nodes; temporal location allows edges to come into existence or cease to exist
For example, nodes could represent identifiable landscape patches of known locations
in a particular year, with their areas as weights; the edges could be movement corridors
with weights related to how frequently or how easily the routes can be used for dispersal.
This introductory chapter describes the concepts and terminology that form the foun-
dations of a tour through graph theory and the smart ways to use it for understanding
ecological phenomena. This tour illustrates the assertion that these graphs are about
structure and the pattern of relationships that are the essence of structure. A subtle dis-
tinction here is that despite the fact that “graph” and “network” have come to be almost
synonymous, “graph theory” is still more about structure and “network theory” is more
about function and flow.
1.1 Graphs as Structure
The branch of mathematics that we know as graph theory has arisen from a number of
different sources, developed to solve problems in diverse fields. The most famous of
these is Euler’s solution in 1736 to the “Königsberg bridge problem,” which concerned
walking routes around two islands in a river with seven bridges over it. By converting the
question into a general problem about graphs, it could be shown that a closed route that
crossed each bridge exactly once was impossible (Euler, as cited in Biggs et al. 1976).
This solution is usually cited as the beginning of graph theory, although Tutte (1998)
has suggested that the discipline might date back to ancient times and the study of
05:35:16 at

Platonic solids (tetrahedron, octahedron, etc.), which are essentially symmetric graphs
on the sphere. Another origin is Kirchhoff’s studies of 1847 (Biggs et al. 1976) on the
flow of electricity through a network of circuits with different characteristics. A third
beginning is Cayley’s work on the combinatorics of the chemical structures of organic
compounds (e.g. butane and its isomer, isobutene) and the structurally different forms
any one chemical might take (Cayley 1857). Other possible sources of the discipline
include studies of map colouring problems (any map can be coloured with only four
colours), interactions between molecules in statistical mechanics and Markov chains
in probability theory (see Harary 1969, Chapter 1). I would, however, add a different,
fourth area to the list of inspirations, and that is the study of networks of positive and
negative interactions between individuals in a social setting, with developments due to
Harary and co-workers from the 1950s.
All these problems are clearly about structure, the structure associated with
1 spatial constraints on physical routes
2 energy flow in a system with alternate pathways and different resistance characteris-
tics
3 physical forms from combinations of component units (atoms)
4 relationships in interaction networks
All these sources of graph theory as a branch of mathematics have close parallels in
ecological research, and all require, and take advantage of, different characteristics and
results developed in that discipline.
In mathematical terms, a graph is an object made up of two sets: nodes (also points or
vertices) and edges (the lines, also called arcs or links) that join pairs of nodes (Harary
1969; West 2001; see Box 1.1). Therefore, graph G can be seen as an ordered pair of
sets V and E:
G = (V, E) with E being pairs of the elements of V.
Less formally,
graph = {nodes} and {edge joining pairs of nodes}; say n nodes and m edges.
The density of edges is measured by the connectance, which is the proportion of
possible edge positions actually occupied; here 2m/n (n − 1). (This is not the same
as a graph being connected, with a path between any two nodes, nor is it the same
as connectivity, which measures how difficult it is to separate a connected graph into
pieces.)
In contemporary usage, the terms “graph” and “network” are used interchangeably
as equivalents (Estrada 2012), although previous practice was to reserve “network” for
graphs or digraphs which had a real number (weight) assigned to each edge (Harary
1969), such as those in trophic networks or transportation systems. Digraph networks,
with directed edges, are frequently used to study the flow of material or information,
one of the most important applications of graph theory, and for such applications, each
edge can have several weights, including capacity, flow and cost (Bang-Jensen Gutin
2009).
05:35:16 at

Box 1.1 Graph Theory: Checklist of Objects
Each term has a sketchy phrase to hint at its meaning, rather than a full definition,
for which see the text and the Glossary. This is not all the graph theory we need but
much of the important material in a concise format. Not everything required will fit
into Chapter 1; more will be introduced as needed.
1.1.1 Graphs
Graph (nodes and edges)
Subgraph (subsets of graph’s nodes and edges)
Induced Subgraph (subset of nodes, and all edges of the original graph joining those
nodes)
Connected Graph (path exists between any two nodes)
Tree (connected with no cycles)
Dendrogram (binary tree, often from cluster analysis)
Complete Graph (all possible edges are included)
Bipartite Graph (nodes in two distinct subsets)
Digraph (directed edges)
Tournament (each pair of nodes has a one-way outcome edge)
Signed Graph and Digraph (edges are positive or negative)
Weighted Graph (nodes or edges have weights)
Weighted Digraph (ditto and edges have directions)
Line Graph (edges become nodes in the derived line graph)
Network (same as graph, or graph with directed weighted edges)
Dynamic Network (changes through time, either edges or their weights)
Spatial Graphs (nodes located in space [vs aspatial])
Temporal Graphs [many names] (nodes located in time [vs atemporal])
Spatio-temporal Graphs (nodes located in time and space)
Planar Graph (can be drawn flat without edges crossing)
Dendrogram (clustering process and levels of joins)
1.1.2 Parts of Graphs
Subgraph (subsets of nodes and edges)
Cut-point (node removal disconnects)
Cut-edge (edges removal disconnects)
Block (maximal connected subgraph with no cut-points)
Walk (sequence of nodes and their edges; may re-use)
Path (sequence of nodes and edges, no re-use)
Closed Walk (ends at its beginning node)
Cycle (path that ends at its beginning node)
Clique (complete subgraph)
Tree “Leaf Node” (degree = 1; “object” in classification dendrogram)
Tree “Branch Node” (degree 1; joins objects into groups in dendrogram)
Spanning Tree (connected subgraph with all nodes, but no cycles)
Clusters or Modules (subgraphs well connected within, few connections out)
Components (maximal connected subgraphs)
05:35:16 at

Graph is disconnected by removal of node K (cut-point)
or edge BK (cut-edge)
A B
D
C
F
G
E
I H
A
I H
A B
D
C
F
G
E
I H
J
B
D
C
F
G
E
J
K
K
K
J
Figure 1.2 Children who play together. The graph is disconnected by removal of node K
(a cut-point) or edge BK (a bridge or cut-edge).
Graph theory is usually introduced by formal development, and we cannot avoid that
altogether; but we will introduce much of the basic terminology through an example,
not trying to cover everything, with more to be introduced in later sections as required.
The introductory narrative will be complemented by a checklist table of terms (Box
1.2) as well as the figures that go with them. This book also has a Glossary that collects
almost all of the terms introduced throughout the chapters in one place.
To start with an instructive and almost-ecological example, consider children on a
playground. Each child is represented by a node of a graph, G, and a simple edge is used
to indicate which children are playing together during an observation period (Figure
1.2). There may be large and small groups, or individuals may play mostly alone. We
can use graph-theoretical properties to evaluate this social structure for average number
of playmates, maximum number of shared-play relationships between any two children
and so on, and to determine the most coherent clusters. Each child has a name, and so
each node has a natural label. The degree of a node is the number of edges attached to
it, the number of nodes that are its neighbours. In the playground example, the degree
is the number of shared-play interactions, ranging from 1 (nodes G or J) to 5 (node E),
averaging around 2.5.
In Figure 1.2, all the children are joined together by at least one sequence of edges
through the graph, so that a rumour that is passed only between these pairs of playmates
will reach all children. That is, the graph is connected, because there is a path along
nodes and edges between any pair of nodes. It will become disconnected, however, if
child K leaves (that node is a cut-point) or if B and K become estranged and no longer
play together (edge BK is a cut-edge) (see Figure 1.2, bottom). There are two obvious
clusters or modules, AIH and BCDE, which are subgraphs of the whole structure. A
05:35:16 at

A B
D
C
F
G
E
I H
60
55
50
45
25
40
70
40
45
J
K
20
Nodes with weights of minutes in playground
30
45
35
15
25
30
35
35
45
35 40
Edges with weights of shared play me (in bold)
Older
Younger
25
45
40
Figure 1.3 Children who play together. (top) Older children indicated by larger nodes. The
subgraph of BCDE is a complete graph or clique. (bottom) Nodes with weights of minutes in
playground; edges with weights of shared play time (in bold).
subgraph of G is itself a graph of which the nodes are a subset of the nodes of G and
the edges are a subset of the edges of G.
Each node can also be categorized by age and gender, and so it can be determined
in which categories the graph is assortative (most edges between nodes in the same
category) or disassortative (most edges between nodes in different categories). In our
playground example, the graph tends to be associative for age, mainly because of the
clique (a complete subgraph, i.e. with all nodes joined to all nodes) of four older children
(B, C, D and E), as shown in Figure 1.3 (top).
Further properties include a weight for each node, such as the total time on the play-
ground, and weights for each edge, such as the total time or proportion of time the two
children play together (Figure 1.3, bottom). The simple graph of nodes and edges in the
figure is aspatial; space is not explicitly included, but the data on which it is based are
probably truly spatial, if they were to be thus recorded. For example, some groupings
may tend to spend their time by the slides and others by the swings. For some purposes,
this spatial information could be included in the graph. Similarly, the graph shown is
atemporal, but an explicitly temporal graph could be created by recording the different
combinations of children at different times of day or by recording the changing links as
friendships form and dissolve, evidenced by shared time on the playground. The latter
approach gives a dynamic graph or network.
Of course, there are many different ways to define the edges of a graph for the same
children in the playground. For example, with children, unlike some of the animals we
study, we can complement the observational data by asking them their opinions of the
05:35:16 at

A B
D
C
F
G
E
I H
A B
D
C
F
G
E
I
H
J
K
K
J
Directed Graph: Who is your best friend?
Figure 1.4 Children who play together. Nodes HKBEF with edges between form a path. Nodes
AIH with three edges form a cycle. Nodes BCDE with their six edges form a subgraph, a
complete graph and a module. The (undirected) graph (top) becomes a digraph (directed graph)
(bottom) based on “Who is your best friend?”
others: Who do they like? Who is their best friend? and so on. This gives edges that have
direction, because B may consider C to be their best friend, but the “best friend” rela-
tionship is not always reciprocated (Figure 1.4). Directional edges allow the inclusion of
asymmetric relationships. They also mean that the degree of each node can be divided
according to “arrow toward” edges, in-degree, and “arrow away” edges, out-degree. (In
the digraph of Figure 1.4, node E has an in-degree of 2 and an out-degree of 1.)
So far only edges of shared play or liking, which are positive edges, have been
included in the graph, but it might also include negative edges indicated pairs that never
play together or that actively avoid each other; this gives signs to the edges creating a
signed graph (Figure 1.5). By allowing asymmetric “like” and “dislike” for any pair of
nodes, the graph then has edges that are signed and directed, allowing A to B to dif-
fer from B to A (see nodes K and E in Figure 1.5b). To refine further to include the
intensity of “like” and “dislike,” the edges may also have quantitative weights. In a real
study of social structure, it would be interesting to compare the graph based on observed
behaviour and the graph based on stated opinion . . .
A child shows up with a bad cold one day, and the cold spreads among the children
from playmate to playmate following the edges of the shared-play graph. How far and
fast the cold spreads will depend in part on the position of the initial carrier in the social
network, how well connected and how central within the whole population (compare
nodes B and J). The spread of the disease will follow a path in that graph consisting of
a series of nodes and the edges joining them. In a path, the elements are not re-used,
and in this case, the disease does not return to a child who has already had it, and so no
cycles are formed. (A cycle is a path that ends where it began, such as A – H – I – A
05:35:16 at

(a) Mutual “like” or “dislike”
Graph: nodes (•) and signed edges: (solid = +ve; doed = -ve).
A
B
D
C
E
(b) Asymmetric “like” or “dislike”
Digraph of direconal edges with signs.
I
F
G
K
J
H
H
I A B
E
C
K
D
Figure 1.5 Playground children: likes and dislikes. (a) Mutual “like” or “dislike.” Graph: nodes
(•) and signed edges: (solid = +ve; dotted = −ve). Two complete subgraph modules: {A,H,I}
{B,C,D,E}.) (b) Asymmetric “like” or “dislike.” Digraph of directional edges with signs. Some
relationships are reciprocal: HK, BC. Some are not; the association of K with E is +ve, but the
association of E with K is −ve.
in Figure 1.2.) A connected graph without cycles is called a tree. The trace of the disease
through the shared-play graph is a subgraph that is a tree (Figure 1.6); the nodes are the
same as in the original graph, but the edges representing the relationships are different.
The edges could be labelled with directions if the actual process of disease spread was
known, and they could also be labelled with dates or the order of infection if those
data were available. The nodes of a tree are called “leaf” nodes if they have degree 1;
“branch” nodes have degree 2 or higher; and the “root” node is a specially designated
node that is functionally unique, such as the common ancestor in a phylogeny or the
river mouth in a drainage basin, with its meaning depending on the application.
As another example of alternate rules for edges, consider the following. On Saturday
morning, each of the four older children is assigned one, two or three of the others to
A B
D
C
F
G
E
I H
K
J
Figure 1.6 A tree made up of shared-play edges showing how a cold may spread. A tree has no
cycles. A, I, G, J, D and C are leaf nodes. H, K, B, E and F are branch nodes. No node is
identified as the root.
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A
B D
C
F
G
E
I H J
K
coaches
students
4 components
disconnected
A B
D
C
E
H
This connected graph is also biparte
(ABCD | EFGH).
F
G
Figure 1.7 Bipartite graph of math coaching in the playground group; this one is disconnected.
The lower connected graph is also bipartite (ABCD | EFGH); it is a ring graph and regular
because all nodes have the same degree (2).
help coach them in their math skills. This creates a new set of edges that can replace
the friendship edges of shared play; with the nodes representing the same individuals
and the edges now representing that coaching relationship (Figure 1.7). Here the edges
all join older to younger children, with no edges within either age cohort, giving what
is called a bipartite graph for obvious reasons. In our example, the graph is discon-
nected (some nodes not joined by a path) and consists of four components (connected
subgraphs).
This narrative has introduced some of the most ecologically important aspects of
graphs. These are the basics only and more terms and concepts are introduced through-
out the chapters that follow. All are provided in the Glossary at the end of the book.
1.2 Graphs and Ecological Relationships
The objects in ecological studies, which are to be the nodes of a graph, are often individ-
ual organisms, populations, communities, or defined spatial areas like habitat patches;
and the objects are linked by physiological, behavioural, physical and dispersal pro-
cesses. The edges between objects vary in weight and in vulnerability versus persis-
tence, according to the nature and intensity of the ecological processes. Research in the
related fields of evolutionary biology, population genetics and epidemiology, have as
the usual objects individual organisms or other units such as taxa, traits, genes, molec-
ular markers and so on. The edges between these nodes are the relationships of evo-
lutionary history, functional pathways, measured similarity or ecological interactions.
Graphs of these systems have the objects as nodes and their relationships as the edges
(Harary 1969; West 2001; Bang-Jensen Gutin 2009; Lesne 2006; Kolaczyk 2009).
These graphs of relationships can be thought of as “abstracted” structures, because they
05:35:16 at

have been derived from but taken out of the spatial and temporal contexts in which the
information originated. In these studies, organisms and interactions have been modelled
and analyzed for several decades using graphs and networks that are therefore aspatial
and atemporal (Dale 1977a, 1977b; Proulx et al. 2005; Lesne 2006; Mason Verwoerd
2007; Dale Fortin 2010).
Graph theory has seen further significant applications in trophic network studies (Pas-
cual Dunne 2006; Kondoh et al. 2010), conservation ecology (Keitt et al. 1997; James
et al. 2005), epidemiology (Shirley Rushton 2005; Meyers 2007), and mutualisms
(Bascompte Jordano 2014). Graphs are used now in ecology for many applications
depicting physical or functional connectivity among organisms (predation, pollination,
competition and other forms of interactions; Bascompte 2009) or among spatially struc-
tured groupings of local populations (metapopulations [Fagan 2002; Grant et al. 2007],
although the actual locations are not retained for analysis), and they can obviously
be used for more. Where the locations in time or space are not explicitly maintained
for analysis, many of these resulting interaction graphs or networks might be called
abstracted interaction graphs.
1.3 Graphs and Locations: Spatial and Temporal
Much of the ecological data we collect originate each from a particular place at a par-
ticular time, and ecological systems usually have some spatial and temporal structure.
It therefore makes sense to maintain the locations of observations for analysis, although
summarizing over time and space may provide its own insights. This gives rise to graphs
of the ecological systems in which the nodes (and possibly the edges) have locations in
space or in time, or in both. Although many of the phenomena that inspired the devel-
opment of graph theory were actually spatial (walking routes, electric circuits, maps,
etc.), the graphs originally were not, but simplified the problems by removing the spa-
tial context to become simplified combinatorial entities. In many of these applications,
the locations of events in time or space are maintained explicitly for analysis, giving
what might be called “locational graphs.”
1.3.1 Spatial Graphs
In spatial graphs, the nodes have locations that provide an explicit spatial context and
spatial meaning. The end-points of the edges obviously have locations, too, but any
edge may not reflect the trajectory of any thing moving through space but may be an
abstract indication of a relationship between the nodes. For example, an edge might be
the pseudo-trajectory of a seed from its parent tree to where the seedling is eventually
found; that trajectory is usually unknown. In Figure 1.8, the nodes are sites in a land-
scape, the edges represent the shortest set of connections between sites according to the
rule of a Minimum Spanning Tree; the details of the landscape will determine whether
they represent practical routes of dispersal. On the other hand, the edges between nodes
may have physical locations, as well as other characteristics such as length and width,
05:35:16 at

x
y (x,y)
space
Figure 1.8 Spatial graph: nodes have spatial locations (x, y). Edges may (or may not) show
trajectories or physical connections. The edges are a Minimum Spanning Tree: a tree (no cycles)
that includes all nodes and minimizes the total physical length of edges (consider animals
moving between landscape patches).
as in the case of hedgerows being dispersal connections for small mammals across an
agrarian landscape. In this case, the level of abstraction is low and our spatial graph has
become very much like a map of the system it portrays. If the corridors are actually
more important than the patches, the original graph can be reformatted as its line graph;
the original edges are now the nodes and the new edges indicate which pairs of the orig-
inal edges shared nodes (Figure 1.9). In this conversion, the degree of any node created
in the line graph is 2 less than the sum of the degrees of the original end-nodes; for
example, the degree of node BE in the line graph in Figure 1.9 is d(B) + d(E) – 2 = 5.
A great example of where this kind of duality might be of interest comes from studies
of mycorrhizal networks: Southworth et al. (2005) found different network topologies
AC BC
ED
CD
BF
EG
BE
Line Graph
9 nodes; 15 edges.
Original Graph
8 nodes; 9 edges
EH
A B
D
C
F
G
E H
FH
C B F
E
D
Figure 1.9 Line graph. Focus on the connections: converting a graph to its line graph. Edges
become the new nodes; new edges determined by shared nodes in the original graph. Some are
indicated by single letter labels.
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A B
D
C
F
G
E
A B
D
C
F
G
E
Aspaal
graph
Spaal
graph
10
12
11
10
6
12
9
A B
D
C
F
G
E
Spaal
digraph
10
12
11
10
6
12
9
δ(A,F) = 3
δ(A,G) = 4
Path length distances
dp(A,F) = 31
Path length distances
dp(A,G) = 37
dAG = 26
dp(D,E) = 33
dp(E,D) = 6
26
Figure 1.10 Distance and path length: an aspatial graph, a spatial graph and a spatial digraph.
when the Quercus garryana trees were the nodes and the fungi were the links com-
pared to the fungi being the nodes and the trees the links. In the first case, the network
appeared random with a short tail to the distribution of edges per node; in the second,
the results were consistent with a “scale-free” model (more on this later) suggesting that
some of the fungal species act as hubs in the network.
A key characteristic of edges in a spatial graph is the meaning of “distance.” A path
is an alternating sequence of nodes and edges joining them from u to v that uses no ele-
ment more than once. The basic measure of distance between two nodes is the smallest
number of edges in a path between them, called the geodesic distance. These distances
are not always symmetric; in a digraph, for example, δ(u,v) and δ(v,u) may be differ-
ent. If the edges have weights, the graph theory distance between two nodes δ(u,v) is
the smallest total of weights in any path from node u and to node v. (A walk is also an
alternating series of nodes and their edges that lead from node u to v; but its compo-
nents may be used more than once, whereas a path may use an element only once [see
Glossary].)
In a spatial graph, the edge euv has a weight that is the spatial distance between the
nodes, call it duv or ds(u,v) for clarity. This means that there are two distances between
nodes: the simple spatial distance duv between the locations of the two nodes (which is
the same as ds(u,v) if there is an edge between them), and the spatial distance along the
shortest path, which is the sum of the spatial lengths of the edges in that path; call it
dp(u,v) with “p” for “path.” Figure 1.10 illustrates these meanings of distance. Again,
digraphs are different because of possible asymmetries; in fact in sparse digraphs, dis-
tance may be “infinite” because there is no path between some pairs of nodes. In the
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Objects
Time interval
A
B
C
D
E
t = 5 10 15 20
Figure 1.11 Temporal or time-ordered or time-only graph. The dashed lines serve as edges of
identity, joining instances of the same node. Pairwise contacts are recorded in discrete time
intervals as indicated by the temporal “locations” of the edges.
lowest panel of Figure 1.10, the distance from D to E is 33, but it is only 6 from E to D.
The distance from B to A is 38, but B cannot be reached from A.
1.3.2 Temporal and Time-Only Graphs
There are many terms for graphs with nodes that have locations in time but not in space;
“temporal graphs” is obviously one, and “time-varying” and “time-ordered” are other
possibilities, as is “time-only graph” where it is the correct description. The nodes are
located in time, and they may come into being or cease to be at particular points or
in particular periods in time. Graphs in which the nodes come and go may be “time-
ordered,” but the same description can apply where the nodes persist but the edges
appear and disappear. Figure 1.11 shows the example of a time-ordered graph of five
persistent nodes observed over more than 20 time periods; the edges represent short-
lived contacts between pairs of nodes, at most one per time period. Obviously the flow
of information or material through the system depends on the order in which the nodes
or edges form and disappear. In a system of three nodes and two time-ordered edges,
A–B–C, information or disease cannot flow from A to C if the B–C edge ceases to exist
before the A–B edge is formed. It is not common for a graph of an ecological study
to include only temporal locations without space, but these do occur, and they have a
chapter of their own (Chapter 8).
Some graphs change their structure through time, or they may document a struc-
ture that controls or influences how a system behaves through time. These are called
“dynamic graphs” (Harary Gupta 1997) or more commonly “dynamic networks”
(Casteigts et al. 2011), described in Section 1.4 which follows.
1.3.3 Spatial-Temporal Graphs
In spatio-temporal graphs, the nodes have locations both in time and in space, as the
term suggests. The comments about the locations of edges apply in this spatio-temporal
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x
y
(x,y)
(x,y,t)
me
Figure 1.12 Spatio-temporal graph: nodes have temporal and spatial locations. Edges may (or may
not) show trajectories.
context, too, just as in the purely spatial case. They exist in a spatial and temporal con-
text and their end-points have locations, but they themselves may not portray actual tra-
jectories (Figure 1.12), standing as “pseudo-trajectories” perhaps. Both time and space
may each be either continuous or discrete, although most applications in ecology and
related fields divide time into discrete units like days or years.
1.3.4 Aspatial and Atemporal Graphs
In aspatial graphs, space has no explicit role in the structure and presentation, so that
the positions of the nodes and edges convey no meaning. The subtlety is that “aspatial”
is not identical to “non-spatial”; it is a more neutral term rather than negative. The infor-
mation in an aspatial graph may be derived from truly spatial data; for example, a graph
of pairwise species-to-species neighbour associations is derived from the frequencies of
neighbour occurrences in the spatial context of a plant community, but spatial relations
are summarized, not retained, in the resulting graph. A non-spatial graph has no spatial
component, neither explicitly nor implicitly. To parallel “aspatial,” “atemporal graphs”
are those in which the data may have a temporal component, but it is not explicit in the
graphs.
1.3.5 Spatial Statistics and Local Statistics on Graphs
In the preceding sections, graphs have been discussed relative to locations in space and
in time, with those dimensions being prior and external to the graph. A graph can also
be seen as creating its own “space,” with its own measure(s) of location, which can be
very useful in studies of ecological networks that are abstracted from physical dimen-
sions. This is not the same as embedding a graph in Euclidean n-space (see Erdös et al.
1965), but rather using the simple measure of geodesic distance (Figure 1.10) to provide
the “space.” Within this defined space, many methods based on standard spatial analysis
or spatial statistics can be applied, even if the locations are not spatial in the traditional
sense and the familiar rules of Euclidean space may be violated. This is different from
the approach described by Okabe and co-authors, who apply familiar spatial analysis
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1
1
2
2
2
2
3
3
3
3
4
4
Correlaons
by distance
r(1) = 0.45
r(2) = – 0.45
r(3) = – 1.0
r(4) = -̶ 0.45
r(5) = 0.45
Figure 1.13 A ring graph with weights on the nodes, showing the correlations of node weights at
various distances.
(such as Ripley’s K function for point pattern analysis) to events that are constrained
to occur on networks of linear structures that are embedded in “real space” (see Okabe
Sugihara 2012). The transfer of techniques from Euclidean space to “graph space”
requires caution because of potential violations of the usual assumptions; however, with
the correct adaptation and interpretation, almost any analysis that might be performed
on grid or lattice data can be used for the irregular structure created by a non-lattice
graph. For example, the observed node weights in a graph can be analyzed for autocor-
relation using any appropriate statistic (e.g. Moran’s I) with the distance classes defined
by geodesic distance. Figure 1.13 provides the simple example of a ring graph with node
weights of 1 to 4; Pearson’s correlation is positive for distances of 1 and 5, and nega-
tive for distances of 2, 3 and 4. This allows an ecologist access to a rich set of familiar
analytical tools to explore the structural characteristics of abstracted networks as if they
were spatial data.
For spatial statistics and indices of spatial structure, measures of characteristics in
“graph space” can be created both in the global form, summarizing for the entire graph,
and in the local form, focussed on one particular part of the graph rather than the whole.
For example, from network analysis comes the concept of assortativity: the graph is
assortative if nodes with similar characteristics tend to be neighbours, joined by edges
or short paths. This is positive “spatial” autocorrelation. The opposite is a graph that is
disassortative (having negative autocorrelation): neighbours tend to be less similar. This
property can be assigned to the whole graph as a global measure, or it may be regional
within the graph, with some regions highly assortative, and others disassortative (see
Piraveenan et al. 2008; Thedchanamoorthy et al. 2014). As an example, Figure 1.14
shows a graph in which the left half is assortative for the category of the nodes (many
edges join nodes of the same colour), but the right half is disassortative (many edges join
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1.4 Networks and Dynamics 17
assortave
edges mainly between like nodes
disassortave
edges mainly between unlike nodes
Figure 1.14 A graph that is locally assortative or disassortative: in one part, it is assortative, with
edges mainly between like nodes, and in another, it is disassortative, with edges mainly between
unlike nodes.
nodes of different colours). The concept of local versus global evaluation can prove very
useful in assessing the information in a graph or network that is itself heterogeneous
(e.g. species interaction networks that are divided into compartments), or subject to
non-stationarity (e.g. time-only graphs of interactions over a long time scale). Again,
what can be done for information in graph-defined space mirrors closely the analysis
options available for standard spatial data, which may prove especially useful as a route
to understanding the structure of long-term or “big” data such as relating community
composition and phylogeny, or modularity in microbial consortia responding to changes
in hosts and environmental conditions.
1.4 Networks and Dynamics
With the growing popularity of networks for many and varied ecological applications,
the term “network” has entered our ecological vocabulary from a variety of sources,
so much so that it has become informally equivalent to “graph” as a general term for
these abstracted structural models. Sometimes “network” is still used in a more narrow
sense, referring to a graph with directed and weighted edges, used to depict or to ana-
lyze system function. These networks are often dynamic, changing in structure or with
quantitative characteristics that change through time. These two cases are not identical,
but obviously related. In some applications, the network is an atemporal (and frequently
aspatial) summary of system function; think of a trophic network with average annual
energy flows associated with each edge linking an organism of one trophic level to a
consumer in the level one above it. In general, these are abstracted graphs, removed
from the spatial and temporal context in which the data originally resided.
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Of particular interest to ecologists are those graphs or networks in which the structure
affects the dynamic function of the system it describes, as you would expect, but the
processes that occur on the structure are able to change the structure itself. This is a
variant on the familiar “pattern and process” or “structure and function” interaction
seen in many areas of ecological study. Because these structures permit considerable
complexities, they deserve a chapter all their own, and that is Chapter 11, far ahead,
although the theme will recur in the intervening material.
1.5 Graphs and Data
Typically, ecological studies gather data, synthesize information from the data, analyze
that information and interpret the results. To take advantage of graph theory, the data
have to be converted to a graph (of course). How this is done depends on the nature of
the data and the analysis method; the latter may determine the format of the graph. A
graph portrayed as a diagram is good for intuitive understanding (hence so many figures
in this book), and in fact one primary function of data as a graph is to facilitate the
visualization and exploration of the data, including scanning for mistakes and anoma-
lies (see Raymond Hosie 2009). Beyond visual presentation and interpretation, most
analyzes require computation and that requires a digital format for the information, and
hence a good reason for the representation of graphs by matrices. Matrices for graphs
can be generated in many ways, each emphasizing different features of the graph, or
facilitating particular calculations.
Creating and handling these matrices becomes especially important for very large, or
“massive,” data sets (e.g. the Internet; see Newman 2010, Plate 1), which are just too
large to be appreciated intuitively (see Hampton et al. 2013). Just displaying such large
data sets as graphs requires specialized software (see Kolaczyk Csárdi 2014). Some
ecological applications may include huge numbers of nodes, especially where there are
long time series and large spatio-temporal data sets (e.g. the exploratory analysis of
sea surface temperature records, illustrated in Cressie Wikle 2011, Chapter 5). Very
large graphs provide great opportunities, rich with information and outcomes, but they
present challenges for data conversion and processing, and visualization may require
extra effort to detect the important patterns in the structure.
The descriptions of converting ecological data into graphs and of representing graphs
as matrices can be presented in either order, but the data-to-graph procedure may depend
on the graph format required, and so it makes sense to begin with the graph-as-matrix
material. Of course, knowing the format required for the analysis should inform the
development of the sampling or experimental procedure before it is carried out (the
usual advice, ignored at our peril!).
1.5.1 Graphs as Matrices
This is the technical part of the discussion, but it is necessary to understand the basics
of how the graphs are represented and analyzed. A matrix is an array of numbers,
usually rectangular and often square, where the meaning of an entry is determined by its
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5
2
1
4 3
6
5
2
1
4
3
6
Graph G1
…recast as a digraph, D1
Figure 1.15 Example graph, G1, with six nodes and six edges, then recast as a digraph, D1, to
illustrate the matrices used to represent graphs.
position in the array. It conveys numerical information in a compact format that facil-
itates manipulation and analysis, and matrices are fundamental to computer programs
and software packages.
Most ecologists are familiar with matrices as commonly used in multivariate analysis
such as principle components analysis or multivariate regression analysis. The matrices
for graphs should seem less daunting because the entries in the matrices are often easy
integers, and frequently just 0s and 1s (and −1s).
Given that a basic graph can be defined as the ordered pair of sets of nodes and edges,
themselves ordered pairs of the nodes
graph = ({nodes} , {edges joining pairs of nodes}) = ({vi}, {(vi, vj )}),
a simple data structure is a list of the pairs of nodes joined by the edges. If the n nodes
are identified by integer labels, 1 to 6 in Figure 1.15, a list of the m edges can be created
and the list is essentially a 2 × m matrix: [(1,2), (1,3), (1,5), (2,3), (2,4), (3,5)]. An
alternative is an adjacency matrix, A, which is a square n × n matrix with elements
aij = 1 when nodes i and j are joined by an edge, and 0 otherwise. For the graph in 1.15,
the adjacency matrix A1 is
Matrix 1.1 Adjacency matrix A1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 1 1 0 1 0
1 0 1 1 0 0
1 1 0 0 1 0
0 1 0 0 0 0
1 0 1 0 0 0
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
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For a basic simple graph, a node cannot be adjacent to itself and so the entries on
the main diagonal are not just zeros but structural zeros, which means that by definition
they cannot be anything else but 0, and they are indicated by bold font.
One interpretation of the matrix A is that it shows the number of walks of length
1 between nodes i and j. What is helpful about this interpretation is that A2
gives the
number of walks of length 2 between nodes i and j, A3
the walks of length 3 and so
on. These counts include walks that begin and end at the same node and may re-use
other elements; so these are not true “paths” in graph theory terminology. The result is
that the major diagonal is no longer structural zeros, but the number of closed walks of
the designated length that begin and end at node i. In A2
these are “cycles” of length 2,
which begin and end at the same node by going out and back on the same edge, therefore
also giving the node’s degree, which is just the number of such edges.
A1
2
is
Matrix 1.2 Self-product of adjacency matrix A1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
3 1 2 1 1 0
1 3 1 0 2 0
2 1 3 1 1 0
1 0 1 1 0 0
1 2 1 0 2 0
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
The same basic information from the adjacency matrix, A, can be contained in the
n × m incidence matrix, B, in which the rows are the nodes and the columns are the
edges, with elements bik = 1 when node i is an end-point of edge k, and 0 otherwise. By
convention, the edges are labelled in the lexicographical order of their nodes pairs (i.e.
in order first by the lower label, and by the higher label if the lower labels are tied). For
a digraph (a graph with directed edges), the incidence matrix has values 1, 0 and −1 as
follows: bik = 1 when node i is the source of edge k, bik = −1 when node i is the sink
of edge k, and 0 otherwise.
B1 is
Matrix 1.3 Incidence matrix B1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 1 1 0 0 0
1 0 0 1 1 0
0 1 0 1 0 1
0 0 0 0 1 0
0 0 1 0 0 1
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
For the digraph version (Figure 1.15), BD is
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Box 1.2 Graph Theory: Checklist of Properties
Node Labels (individual or class identification)
Node Weights (quantitative, possible dynamic)
Node Locations (usually as coordinates in time or space)
Node Degree (number of edges; in-degree vs out-degree for digraphs)
Node Degree distributions (for model comparison)
Node Degree joint distributions (degree autocorrelation; by neighbour category)
Edge Weights (quantitative, possibly dynamic, possibly with equation)
Path Length (geodesic or physical)
Graph Diameter (maximum shortest path)
Connectance (proportion of possible edge positions occupied)
Connectivity (how difficult to disconnect)
Clustering coefficient (frequency of third edge of a triangle)
Node Centrality [many versions] (importance in shortest paths)
Associative (positive correlation of adjacent node properties)
Disassociative (negative correlation of adjacent node properties)
Matrix 1.4 Digraph incidence matrix BD
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
1 1 1 0 0 0
−1 0 0 1 −1 0
0 1 0 −1 0 1
0 0 0 0 1 0
0 0 −1 0 0 −1
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
The adjacency matrix for the same digraph is AD; the entries of “1” show the direction
of the edge by their position in the matrix, which is no longer symmetric about the main
diagonal.
Matrix 1.5 Digraph adjacency matrix AD
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
0 0 0 0 1 0
1 0 0 0 0 0
1 1 0 0 0 0
0 1 0 0 0 0
0 0 1 0 0 0
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
Many of the matrices listed in Box 1.3 are fairly obvious in meaning based on their
relation with terms in the Glossary, but the Laplacian matrix needs something more by
way of explanation. The degree matrix, D, has 0s everywhere except on the main diag-
onal, which contains the degree of each node in their standard order. (For a digraph we
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Box 1.3 Graph Theory: Checklist of Matrices
This is a very small subset: there are lots and lots of matrices in graph theory, most
arranged in lexicographical order, but not always square and n × n . . .
Edge list (2 × m; joined node pairs, edge by edge)
Adjacency matrix (n × n; symmetric, joined node pairs by row and column)
Incidence matrix (n × m; nodes with incident edges)
Degree matrix (n × n; nodes’ edge totals on main diagonal)
Laplacian matrix (n × n; symmetric, combines adjacency and degree matrices)
Digraph and network matrices (n × n; asymmetric from directed edges)
have a choice of using the in-degree or the out-degree, which will affect the interpreta-
tion.) The Laplacian, L, is then D – A.
Starting with the same adjacency matrix for G1 in Figure 1.15, the degree matrix D1
is
Matrix 1.6 Degree matrix D1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
3 0 0 0 0 0
0 3 0 0 0 0
0 0 3 0 0 0
0 0 0 1 0 0
0 0 0 0 2 0
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
and the Laplacian matrix L1 = D1 − A1 is
Matrix 1.7 Laplacian matrix L1
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
3 −1 −1 0 −1 0
−1 3 −1 −1 0 0
−1 −1 3 0 0 0
0 −1 0 1 0 0
−1 0 −1 0 2 0
0 0 0 0 0 0
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
This matrix is important in graph theory because it can be used to determine certain
characteristics, such as the number of spanning trees (subgraphs with no cycles that join
all nodes) or the number of connected components (see Newman 2010, Chapter 6). It
is also an important approach to understanding the properties of random walks on the
original graph; these are walks that develop by iteratively choosing the next node to be
included at random from those available by edges from its current node (Figure 1.22; see
also Section 1.7.5). The various matrices related to the adjacency matrix provide clear
algorithmic alternatives for exploring a graph and determining its properties (e.g. cycles,
blocks, random walk . . . ), often based on the eigenvalues of a matrix. These properties
will be discussed at greater length as they come up in the chapters that follow.
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1.5.2 From Data to Graphs
The advice always given is to determine the analysis to be performed on the data to be
collected before sampling or running experiments to obtain the data. Good advice is not
always taken, but the form of the data limits the form of the graph that can be derived;
for example, sampling that is itself symmetric cannot give asymmetric data or directed
graphs (e.g. “A then B” is asymmetric information, but “A and B” is symmetric) whereas
sampling that is asymmetric produces asymmetric data, which can be aggregated to
symmetric outcomes, thus allowing either directed or non-directed graphs. The match
or mismatch of spatial and temporal scales of the phenomenon being investigated and
the sampling and analysis being carried out also has to be considered as an important
part of the relationship between phenomenon and data, and between data and analysis
(see Dale Fortin 2014, Figure 1.8).
A few examples can illustrate the relationship between design and outcome: the data,
the analyses and the graphs.
1. The first is from plant ecology, where a common approach is to investigate the com-
munity structure of positive and negative associations between pairs of species using
density or presence-absence data. Plants of two species may tend to occur close together
because they have similar ecological properties or because of some positive influence
of one on the other. Plants of different species may tend to occur farther apart because
they have divergent ecological properties or because there is some negative influence
of one species on the other. Some of these causes are most likely symmetric (shared or
divergent ecological properties) and others are more likely to be asymmetric (positive
or negative influence), and the sampling scheme may be designed to account for that
fact.
A standard approach is to record the presence and absence of all species in each of
many small quadrats, randomly or systematically arranged. The data are used to create
graphs of the inter-specific associations of pairs of species (also known as constella-
tion diagrams or phytosociological structure). For each pair of species, A and B, any
quadrat belongs to one of four categories: both present; A present, B absent; A absent,
B present; both absent. The counts of quadrats in the four categories form a 2 × 2 contin-
gency table and each pair of species can be assessed using a goodness-of-fit test with the
X2
or G statistic compared with the χ2
distribution. There are many problems in treat-
ing this as a reliable statistical test, because of various sources of non-independence. It
does, however, provide a standard by which edges between species (the nodes) can be
determined for a graph of “significant” inter-specific associations; this graph has nodes
representing species and signed edges joining some pairs, and so the result resembles
Figure 1.5a. The edges are sometimes shown as lines of different thicknesses indicat-
ing the strength of the associations. The symmetric sampling design will not permit
distinguishing between possible effects of A on B from possible effects of B on A.
To determine asymmetric associations for graphs with directed edges, the sampling
must have asymmetry in its design. One such is “point-contact” sampling. Dimension-
less points are set out in random or regular arrangements and at each such location
the first species contacted by the point sample is recorded, together with the species
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Figure 1.16 Neighbour samples for inter-specific tests: four species in a planar graph (can be
drawn with no edges crossing).
closest to the initial contact that is different from the initial species. The resulting data
set consists of ordered pairs of species frequencies: (initial contact, nearest neighbour to
contact point). These data can be evaluated using a goodness-of-fit test, although there
are technical details to consider (Dale et al. 1991). These data provide the basis for an
asymmetric graph of pairwise associations for further analysis; the nodes are species
with signed directed edges joining some pairs, thus resembling Figure 1.5b.
The third method for deriving association graphs requires identifiable individuals that
can be mapped, like tree stems in a forest. Given the location and species of each stem,
the counts of the species of neighbours, variously defined, can be determined for each
species and analysis will detect unusually rare or common neighbours for each. Neigh-
bours can be defined in a number of different ways, whether based on a quantitative dis-
tance threshold (e.g. 1.5 m), or using “topological” definitions of neighbours in spatial
graphs (Figure 1.16 provides an example), such as the Minimum Spanning Tree (Figure
1.8), or other such rules for spatial graphs described below. Whatever the definition of
neighbours, to produce a spatial graph of the site, an abstracted association graph for
the community is based on significant deviation of neighbour frequencies from those
expected based on complete spatial randomness.
2. The second example is of the interactions between two identifiable groupings of
species, such as plants and their pollinators or herbivores and their predators, so that
the resulting graph is bipartite, with two non-overlapping subsets of nodes, and with
the graph’s edges running between the subsets, not within (Figure 1.17). One intriguing
question about these interactions is whether specialist species on one side tend to pair
up with specialists on the other side of the interaction or whether there is a tendency for
specialists to pair only with generalists (a “nested” arrangement, see Figure 1.17). The
strength of any of these interactions is usually determined from some “surrogate” that is
easier to observe (see Bascompte Jordan 2014, Figure 3.7; Vázquez et al. 2005a). The
data from which these graphs are produced are usually (1) counts of interaction events,
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Pollinators
Plants
4
4
3
3
3
3
2
2
1
1
5
1
4
2
3
3
2
4
1
5
Matching: specialists with specialists
and generalists with generalists
5
5
node degrees
node degrees
Nested: specialists only with generalists
and generalists with any
Figure 1.17 Bipartite graphs for ecological interactions between two distinct categories of
organisms, such as plants and their pollinators. On the left, there is matching of specialists with
specialists and generalists with generalists. On the right, there is nesting so that specialists are
only with generalists and generalists with specialists or generalists.
such as the number of visits of pollinators of species X to flowers of species Y or (2)
quantitative measures, such as the amount of biomass of herbivore Y removed or con-
sumed by predator Z. Most such data are derived from in situ field work of observation
or sampling.
The count data can be converted into complete bipartite graphs, with all possible
edges included, each weighted by the count of the pairwise-specific events. An alterna-
tive is to convert the count frequency data, call it fXY, into proportions p•Y or pX• by
dividing by the plant total or by the pollinator total. These values, associated with all
possible edges, can then be used to determine which edges to include in the graph using
a threshold value (Figure 1.17).
3. The third example concerns food webs, which are trophic interaction networks.
These are generally intended to be quantitative and attempt to be complete, both in the
interactions that are quantified and in the species that are included. Taxonomically dif-
ficult groups are often aggregated, giving large groupings of taxa such as “grasses” or
“centric diatoms” or “parids”; an alternative to taxonomic groupings is to aggregate by
function, such as “grazers” versus “browsers.” The data that contribute to the construc-
tion of a trophic network graph include biomass estimates from destructive sampling,
observations of encounter rates, trapping, feeding preferences, stomach content analy-
sis, calorific measurements and so on. While demanding, the data collected produce a
network graph of great richness and clear value for further study. Figure 1.18 shows a
simplified trophic network for the Kluane boreal ecosystem based on a truly enormous
amount of effort over decades of study and a very wide range of estimates, measures
and observations (see Krebs 2010 and references therein).
To conclude this section, Table 1.1 lists some of the features of the three examples
just discussed as a summary and with added notes or comments for consideration. A
specific comment at the end of the table offers consoling advice on how to deal with the
common problem of incomplete data, always a concern for field-based studies.
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Table 1.1 Data to Graphs: Features and Considerations
Plant communities for inter-specific associations
Standard quadrat sampling
Symmetric “constellation” graphs of association from frequency tables (many caveats!)
Point-contact sampling
Asymmetric association digraphs from frequency tables
Stem mapping (trees)
Many choices for neighbour definitions
Associations from neighbour frequency
Bipartite graphs of pairwise interactions
Counts of interaction events
Flower visits
Foraging stops
Quantitative measures
Pollen biomass loads
Fruit consumed
All edges with weights, or only strongest edges by threshold
Multi-level food webs of predation (trophic networks by taxonomic or functional group)
Biomass estimates for taxonomic or trophic groups
Encounter rates by species or group pairs
Stomach content analysis
Calorific measurements
Comment on Incomplete Data
No study is likely to be sufficiently intensive and extensive to determine all the individ-
uals or to detect all the species that should be included, nor all the possible interactions
of the species that are encountered. This means that in these studies, as in much of
goshawk
great-horned owl lynx coyote
fox
snowshoe
hare
ground squirrel red squirrel small rodents
forbs grasses birch willow spruce
Figure 1.18 Simplified food web or trophic network: the Kluane boreal ecosystem (redrawn from
Krebs 2010). Arrows are directed from prey to predator, indicating the transfer of energy and
matter; some conventions use the reverse.
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1.6 Ecological Hypotheses and Graph Theory 27
ecological research, we have only partial and incomplete information on which to base
our conclusions. Some of these are data zeros, not structural zeros; that is, there is
no observation although one is possible (structural zeros occur where no observa-
tion is possible). One approach to evaluating the importance of this incompleteness
is sub-sampling known-to-be incomplete data, and by calculating the measures of inter-
est for each sub-sample to determine the findings’ robustness to missing information
(Naujokaitis-Lewis et al. 2013). This is much like bootstrapping, but not intended to
determine statistical significance (for more on this technique in other applications, see
Efron Tibshirani 1993; Manly 2006). This approach is not the same as sampling a
graph or network, to be followed by making inferences about the whole from the sam-
ple (see Kolaczyk 2009), nor is it the same as techniques to identify missing or spurious
edges and thus to reconstruct its graph and refine the determination of properties (see
Giumerà Sales-Pardo 2009).
The three chosen examples are all of abstracted graphs, rather than locational graphs
for which the locations of nodes in time or space are retained explicitly. The relationship
between data and graph is conceptually simple for locational cases. The nodes represent
objects or events that have coordinates in time or space, and you just have to record
them (sounds easy!). Of course, the practicalities of actually doing this may be daunting,
expensive or overwhelmingly detailed. The maps of tree stems are locational graphs, as
would be the space-time coordinates of bees’ visits to orchid flowers, or of goshawk-hare
encounters (labelled as to outcome). Much locational data is now collected by automatic
tracking systems, such as radio-collared elk or “tagged” sharks, but the older methods
of mark-recapture or trapping also provide data for location-specific graphs for further
analysis. Locational data are essential for studies related to diversity and conservation,
examining features such as community composition, genetic structure and patterns of
dispersal through fragmented landscapes, all of which can be helped by the use of graph
theory as will be detailed in Chapter 9 (see, among many others, Urban Keitt 2001;
Saura et al. 2014; Watts et al. 2015; Rayfield et al. 2011; Fall et al. 2007; James et al.
2005).
1.6 Ecological Hypotheses and Graph Theory
The relationship between hypotheses and analysis is often iterative in ecological
research. The hypothesis determines the data required and the analysis that is carried
out; but usually the results of the analysis generate new hypotheses which lead to further
and refined studies. Graph theory lends itself well to this iterative process, particularly
because it provides a natural approach to hierarchical analysis.
Hierarchical frameworks for hypothesis testing work best when each level of evalua-
tion is independent of the previous levels, but this is not always possible and our hierar-
chical layers of tests may not be mutually independent. The most challenging task can
be the translation of the ecological hypothesis into a well-formulated and testable for-
mal hypothesis in the language of statistics or graph theory. The ecological hypothesis
determines the formal hypothesis and the data and other information that are required to
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(a) (b) (c)
Figure 1.19 Proximity graphs or spatial neighbour networks. (a) Networks 1 and 2: First Nearest
Neighbours and First Mutually Nearest Neighbours in bold. (b) Networks 3 and 4: Least
Diagonal Neighbours with Minimum Spanning Tree in bold. (c) Networks 5 and 6: Delaunnay
triangulation with Gabriel graph in bold.
test it. These then determine the analytical approach, and the results inform the decision
to reject the formal hypothesis or not. This decision is then used to evaluate the original
ecological hypothesis.
In the many publications on applications of graph theory in non-mathematics (e.g.
biology, technology or sociology), it is interesting how rarely the testing of hypotheses
or evaluations of statistical significance are included explicitly. The many discussions of
measures of graph or network properties, and many comparisons with various random
“null models,” are rarely formulated in these terms.
As an introductory example, consider the hierarchy of spatial neighbour networks
described in Dale and Fortin (2014) and shown in Figures 1.19a, 1.19b and 1.19c. These
spatial graphs form a series from mutually nearest neighbours with few edges per node
(averaging 0.62), through the Minimum Spanning Tree (about 2.0) to the Delaunay tri-
angulation with many (about 6.0); each graph in the series is a subgraph of the graph that
follows (see Chapter 9). A spatio-temporal hierarchy can be more complicated, includ-
ing only temporal neighbours (history explains it all), only spatial neighbours (location
is everything), or some of both (Figure 1.20), with the critical question being the spa-
tial and temporal distances at which neighbours have an influence. Similar hierarchies
of inclusive subgraphs can be created in aspatial and atemporal applications, such as
trophic networks, by using different threshold values for including the directed edges
based on their transfer rates or feeding preferences: each change in threshold having the
potential to produce a different graph that includes all the edges of the previous version.
In all cases, the hierarchy allows a hierarchical series of hypothesis tests, admittedly not
independent, that will permit an evaluation of ecologically interesting hypotheses.
One major goal of this book is to help ecologists understand the wide array of “smart
things” that can be done with graph theory, providing some guidance on the range of
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t + 1
t − 1
1o spatial neighbour
nodes at time t
t
Focal node at time t + 1
1o spatial neighbour
nodes at time t - 1
Figure 1.20 Ecological memory and neighbour influence: state of focal node at time t + 1
depends on its own history (times t and t − 1), or on its contemporary neighbours or on
neighbours and histories.
approaches to evaluating hypotheses that graph theory affords. That range is broad and
deep.
1.7 Statistical Tests and Hypothesis Evaluation
When ecological hypotheses are translated into statistical hypotheses, we need the
appropriate statistics and test procedures to assess significance. The general approach
is to compare a statistic calculated from the data to the appropriate reference distribu-
tion. When the reference distribution is known and tabulated, parametric tests may be
used, but these significance procedures require independence of the data. When the ref-
erence distribution is not known, but the characteristics of interest can be studied by
the enumeration of graph structures, enumeration can be used to provide the frequency
distributions upon which tests can be based. If parametric methods are not available,
and enumeration is not an option, then randomization procedures are a good alternative
to generate a reference distribution from the data (Edgington 1995; Manly 2006). When
inference to the population level is required, bootstrap procedures and Monte Carlo sim-
ulations can be used (Efron Tibshirani 1993; Manly 2006). Randomization tests are
an attractive approach because significance is evaluated by comparison with empirical
distributions generated from the data, which is especially appealing for small data sets
that do not meet the assumptions of parametric tests.
1.7.1 Parametric Tests
Parametric tests may be available only rarely for the applications described here, partic-
ularly as any source of lack of independence, and there may be several, invalidate the
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application of these familiar test procedures, but a common comparison for observed
properties is the same characteristic in random graphs (Erdös Rényi 1960). Their
simple model for random graphs is to start with n nodes and to include any one of all
possible edges independently and with constant probability p. Under those conditions, a
number of characteristics follow binomial, Poisson or (approximately or asymptotically)
Normal distributions. For example, the degree of any vertex follows a binomial distribu-
tion, and the distribution of the number of vertices of a given degree, d, is asymptotically
Normal (see Barbour et al. 1989). For large graphs, this property enables parametric test-
ing as a good approximation where appropriate. The big question, of course, is when a
random graph of this type is actually a good null hypothesis for comparison.
These Erdös-Rényi random graphs have been well studied since they were introduced,
and much is known about them. For example, threshold values for p and thus for the den-
sity of edges have been determined related to the appearance of certain substructures in
the random graph, such as trees of a given size, cycles of a given size or complete
subgraphs of a given size (Newman 2010). This knowledge can also be useful for eval-
uating the subgraph characteristics of an observed graph. It is because the edges in
these random graphs are placed independently and with constant probability that many
distributions derived from them can be assumed to converge to known parametric dis-
tributions.
1.7.2 Enumeration and Probability Calculations
Enumeration is a straightforward way to determine whether the observed structural char-
acteristics that are related to a hypothesis of ecological interest are surprising, in the
sense of being significantly different from what would be expected for “randomly cho-
sen” graphs of the same kind. Rather than really using randomly chosen or randomly
constructed graphs, it may be possible to compare an observed structure with all possi-
ble graphs in the domain of interest. For example, we might want to know how unusual
it would be or it would be for a bifurcating tree, like a cladogram, to have exactly one
branch node of each possible order from 1 to n − 1. We could generate all possible trees
of that kind and determine directly how many have that property. While this approach
may sound “labour-intensive,” computing power now makes it easy. An alternative in
some circumstances is to calculate the probability of a given characteristic, such as the
cladogram example just given, from something like first principles, hoping that we have
got those first principles right (!). The advantage of the “brute force” enumeration of
all possible structures is that limits can be placed on the structures considered. We can
therefore enumerate within subsets defined by particular characteristics, allowing us to
circumvent, or at least understand, the effects of the lack of independence in how the
structures are put together.
For example, consider studying competition in a community of n species by a large
complete experiment that tests the competitive outcome of every pairwise combination.
For each pair of species, one is determined to be the “winner,” based on some crite-
rion evaluating performance. The structure that results from all pairwise tests is called a
tournament for obvious reasons, and can be represented by a directed graph of n nodes
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Labels are node scores
4
2
0
2
2
Figure 1.21 Tournament with n = 5: a complete graph with a directed edge from winner to loser
for all pairs of nodes. Labels are node scores (wins, as here, or sometimes wins – losses). The
node scores are 4, 2, 2, 2 and 0, as indicated.
and n −1 directed edges pointing from “winner” to “loser.” One way to examine the
consistency of competitive outcomes is to evaluate the transitivity of the digraph. A
relationship like competitive dominance is said to be transitive whenever, given edges
A–B and B–C, we also find A–C. To evaluate the competition results, we determine the
frequency of transitive triangles in the graph, compared with the frequency of “paradox-
ical” triangles that show cycles of competitive outcome or a kind of competitive reversal
where we find A–B and B–C, but also C–A. If the tournament is almost completely tran-
sitive, we can work through enumeration and some probability calculations to determine
how expected or unexpected that outcome might be, given all possible tournaments and
an assumption of equal probabilities for all.
There are n(n −1)/2 positions for edges among the n nodes, and each takes one of two
possible directions, A–B or B–A, independently (that’s important!) of the others, and so
there are 2n(n − 1)/2
tournaments. If the competitive outcomes are consistently transitive
throughout the tournament, then there is a strict order of all n species from the strongest
competitor to the weakest. There are exactly n! of these orderings, since all orders of
the n species are possible, and may be considered equally probable. The best competitor
outcompetes all others, and so it has an out-degree (or “score”) of n − 1, the next best a
score of n − 2, and so on, down to 0 for the weakest competitor (Figure 1.21).
For example, the tournament in Figure 1.21, n = 5, and the scores are 4, 2, 2, 2, 0.
While not completely transitive, the competitive relationships among the species are
quite consistent in their transitivity. This example will be discussed further in Chapter
6, but it shows something of the procedure to evaluate the graph-based results.
1.7.3 Other Random Graph Constructions
The Erdös-Rényi model for random graph construction is, of course, not the only model.
The literature describes many others, of which the Watts-Strogatz “small world” is one
of the most often cited, as is the “scale-free” network model (Newman 2010). One char-
acteristic frequently used to differentiate among these models is the distribution of the
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nodes’ degrees; for the small world model, this follows a delta distribution, which is
bell shaped but sharply peaked; for the scale-free models, it is a power function (Albert
Barabási 2002, among many others). Several other measures are used to character-
ize graphs to determine the most likely descriptive model; these include average path
length, clustering coefficient, and correlation of node degrees (Albert Barabási 2002,
among many). The clustering coefficient is essentially the probability that two neigh-
bours of a given node (say j and k which have edges to node i) are themselves neighbours
(joined by ejk). One measure associated with the correlation among node degrees is the
joint degree distribution; this edge-based statistic tabulates the bivariate distribution of
the degrees of nodes joined by an edge; it can be calculated directly from the adjacency
matrix (Newman 2010). It tells more about connectivity than the degree distribution
because it shows whether the degrees of neighbouring nodes are positively or negatively
autocorrelated. The frequency is often converted into a Pearson correlation measure that
runs between −1 and +1, designated r, called the assortativity coefficient because it dis-
tinguishes between assortative networks with r 0 in which high degree nodes tend to
be first-order neighbours (path length 1) of other high degree nodes (giving positive
autocorrelation) and disassortative networks with r 0 in which high degree nodes
tend to be first-order neighbours of low degree nodes (giving negative autocorrelation).
Of course, characteristics other than node degree can be used as the basis for assortative
versus disassortative designations for network graphs (Newman 2010, Section 7.13).
The many random models for graphs and networks may be tailored to the circum-
stances. As with randomization of existing structures, creating random models for com-
parison with the observed data allow us to include constraints on the random version
that reflect constraints in the system being studied. For example, in a study of poten-
tial migrations between landscape patches, we might be interested in the diameter of
the Minimum Spanning Tree as a measure of the shortest distances and fewest “steps”
between patches. In that case, it would be reasonable to create a large number of realiza-
tions of randomly placed nodes of a spatial graph within an equivalent area, determine
the Minimum Spanning Tree for each, and thus determine a frequency distribution for
the diameter. This frequency distribution can then be used to evaluate the observed
value.
1.7.4 Randomization and Restricted Randomization
Randomization tests are based on the hypothesis that all re-arrangements by re-ordering
or pairwise exchanges (“shuffling”) of the data are equally likely. Therefore, although
randomization tests may have fewer assumptions than other forms of testing, there are
still some to consider. In addition, large numbers of randomizations (e.g. 10,000) may
be necessary to achieve the desired level of significance for a particular test. Random-
ization tests do not offer fully the familiar security of parametric statistics, but their
flexibility provides the means to analyze complex ecological data using experimen-
tal or sampling designs for which classical tests have not been developed. Ecologists
can also develop their own statistics, opening up the possibility of testing in novel
situations.
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Any lack of independence in data (due to time, space, behaviour, relatedness, phy-
logeny, . . . ) can impair the application of either parametric or randomization tests. Para-
metric tests require that the errors are independent, so that each observation or data point
brings a full degree of freedom. If the lack of independence is due to spatial relation-
ships, the resulting positive spatial dependence usually makes nearby sampling units
more alike and so a spatially autocorrelated sample does not bring a full degree of free-
dom, but rather a fraction of it, inversely proportional to the autocorrelation in the data
(Legendre 1993; Dale Fortin 2009). Several techniques in sampling design and in sta-
tistical analysis (Legendre Legendre 2012; Dale Fortin 2009) can correct or control
for dependence in the data, so that familiar parametric tests can be used with minor mod-
ification. This issue of non-independent errors is at the core of the analysis of ecological
data but it applies to the development of randomization procedures for dependent data,
complete randomness is not really an appropriate comparator and so forms of random-
ness which incorporate some degree of structure (often spatial or temporal) should be
used (Cressie 1993). These are restricted randomization procedures that include some
of the structure of dependency already in the data (Fortin Payette 2002; Manly 2006),
or at least most of it (see Dale Fortin 2014, Figures 8.12 and 8.13). There are sev-
eral different ways to restrict the randomization on a graph depending on the system.
A simple example would be a map of diseased and healthy plants, with a network of
neighbours imposed upon it. A simple question is whether the diseased plants are clus-
tered or overdispersed, and it is one that is easily answered, but a more useful question
might be whether the diseased plants are clustered or overdispersed given the overall
arrangement of the plants of either kind.
1.7.5 Random Walks on Graphs
A specialized form of randomization for investigating graph properties is the application
of random walks on graphs, closely related to the study of Markov models. What is
random here is not the graph itself or its formation, but a walk, a sequence of alternating
nodes and edges, on the existing graph. The basic random walk begins with a single
randomly chosen node (step 0), and then moves with equal probability to any of the
adjacent nodes by the edge that joins them (step 1). The number of possible next steps
is the degree of the current node, di, and the probability for any one of them being
next is di
−1
. The process is then iterated many times, as illustrated in Figure 1.22, and
in general, the relative frequencies of the nodes in a long random walk approaches a
stable distribution determined by the graph’s structure. Many properties of these walks
have been investigated and they provide important insights into how graphs work. More
specifics on random walks on graphs will be discussed in Chapter 9, related to spatial
graphs and the implications for conservation ecology in fragmented landscapes.
1.7.6 Models
An obvious extension for understanding ecological systems using graphs is the devel-
opment and evaluation of models. It is an easy step to go from restricted randomizations
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2 choices 4 choices
3 choices
t = 0 t = 1 t = 2
0 1
2
3
4
5
t = 3 t = 4 t = 5
2 choices 4 choices
4 choices
Figure 1.22 Random walk on an undirected graph with the choices indicated at each stage of the
iterative process.
for generating the distributions of variables, to Monte Carlo models where new “data
sets” are generated with known characteristics and built-in forms of dependence. This is
closely related to the approach that provided an understanding of random graphs versus
small world graphs versus scale-free graphs; but there, the graphs themselves were gen-
erated by random processes, rather than the “data” on which graphs are subsequently
based. The latter version can be useful for much more highly specified circumstances.
For example, in models of spatio-temporal bipartite graphs studying plant-pollinator
relationships, the creation of “random” edges needs to be limited by the phenologies of
both plant flowering and pollinator activity; and to avoid adding in “forbidden links”
(Olesen et al. 2008; Jordano 1987), pollination interactions that cannot occur, such as
the combination of a long-tubed flower and a pollinator with short mouthparts. Mod-
els can also help in dealing with the complexity of large-scale many-factor systems by
integrating pattern and process (e.g. Peterson et al. 2013) or by modelling spatial deci-
sions for resource exploitation by simulation and evaluation (Walker et al. 2013). More
examples of the use of a range of model types will arise in the following chapters, but
this is another area of applying graph theory in ecological studies that deserves more
emphasis and exploration.
1.7.7 Sampling and Inference
Many of the ecological systems of great interest involve very large numbers of individ-
uals or of taxa or of observations, producing graphs of networks that have many nodes
and many edges. Very large graphs present serious challenges for data collection, data
05:35:16 at

analysis, and inference (see e.g. Ahmed et al. 2014). One obvious solution is to choose
a sample of the entire graph or network and use the information from the sample to
make inferences about the whole, whether the questions of interest are about the sys-
tem’s topology and structure or about its function and dynamics (Leskovec Faloutsos
2006; Maiya 2011). There are many different ways of selecting the sample, but a sim-
ple approach is to take a random subset of the nodes of the graph and to include all
edges between pairs of those nodes, giving an “induced subgraph sample” (Kolaczyk
2009). Samples can be based on nodes, as in this case, or on randomly chosen edges,
or on a node’s edges or its neighbour clusters (“star” and “snowball” sampling) with
many variations (Lee et al. 2006). Such a sample can be the basis for estimates of sev-
eral characteristics of the whole graph: average node degree, betweenness centrality, the
clustering coefficient or graph transitivity, and so on (for details, see Lee et al. 2006;
Kolaczyk 2009, Chapter 5; Ahmed et al. 2014). Most ecologists are familiar with the
effect of spatial autocorrelation on the analysis of locational data (see Legendre 1993;
Dale Fortin 2009); similar effects are to be expected within the “graph space” of
network graphs, due to the positive autocorrelation called “homophily” or “assortative-
ness” or the negative form called “heterophily” or “disassortativeness,” but much of
the emphasis in network graph sampling is on estimation rather than on significance
levels.
1.8 Concluding Comments
The purpose of this chapter was to introduce the most important concepts of graph
theory as they can be applied in ecological studies, without duplicating the formality of
standard graph theory texts. A second criterion was not to include everything that would
be needed throughout the book. That would seem overwhelming. More objects and
properties will be introduced throughout the chapters that follow as they are required.
Some themes and concepts are included more than once throughout the book; the rep-
etition is intentional because the reader is not expected to work through the material
cover to cover and in the order imposed by the chapters.
The basic message for the reader from the material of Chapter 1 on applying graph
theory to their own ecological research is “You can do this . . . ” The chapters that follow
provide the next important level, showing some of the details of “ . . . and here’s how.”
Many of the investigations that we may wish to pursue can be improved or facilitated by
the application of graph theory, and there are many which would be impossible without
graph theory to develop and test the hypotheses of interest.
I do not apologize for the lack of “real” examples for some of the suggested applica-
tions. The whole point is that these are things that can be done, but many have not. We
should not wait for others to complete their studies and present us with their mature and
considered results before following up on these promising approaches. In some cases,
there have been applications in other fields of endeavour that are sufficiently similar
to act as models for ecological applications to follow. There is a long and varied (and
enticing) list of ideas for us to explore!
05:35:16 at

That’s it for the introduction! There is lots to learn, but worth the effort. The long list
of the subjects of ecological studies is matched by a long list of the ways in which graph
theory can be employed to support and facilitate, or even to direct those studies, and that
is the focus of the rest of this volume. The next chapter is designed to complement this
chapter’s introductory material by reviewing the range that graph theory covers from a
different angle: the shapes of graphs, from trees to triangles.
05:35:16 at

2 Shapes of Graphs: Trees to Triangles
Introduction
This chapter is designed to help “unpack” many of the concepts introduced in Chapter 1,
with a more in-depth treatment and some pre-figuring of material that will be covered in
more detail by later chapters. Complementary presentations should ease the intense list
of concepts and details, and help with getting into the mode of “thinking with graphs.”
The main topic is a quick tour through a range of different shapes or topologies for
graphs and some of the variations on these basic forms that are helpful in ecological
applications.
One theme is how seemingly small changes in the rules that control how graphs are
created have large effects on the graph that results. For example, what happens if graphs
are not allowed to have cycles in their structure? What is the effect of having directions
to the edges? If the edges occur at random positions, how many triangles should the
graph have? There are good reasons for starting with the first question and considering
the consequences of “acyclic” graphs.
2.1 Acyclic Graphs
A graph with undirected edges and no cycles (i.e. no paths that form closed loops) is
a “tree” when it is connected, and (of course) “a forest” when it is not (Figure 2.1).
Trees are common structural forms in almost all branches of science (pun intended),
from hydrocarbon molecules and river basin drainage networks to evolutionary diver-
sification, to data storage and retrieval and to modelling clonal growth (see Box 2.1).
Compared with graphs of interaction networks, these are simple structures with simple
rules for associated functions, but they are both useful and powerful for applications in
the ecological context.
Considering trees as spatial structures, one feature is that while there is always a
path through the tree between any two nodes, there is only one path, with no detours or
alternative routes between nodes. This fact has implications for tree-shaped structures
in a natural setting based on their vulnerability to disconnection or disruption.
In ecology, as in other fields, classification is one way of organizing multivariate data
sets; for example, using species abundance data to group quadrats into a hierarchy of
clusters based on similarity. When the objects or clusters are joined two at a time, the
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38 2 Shapes of Graphs: Trees to Triangles
Graph
(with cycles)
Tree
(no cycles, connected)
Forest
(no cycles, not connected)
Figure 2.1 Graphs and cycles: a graph may have cycles (or not) and may be connected (or not);
here connected with cycles. A tree is connected and has no cycles. A forest has no cycles and is
not connected.
clustering process can be depicted as a dendrogram, which is a binary tree (Figure 2.2).
It is often of interest to ecologists (1) to evaluate the shape of the dendrograms and (2) to
determine the similarity of two dendrograms. Graph theory can be helpful in providing
guidance to both these evaluations.
Dendrograms are rooted trees (see Box 2.2), with the “root” node representing the
grouping of all objects (a node of degree one), “leaf” nodes that represent the original n
objects (nodes of degree one), and n − 1 “branch” nodes of degree three, that represent
where the groups are joined (Figure 2.3).
2.1.1 Shape
The “shape” of a dendrogram can be defined in different ways, but one difficulty is the
fact that, for a single data set, the dendrograms that result from clustering can be very
Box 2.1 Trees
A tree is a connected undirected graph with no cycles. Trees can be used in ecol-
ogy to provide a spatial framework for locations (e.g. Minimum Spanning Tree or
radial spanning tree), to model the growth of branching organisms (e.g. trees [of
course!], corals or clonal herbs and grasses) and to describe the evolutionary history
of related organisms (a phylogenetic tree). Trees are also fundamental as data and
computational structures, important to ecologists as well through the analyses they
do! In some cases, one node is designated as the “root” of the tree; this gives an
implicit directionality to the structure.
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2.1 Acyclic Graphs 39
e
u
l
a
v
y
t
i
r
a
l
i
m
i
s
n
i
o
j
join sequence
1.0
0.0
0.5
1
2 3 4
5 6
root
n = 7 leaf nodes
6 branch nodes
Similarity levels of joins Sequence of joins
Figure 2.2 A dendrogram (a binary tree) is a graph that shows the process of a cluster analysis.
Usually it records more than shape, often values at which joins occur or their order.
different, depending on the similarity measure and clustering algorithm on which the
dendrogram is based. This fact needs to be considered in interpreting the results. One
description of shape is the frequency distribution of the sizes of the subtrees (the number
of leaf nodes) at each branch node: mk is the number of branch nodes with k + 1 leaf
nodes (Dale Moon 1988). In the example in Figure 2.3, there are three groups of two
objects, one of three and one of four. The subtree shape is M = (m1, m2, . . . , mn − 1); in
Figure 2.3, M = (3, 1, 1, 0, 0, 1).
Another shape characteristic is the number of “terminal singles,” single objects that
are joined to the all-inclusive group only at the very end of the clustering process (Dale
Moon 1988). In Figure 2.3, the dendrogram has no terminal singles; in Figure 2.4,
there are four in the first example, and two in the second. Terminal singles are interesting
Box 2.2 Dendrograms
Dendrograms are trees. They are binary rooted trees, used to depict and analyze
the results of classification procedures or cluster analysis; n objects classified pro-
duce a dendrogram of n leaf nodes and n − 1 branch nodes. The branch nodes may
be located in the diagram at the similarity level of the join represented, or may be
ordered by the size of the subtree. The shape of dendrograms can be characterized
by the frequency distribution of the numbers of subtrees of a given size (mk subtrees
with k leaf nodes) and by the number of “terminal singles” which are leaf nodes
that are joined as singletons at the end of the process. Of course, by some mea-
sures, all dendrograms are the same “shape” for any given n, with the same number
of leaf nodes, branch nodes and edges. Dendrograms’ shapes can be compared to
determine similarity based on probability calculations or on more recently devel-
oped approaches that use graph kernels.
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m1 = 3
m2 = 1
m3 = 1
n = 7 leaf nodes
m6 = 1
root
6 branch nodes
and their orders
Figure 2.3 Dendrogram shape: one evaluation of shape is the numbers of subtrees of any given
size (the number of leaf nodes included), often recorded as the orders of the branch nodes. Here
the tree is drawn with equal heights for same-size groupings in triangular form.
because they represent species or sites that are not closely related to any group and
should be more common in data that are not strongly structured into sub-groupings of
the objects. Calculating the frequency distribution of the number of terminal singles, S,
provides the expected values, and S is significantly larger than expected if S 1 for n
10, and (surprisingly) if S 0 once n is greater than 40. That means that, for larger
values of n, terminal singles should be very rare.
4
2
(a)
4 terminal singles
(b) 2 terminal singles
Figure 2.4 Terminal singles in dendrograms: single leaf nodes joined to the whole group at the
end of the clustering process. Here the dendrograms are presented in their triangular versions.
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2.1.2 Comparison
To compare two dendrograms of the same objects based on similar or different data,
there are several methods, but the general approach was developed in the early 1960s
(Sokal Rohlf 1962), which uses the correlation of the values of the joins of the same
pair of objects in the two classifications. One variant of this method is to use the correla-
tion of the subtree sizes in which the pair of objects first occur together. The significance
of the outcome can be determined by a simple randomization procedure that maintains
the overall structures but re-labels the leaf nodes at random.
A more recent approach is to compare two dendrograms by a technique based on “ker-
nels” or kernel functions, which was developed for trees in general. Kernels are math-
ematical functions used extensively in pattern analysis and machine learning (Shawe-
Taylor Cristianini 2004). A kernel is a function of two mathematical objects (like two
graphs) that measures their similarity, based on an inner product or on a direct product
of the two objects to map them into a more usable space (Shervashidze et al. 2011).
Chapter 9 of this book describes one such kernel method for determining similarity that
is based on random walks on the graphs. Imagine comparing a set of random walks (as
described in Chapter 1) on dendrogram 1 with a set of random walks on dendrogram 2,
more similar graphs will have more similar sets of walks. Interestingly, Oh et al. (2006)
used the kernel approach in determining phylogenetic trees based on metabolic networks
to quantify similarity from the original data and based their classification on those val-
ues. This approach has also been used in the analysis of natural languages (Moschitti
2006; Sun et al. 2011) and to compare characteristics of phylogenies of RNA viruses
(Poon et al. 2013). In those phylogenetic trees, the patterns discerned in the analysis
seemed to reflect modes of transmission and pathogenesis, and the authors concluded
that the kernel approach represents an important new tool for characterizing evolution
and epidemiology of viruses (Poon et al. 2013). That paper provides the information
needed to apply this technique to comparing trees from ecological data, for example in
studies related to functional diversity (see Petchey Gaston 2002; Poos et al. 2009) and
it seems like a smart thing to try!
2.2 Digraphs and Directed Acyclic Graphs
The next change in the rules of graph construction to be considered is allowing the
edges to have directions, giving digraphs (Box 2.3) and other directed structures.
With digraphs, some of the familiar concepts needed adjusting. For example, a non-
directed graph is connected when there is a path between any two nodes. In a digraph,
if there is a (directed) path between any two nodes, it is strongly connected; if there
are only semi-paths between some nodes with paths between all others, it is weakly
connected. (The tree in Figure 2.5 is weakly connected because there are only semi-
paths to the root node.) The other adjustment is in the possible existence of an edge in
each direction between a pair of nodes, which is easy to see for symmetric interactions
like associations, but it can apply to asymmetric interactions too. There can be edges
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Box 2.3 Digraphs
Digraphs have edges that have direction, and any pair of nodes can have two edges
between them: A to B as well as B to A. Properly, a cycle in a digraph is a directed
cycle: that is a directed path from a node back to itself following the directions of
the arrows on the edges. Not following the edge directions yields a semi-path and
thus a semi-cycle. Digraphs have obvious applications for asymmetric relationships,
such as the influence of one organism on another (whether positive or negative),
but they can also be used to signal reciprocal relationships explicitly in contrast
to unidirectional affects. Predation, competition, facilitation and other asymmetric
interactions are portrayed by digraphs, but so are systems with physical routes for
movement, flow or transportation; in those digraphs the edges often have weights to
indicate capacity, rates or distances.
representing predation between the nodes representing species; consider the situation in
which large fish of species A eat small fish of species B and large fish of species B eat
small fish of species A. Schematically:
A ↔ B because A ← B and A → B.
A system that has directionality and directed edges and no cycles combines some of
the features of trees with those of digraphs. A directed acyclic graph (DAG, see Box
2.4) is a digraph that contains no directed cycles but may have semi-cycles (Harary
1969) (i.e. which would be cycles if the direction did not count); this is, its underlying
graph may not be a tree and may have non-directed cycles (cf. Newman 2010, Section
6.4.2). Figure 2.5 shows a directed rooted tree, sometimes called an arborescence, and
a directional acyclic graph. In the directed tree, there is exactly one directed path from
the root to any other node, and at most one directed path between any two nodes; in the
acyclic digraph, there can be several. Every arborescence is a directed acyclic graph but
root
Figure 2.5 Directed and directional graphs. (left) A directed rooted tree is an arborescence and its
underlying graph (undirected version) has no cycles. (right) A directed acyclic graph (DAG) is
similar, but its underlying graph has cycles. Both are partial orders.
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Box 2.4 Directed Acyclic Graphs
These DAGs are digraphs that have no directed cycles, and so are acyclic like trees.
Unlike trees, however, they can have more than one path between any two nodes, but
not all pairs of nodes will have a path between them. Their chief applications are
in causal and data structures, but their ecological application can include the spatial
context for physical paths like braided streams, anastomosing hyphal systems, or the
trails or movements of animals (like transportation systems).
not every acyclic graph is an arborescence. Any directed acyclic graph has at least one
source node of in-degree 0, and at least one sink node of out-degree 0.
Obvious examples of directed acyclic graphs are deltas and braided streams in hydro-
logical systems, or the anastomosing networks of fungal colonies. These may have mul-
tiple paths between nodes, not just one as in a tree, but they can also have more than one
source and more than one sink. Figure 2.6 shows a system of braided trails that form a
graph with cycles (top), but the migration of a herd along those trails form an acyclic
digraph (Figure 2.6a, bottom). The acyclic digraph does not need to be a connected
graph (Figure 2.6b). Directed trees and acyclic digraphs arise naturally in many applica-
tions, including causal structures in epidemiology (Greenland et al. 1999); genealogical,
phylogenetic, and recombinant networks (e.g. Strimmer Moulton 2000); and search
and topological ordering algorithms in computer science (Cormen et al. 2009). They
are also key structures in the study of correlation, causality and the development of
structural models (Mitchell 1992; Shipley 2000, 2009; Pearl 2009). In a rooted tree, the
structure creates a “partial ordering” (see Box 2.5) of the nodes determined by (path)
distance from the root (see Figure 2.7); similarly, DAGs give partial order to the nodes
(Figure 2.6b).
As remarked, trees are also a standard format for data storage and retrieval, with a
number of different techniques for organizing and finding information. A distinguish-
ing feature for examining any of these trees is whether the nodes represent the same
or different things. In a Minimum Spanning Tree, which provides a skeleton that joins
spatially located nodes, the nodes are all the same (e.g. locations of individual organ-
isms); but in a dendrogram, the leaf nodes and the branch nodes represent different
things (e.g. individual units vs joins forming groupings of such units), and thus have
different roles. The difference between the roles is reflected in their graph theory prop-
erties; for example, leaf nodes have degree 1 and branch nodes have degree 3 (or more).
In digraphs, generally, and in DAGs, each node has both an in-degree (upstream neigh-
bours) and out-degree (downstream neighbours). Some nodes will have only outgoing
edges (sources) and some will have only incoming edges (sinks). In the functioning of
the ecological system depicted by a digraph, sources and sinks obviously have different
roles, and for intermediate nodes, the ratio or difference between in-degree and out-
degree is expected to indicate role differentiation. Further on in the discussion of graph
properties and system function, the topic of what can be understood about function
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Exploring the Variety of Random
Documents with Different Content

CHAPTER V.
Clerval then put the following letter into my hands.
“To V. Frankenstein.
“My Dear Cousin,
“I cannot describe to you the uneasiness we have all felt concerning
your health. We cannot help imagining that your friend Clerval
conceals the extent of your disorder: for it is now several months
since we have seen your hand-writing; and all this time you have
been obliged to dictate your letters to Henry. Surely, Victor, you must
have been exceedingly ill; and this makes us all very wretched, as
much so nearly as after the death of your dear mother. My uncle was
almost persuaded that you were indeed dangerously ill, and could
hardly be restrained from undertaking a journey to Ingolstadt.
Clerval always writes that you are getting better; I eagerly hope that
you will confirm this intelligence soon in your own hand-writing; for
indeed, indeed, Victor, we are all very miserable on this account.
Relieve us from this fear, and we shall be the happiest creatures in
the world. Your father’s health is now so vigorous, that he appears
ten years younger since last winter. Ernest also is so much improved,
that you would hardly know him: he is now nearly sixteen, and has
lost that sickly appearance which he had some years ago; he is
grown quite robust and active.
“My uncle and I conversed a long time last night about what
profession Ernest should follow. His constant illness when young has
deprived him of the habits of application; and now that he enjoys
good health, he is continually in the open air, climbing the hills, or
rowing on the lake. I therefore proposed that he should be a farmer;
which you know, Cousin, is a favourite scheme of mine. A farmer’s is

a very healthy happy life; and the least hurtful, or rather the most
beneficial profession of any. My uncle had an idea of his being
educated as an advocate, that through his interest he might become
a judge. But, besides that he is not at all fitted for such an
occupation, it is certainly more creditable to cultivate the earth for
the sustenance of man, than to be the confidant, and sometimes the
accomplice, of his vices; which is the profession of a lawyer. I said,
that the employments of a prosperous farmer, if they were not a
more honourable, they were at least a happier species of occupation
than that of a judge, whose misfortune it was always to meddle with
the dark side of human nature. My uncle smiled, and said, that I
ought to be an advocate myself, which put an end to the
conversation on that subject.
“And now I must tell you a little story that will please, and perhaps
amuse you. Do you not remember Justine Moritz? Probably you do
not; I will relate her history, therefore, in a few words. Madame
Moritz, her mother, was a widow with four children, of whom Justine
was the third. This girl had always been the favourite of her father;
but, through a strange perversity, her mother could not endure her,
and, after the death of M. Moritz, treated her very ill. My aunt
observed this; and, when Justine was twelve years of age, prevailed
on her mother to allow her to live at her house. The republican
institutions of our country have produced simpler and happier
manners than those which prevail in the great monarchies that
surround it. Hence there is less distinction between the several
classes of its inhabitants; and the lower orders being neither so poor
nor so despised, their manners are more refined and moral. A
servant in Geneva does not mean the same thing as a servant in
France and England. Justine, thus received in our family, learned the
duties of a servant; a condition which, in our fortunate country, does
not include the idea of ignorance, and a sacrifice of the dignity of a
human being.
“After what I have said, I dare say you well remember the heroine of
my little tale: for Justine was a great favourite of your’s; and I

recollect you once remarked, that if you were in an ill-humour, one
glance from Justine could dissipate it, for the same reason that
Ariosto gives concerning the beauty of Angelica—she looked so
frank-hearted and happy. My aunt conceived a great attachment for
her, by which she was induced to give her an education superior to
that which she had at first intended. This benefit was fully repaid;
Justine was the most grateful little creature in the world: I do not
mean that she made any professions, I never heard one pass her
lips; but you could see by her eyes that she almost adored her
protectress. Although her disposition was gay, and in many respects
inconsiderate, yet she paid the greatest attention to every gesture of
my aunt. She thought her the model of all excellence, and
endeavoured to imitate her phraseology and manners, so that even
now she often reminds me of her.
“When my dearest aunt died, every one was too much occupied in
their own grief to notice poor Justine, who had attended her during
her illness with the most anxious affection. Poor Justine was very ill;
but other trials were reserved for her.
“One by one, her brothers and sister died; and her mother, with the
exception of her neglected daughter, was left childless. The
conscience of the woman was troubled; she began to think that the
deaths of her favourites was a judgment from heaven to chastise her
partiality. She was a Roman Catholic; and I believe her confessor
confirmed the idea which she had conceived. Accordingly, a few
months after your departure for Ingolstadt, Justine was called home
by her repentant mother. Poor girl! she wept when she quitted our
house: she was much altered since the death of my aunt; grief had
given softness and a winning mildness to her manners, which had
before been remarkable for vivacity. Nor was her residence at her
mother’s house of a nature to restore her gaiety. The poor woman
was very vacillating in her repentance. She sometimes begged
Justine to forgive her unkindness, but much oftener accused her of
having caused the deaths of her brothers and sister. Perpetual
fretting at length threw Madame Moritz into a decline, which at first

increased her irritability, but she is now at peace for ever. She died
on the first approach of cold weather, at the beginning of this last
winter. Justine has returned to us; and I assure you I love her
tenderly. She is very clever and gentle, and extremely pretty; as I
mentioned before, her mien and her expressions continually remind
me of my dear aunt.
“I must say also a few words to you, my dear cousin, of little darling
William. I wish you could see him; he is very tall of his age, with
sweet laughing blue eyes, dark eye-lashes, and curling hair. When
he smiles, two little dimples appear on each cheek, which are rosy
with health. He has already had one or two little wives, but Louisa
Biron is his favourite, a pretty little girl of five years of age.
“Now, dear Victor, I dare say you wish to be indulged in a little
gossip concerning the good people of Geneva. The pretty Miss
Mansfield has already received the congratulatory visits on her
approaching marriage with a young Englishman, John Melbourne,
Esq. Her ugly sister, Manon, married M. Duvillard, the rich banker,
last autumn. Your favourite schoolfellow, Louis Manoir, has suffered
several misfortunes since the departure of Clerval from Geneva. But
he has already recovered his spirits, and is reported to be on the
point of marrying a very lively pretty Frenchwoman, Madame
Tavernier. She is a widow, and much older than Manoir; but she is
very much admired, and a favourite with every body.
“I have written myself into good spirits, dear cousin; yet I cannot
conclude without again anxiously inquiring concerning your health.
Dear Victor, if you are not very ill, write yourself, and make your
father and all of us happy; or——I cannot bear to think of the other
side of the question; my tears already flow. Adieu, my dearest
cousin.”
“Elizabeth Lavenza.
“Geneva, March 18th, 17—.”

“Dear, dear Elizabeth!” I exclaimed when I had read her letter, “I will
write instantly, and relieve them from the anxiety they must feel.” I
wrote, and this exertion greatly fatigued me; but my convalescence
had commenced, and proceeded regularly. In another fortnight I was
able to leave my chamber.
One of my first duties on my recovery was to introduce Clerval to the
several professors of the university. In doing this, I underwent a kind
of rough usage, ill befitting the wounds that my mind had sustained.
Ever since the fatal night, the end of my labours, and the beginning
of my misfortunes, I had conceived a violent antipathy even to the
name of natural philosophy. When I was otherwise quite restored to
health, the sight of a chemical instrument would renew all the agony
of my nervous symptoms. Henry saw this, and had removed all my
apparatus from my view. He had also changed my apartment; for he
perceived that I had acquired a dislike for the room which had
previously been my laboratory. But these cares of Clerval were made
of no avail when I visited the professors. M. Waldman inflicted
torture when he praised, with kindness and warmth, the astonishing
progress I had made in the sciences. He soon perceived that I
disliked the subject; but, not guessing the real cause, he attributed
my feelings to modesty, and changed the subject from my
improvement to the science itself, with a desire, as I evidently saw,
of drawing me out. What could I do? He meant to please, and he
tormented me. I felt as if he had placed carefully, one by one, in my
view those instruments which were to be afterwards used in putting
me to a slow and cruel death. I writhed under his words, yet dared
not exhibit the pain I felt. Clerval, whose eyes and feelings were
always quick in discerning the sensations of others, declined the
subject, alleging, in excuse, his total ignorance; and the
conversation took a more general turn. I thanked my friend from my
heart, but I did not speak. I saw plainly that he was surprised, but
he never attempted to draw my secret from me; and although I
loved him with a mixture of affection and reverence that knew no
bounds, yet I could never persuade myself to confide to him that

event which was so often present to my recollection, but which I
feared the detail to another would only impress more deeply.
M. Krempe was not equally docile; and in my condition at that time,
of almost insupportable sensitiveness, his harsh blunt encomiums
gave me even more pain than the benevolent approbation of M.
Waldman. “D—n the fellow!” cried he; “why, M. Clerval, I assure you
he has outstript us all. Aye, stare if you please; but it is nevertheless
true. A youngster who, but a few years ago, believed Cornelius
Agrippa as firmly as the gospel, has now set himself at the head of
the university; and if he is not soon pulled down, we shall all be out
of countenance.—Aye, aye,” continued he, observing my face
expressive of suffering, “M. Frankenstein is modest; an excellent
quality in a young man. Young men should be diffident of
themselves, you know, M. Clerval; I was myself when young: but
that wears out in a very short time.”
M. Krempe had now commenced an eulogy on himself, which happily
turned the conversation from a subject that was so annoying to me.
Clerval was no natural philosopher. His imagination was too vivid for
the minutiæ of science. Languages were his principal study; and he
sought, by acquiring their elements, to open a field for self-
instruction on his return to Geneva. Persian, Arabic, and Hebrew,
gained his attention, after he had made himself perfectly master of
Greek and Latin. For my own part, idleness had ever been irksome
to me; and now that I wished to fly from reflection, and hated my
former studies, I felt great relief in being the fellow-pupil with my
friend, and found not only instruction but consolation in the works of
the orientalists. Their melancholy is soothing, and their joy elevating
to a degree I never experienced in studying the authors of any other
country. When you read their writings, life appears to consist in a
warm sun and garden of roses,—in the smiles and frowns of a fair
enemy, and the fire that consumes your own heart. How different
from the manly and heroical poetry of Greece and Rome.

Summer passed away in these occupations, and my return to
Geneva was fixed for the latter end of autumn; but being delayed by
several accidents, winter and snow arrived, the roads were deemed
impassable, and my journey was retarded until the ensuing spring. I
felt this delay very bitterly; for I longed to see my native town, and
my beloved friends. My return had only been delayed so long from
an unwillingness to leave Clerval in a strange place, before he had
become acquainted with any of its inhabitants. The winter, however,
was spent cheerfully; and although the spring was uncommonly late,
when it came, its beauty compensated for its dilatoriness.
The month of May had already commenced, and I expected the
letter daily which was to fix the date of my departure, when Henry
proposed a pedestrian tour in the environs of Ingolstadt that I might
bid a personal farewell to the country I had so long inhabited. I
acceded with pleasure to this proposition: I was fond of exercise,
and Clerval had always been my favourite companion in the rambles
of this nature that I had taken among the scenes of my native
country.
We passed a fortnight in these perambulations: my health and spirits
had long been restored, and they gained additional strength from
the salubrious air I breathed, the natural incidents of our progress,
and the conversation of my friend. Study had before secluded me
from the intercourse of my fellow-creatures, and rendered me
unsocial; but Clerval called forth the better feelings of my heart; he
again taught me to love the aspect of nature, and the cheerful faces
of children. Excellent friend! how sincerely did you love me, and
endeavour to elevate my mind, until it was on a level with your own.
A selfish pursuit had cramped and narrowed me, until your
gentleness and affection warmed and opened my senses; I became
the same happy creature who, a few years ago, loving and beloved
by all, had no sorrow or care. When happy, inanimate nature had
the power of bestowing on me the most delightful sensations. A
serene sky and verdant fields filled me with ecstacy. The present
season was indeed divine; the flowers of spring bloomed in the

hedges, while those of summer were already in bud: I was
undisturbed by thoughts which during the preceding year had
pressed upon me, notwithstanding my endeavours to throw them
off, with an invincible burden.
Henry rejoiced in my gaiety, and sincerely sympathized in my
feelings: he exerted himself to amuse me, while he expressed the
sensations that filled his soul. The resources of his mind on this
occasion were truly astonishing: his conversation was full of
imagination; and very often, in imitation of the Persian and Arabic
writers, he invented tales of wonderful fancy and passion. At other
times he repeated my favourite poems, or drew me out into
arguments, which he supported with great ingenuity.
We returned to our college on a Sunday afternoon: the peasants
were dancing, and every one we met appeared gay and happy. My
own spirits were high, and I bounded along with feelings of
unbridled joy and hilarity.

CHAPTER VI.
On my return, I found the following letter from my father:—
“To V. Frankenstein.
“My Dear Victor,
“You have probably waited impatiently for a letter to fix the date of
your return to us; and I was at first tempted to write only a few
lines, merely mentioning the day on which I should expect you. But
that would be a cruel kindness, and I dare not do it. What would be
your surprise, my son, when you expected a happy and gay
welcome, to behold, on the contrary, tears and wretchedness? And
how, Victor, can I relate our misfortune? Absence cannot have
rendered you callous to our joys and griefs; and how shall I inflict
pain on an absent child? I wish to prepare you for the woeful news,
but I know it is impossible; even now your eye skims over the page,
to seek the words which are to convey to you the horrible tidings.
“William is dead!—that sweet child, whose smiles delighted and
warmed my heart, who was so gentle, yet so gay! Victor, he is
murdered!
“I will not attempt to console you; but will simply relate the
circumstances of the transaction.
“Last Thursday (May 7th) I, my niece, and your two brothers, went
to walk in Plainpalais. The evening was warm and serene, and we
prolonged our walk farther than usual. It was already dusk before
we thought of returning; and then we discovered that William and
Ernest, who had gone on before, were not to be found. We
accordingly rested on a seat until they should return. Presently

Ernest came, and inquired if we had seen his brother: he said, that
they had been playing together, that William had run away to hide
himself, and that he vainly sought for him, and afterwards waited for
him a long time, but that he did not return.
“This account rather alarmed us, and we continued to search for him
until night fell, when Elizabeth conjectured that he might have
returned to the house. He was not there. We returned again, with
torches; for I could not rest, when I thought that my sweet boy had
lost himself, and was exposed to all the damps and dews of night:
Elizabeth also suffered extreme anguish. About five in the morning I
discovered my lovely boy, whom the night before I had seen
blooming and active in health, stretched on the grass livid and
motionless: the print of the murderer’s finger was on his neck.
“He was conveyed home, and the anguish that was visible in my
countenance betrayed the secret to Elizabeth. She was very earnest
to see the corpse. At first I attempted to prevent her; but she
persisted, and entering the room where it lay, hastily examined the
neck of the victim, and clasping her hands exclaimed, ‘O God! I have
murdered my darling infant!’
“She fainted, and was restored with extreme difficulty. When she
again lived, it was only to weep and sigh. She told me, that that
same evening William had teazed her to let him wear a very valuable
miniature that she possessed of your mother. This picture is gone,
and was doubtless the temptation which urged the murderer to the
deed. We have no trace of him at present, although our exertions to
discover him are unremitted; but they will not restore my beloved
William.
“Come, dearest Victor; you alone can console Elizabeth. She weeps
continually, and accuses herself unjustly as the cause of his death;
her words pierce my heart. We are all unhappy; but will not that be
an additional motive for you, my son, to return and be our
comforter? Your dear mother! Alas, Victor! I now say, Thank God she

did not live to witness the cruel, miserable death of her youngest
darling!
“Come, Victor; not brooding thoughts of vengeance against the
assassin, but with feelings of peace and gentleness, that will heal,
instead of festering the wounds of our minds. Enter the house of
mourning, my friend, but with kindness and affection for those who
love you, and not with hatred for your enemies.
“Your affectionate and
afflicted father,
“Alphonse
Frankenstein.
“Geneva, May 12th, 17—.”
Clerval, who had watched my countenance as I read this letter, was
surprised to observe the despair that succeeded to the joy I at first
expressed on receiving news from my friends. I threw the letter on
the table, and covered my face with my hands.
“My dear Frankenstein,” exclaimed Henry, when he perceived me
weep with bitterness, “are you always to be unhappy? My dear
friend, what has happened?”
I motioned to him to take up the letter, while I walked up and down
the room in the extremest agitation. Tears also gushed from the
eyes of Clerval, as he read the account of my misfortune.
“I can offer you no consolation, my friend,” said he; “your disaster is
irreparable. What do you intend to do?”
“To go instantly to Geneva: come with me, Henry, to order the
horses.”

During our walk, Clerval endeavoured to raise my spirits. He did not
do this by common topics of consolation, but by exhibiting the truest
sympathy. “Poor William!” said he, “that dear child; he now sleeps
with his angel mother. His friends mourn and weep, but he is at rest:
he does not now feel the murderer’s grasp; a sod covers his gentle
form, and he knows no pain. He can no longer be a fit subject for
pity; the survivors are the greatest sufferers, and for them time is
the only consolation. Those maxims of the Stoics, that death was no
evil, and that the mind of man ought to be superior to despair on
the eternal absence of a beloved object, ought not to be urged.
Even Cato wept over the dead body of his brother.”
Clerval spoke thus as we hurried through the streets; the words
impressed themselves on my mind, and I remembered them
afterwards in solitude. But now, as soon as the horses arrived, I
hurried into a cabriole, and bade farewell to my friend.
My journey was very melancholy. At first I wished to hurry on, for I
longed to console and sympathize with my loved and sorrowing
friends; but when I drew near my native town, I slackened my
progress. I could hardly sustain the multitude of feelings that
crowded into my mind. I passed through scenes familiar to my
youth, but which I had not seen for nearly six years. How altered
every thing might be during that time? One sudden and desolating
change had taken place; but a thousand little circumstances might
have by degrees worked other alterations which, although they were
done more tranquilly, might not be the less decisive. Fear overcame
me; I dared not advance, dreading a thousand nameless evils that
made me tremble, although I was unable to define them.
I remained two days at Lausanne, in this painful state of mind. I
contemplated the lake: the waters were placid; all around was calm,
and the snowy mountains, “the palaces of nature,” were not
changed. By degrees the calm and heavenly scene restored me, and
I continued my journey towards Geneva.

The road ran by the side of the lake, which became narrower as I
approached my native town. I discovered more distinctly the black
sides of Jura, and the bright summit of Mont Blânc; I wept like a
child: “Dear mountains! my own beautiful lake! how do you welcome
your wanderer? Your summits are clear; the sky and lake are blue
and placid. Is this to prognosticate peace, or to mock at my
unhappiness?”
I fear, my friend, that I shall render myself tedious by dwelling on
these preliminary circumstances; but they were days of comparative
happiness, and I think of them with pleasure. My country, my
beloved country! who but a native can tell the delight I took in again
beholding thy streams, thy mountains, and, more than all, thy lovely
lake.
Yet, as I drew nearer home, grief and fear again overcame me.
Night also closed around; and when I could hardly see the dark
mountains, I felt still more gloomily. The picture appeared a vast and
dim scene of evil, and I foresaw obscurely that I was destined to
become the most wretched of human beings. Alas! I prophesied
truly, and failed only in one single circumstance, that in all the
misery I imagined and dreaded, I did not conceive the hundredth
part of the anguish I was destined to endure.
It was completely dark when I arrived in the environs of Geneva; the
gates of the town were already shut; and I was obliged to pass the
night at Secheron, a village half a league to the east of the city. The
sky was serene; and, as I was unable to rest, I resolved to visit the
spot where my poor William had been murdered. As I could not pass
through the town, I was obliged to cross the lake in a boat to arrive
at Plainpalais. During this short voyage I saw the lightnings playing
on the summit of Mont Blânc in the most beautiful figures. The
storm appeared to approach rapidly; and, on landing, I ascended a
low hill, that I might observe its progress. It advanced; the heavens
were clouded, and I soon felt the rain coming slowly in large drops,
but its violence quickly increased.

I quitted my seat, and walked on, although the darkness and storm
increased every minute, and the thunder burst with a terrific crash
over my head. It was echoed from Salêve, the Juras, and the Alps of
Savoy; vivid flashes of lightning dazzled my eyes, illuminating the
lake, making it appear like a vast sheet of fire; then for an instant
every thing seemed of a pitchy darkness, until the eye recovered
itself from the preceding flash. The storm, as is often the case in
Switzerland, appeared at once in various parts of the heavens. The
most violent storm hung exactly north of the town, over that part of
the lake which lies between the promontory of Belrive and the
village of Copêt. Another storm enlightened Jura with faint flashes;
and another darkened and sometimes disclosed the Môle, a peaked
mountain to the east of the lake.
While I watched the storm, so beautiful yet terrific, I wandered on
with a hasty step. This noble war in the sky elevated my spirits; I
clasped my hands, and exclaimed aloud, “William, dear angel! this is
thy funeral, this thy dirge!” As I said these words, I perceived in the
gloom a figure which stole from behind a clump of trees near me; I
stood fixed, gazing intently: I could not be mistaken. A flash of
lightning illuminated the object, and discovered its shape plainly to
me; its gigantic stature, and the deformity of its aspect, more
hideous than belongs to humanity, instantly informed me that it was
the wretch, the filthy dæmon to whom I had given life. What did he
there? Could he be (I shuddered at the conception) the murderer of
my brother? No sooner did that idea cross my imagination, than I
became convinced of its truth; my teeth chattered, and I was forced
to lean against a tree for support. The figure passed me quickly, and
I lost it in the gloom. Nothing in human shape could have destroyed
that fair child. He was the murderer! I could not doubt it. The mere
presence of the idea was an irresistible proof of the fact. I thought
of pursuing the devil; but it would have been in vain, for another
flash discovered him to me hanging among the rocks of the nearly
perpendicular ascent of Mont Salêve, a hill that bounds Plainpalais
on the south. He soon reached the summit, and disappeared.

I remained motionless. The thunder ceased; but the rain still
continued, and the scene was enveloped in an impenetrable
darkness. I revolved in my mind the events which I had until now
sought to forget: the whole train of my progress towards the
creation; the appearance of the work of my own hands alive at my
bed side; its departure. Two years had now nearly elapsed since the
night on which he first received life; and was this his first crime?
Alas! I had turned loose into the world a depraved wretch, whose
delight was in carnage and misery; had he not murdered my
brother?
No one can conceive the anguish I suffered during the remainder of
the night, which I spent, cold and wet, in the open air. But I did not
feel the inconvenience of the weather; my imagination was busy in
scenes of evil and despair. I considered the being whom I had cast
among mankind, and endowed with the will and power to effect
purposes of horror, such as the deed which he had now done, nearly
in the light of my own vampire, my own spirit let loose from the
grave, and forced to destroy all that was dear to me.
Day dawned; and I directed my steps towards the town. The gates
were open; and I hastened to my father’s house. My first thought
was to discover what I knew of the murderer, and cause instant
pursuit to be made. But I paused when I reflected on the story that
I had to tell. A being whom I myself had formed, and endued with
life, had met me at midnight among the precipices of an inaccessible
mountain. I remembered also the nervous fever with which I had
been seized just at the time that I dated my creation, and which
would give an air of delirium to a tale otherwise so utterly
improbable. I well knew that if any other had communicated such a
relation to me, I should have looked upon it as the ravings of
insanity. Besides, the strange nature of the animal would elude all
pursuit, even if I were so far credited as to persuade my relatives to
commence it. Besides, of what use would be pursuit? Who could
arrest a creature capable of scaling the overhanging sides of Mont

Salêve? These reflections determined me, and I resolved to remain
silent.
It was about five in the morning when I entered my father’s house. I
told the servants not to disturb the family, and went into the library
to attend their usual hour of rising.
Six years had elapsed, passed as a dream but for one indelible trace,
and I stood in the same place where I had last embraced my father
before my departure for Ingolstadt. Beloved and respectable parent!
He still remained to me. I gazed on the picture of my mother, which
stood over the mantle-piece. It was an historical subject, painted at
my father’s desire, and represented Caroline Beaufort in an agony of
despair, kneeling by the coffin of her dead father. Her garb was
rustic, and her cheek pale; but there was an air of dignity and
beauty, that hardly permitted the sentiment of pity. Below this
picture was a miniature of William; and my tears flowed when I
looked upon it. While I was thus engaged, Ernest entered: he had
heard me arrive, and hastened to welcome me. He expressed a
sorrowful delight to see me: “Welcome, my dearest Victor,” said he.
“Ah! I wish you had come three months ago, and then you would
have found us all joyous and delighted. But we are now unhappy;
and, I am afraid, tears instead of smiles will be your welcome. Our
father looks so sorrowful: this dreadful event seems to have revived
in his mind his grief on the death of Mamma. Poor Elizabeth also is
quite inconsolable.” Ernest began to weep as he said these words.
“Do not,” said I, “welcome me thus; try to be more calm, that I may
not be absolutely miserable the moment I enter my father’s house
after so long an absence. But, tell me, how does my father support
his misfortunes? and how is my poor Elizabeth?”
“She indeed requires consolation; she accused herself of having
caused the death of my brother, and that made her very wretched.
But since the murderer has been discovered——”

“The murderer discovered! Good God! how can that be? who could
attempt to pursue him? It is impossible; one might as well try to
overtake the winds, or confine a mountain-stream with a straw.”
“I do not know what you mean; but we were all very unhappy when
she was discovered. No one would believe it at first; and even now
Elizabeth will not be convinced, notwithstanding all the evidence.
Indeed, who would credit that Justine Moritz, who was so amiable,
and fond of all the family, could all at once become so extremely
wicked?”
“Justine Moritz! Poor, poor girl, is she the accused? But it is
wrongfully; every one knows that; no one believes it, surely,
Ernest?”
“No one did at first; but several circumstances came out, that have
almost forced conviction upon us: and her own behaviour has been
so confused, as to add to the evidence of facts a weight that, I fear,
leaves no hope for doubt. But she will be tried to-day, and you will
then hear all.”
He related that, the morning on which the murder of poor William
had been discovered, Justine had been taken ill, and confined to her
bed; and, after several days, one of the servants, happening to
examine the apparel she had worn on the night of the murder, had
discovered in her pocket the picture of my mother, which had been
judged to be the temptation of the murderer. The servant instantly
shewed it to one of the others, who, without saying a word to any of
the family, went to a magistrate; and, upon their deposition, Justine
was apprehended. On being charged with the fact, the poor girl
confirmed the suspicion in a great measure by her extreme
confusion of manner.
This was a strange tale, but it did not shake my faith; and I replied
earnestly, “You are all mistaken; I know the murderer. Justine, poor,
good Justine, is innocent.”

At that instant my father entered. I saw unhappiness deeply
impressed on his countenance, but he endeavoured to welcome me
cheerfully; and, after we had exchanged our mournful greeting,
would have introduced some other topic than that of our disaster,
had not Ernest exclaimed, “Good God, Papa! Victor says that he
knows who was the murderer of poor William.”
“We do also, unfortunately,” replied my father; “for indeed I had
rather have been for ever ignorant than have discovered so much
depravity and ingratitude in one I valued so highly.”
“My dear father, you are mistaken; Justine is innocent.”
“If she is, God forbid that she should suffer as guilty. She is to be
tried to-day, and I hope, I sincerely hope, that she will be acquitted.”
This speech calmed me. I was firmly convinced in my own mind that
Justine, and indeed every human being, was guiltless of this murder.
I had no fear, therefore, that any circumstantial evidence could be
brought forward strong enough to convict her; and, in this
assurance, I calmed myself, expecting the trial with eagerness, but
without prognosticating an evil result.
We were soon joined by Elizabeth. Time had made great alterations
in her form since I had last beheld her. Six years before she had
been a pretty, good-humoured girl, whom every one loved and
caressed. She was now a woman in stature and expression of
countenance, which was uncommonly lovely. An open and capacious
forehead gave indications of a good understanding, joined to great
frankness of disposition. Her eyes were hazel, and expressive of
mildness, now through recent affliction allied to sadness. Her hair
was of a rich, dark auburn, her complexion fair, and her figure slight
and graceful. She welcomed me with the greatest affection. “Your
arrival, my dear cousin,” said she, “fills me with hope. You perhaps
will find some means to justify my poor guiltless Justine. Alas! who is
safe, if she be convicted of crime? I rely on her innocence as
certainly as I do upon my own. Our misfortune is doubly hard to us;

we have not only lost that lovely darling boy, but this poor girl,
whom I sincerely love, is to be torn away by even a worse fate. If
she is condemned, I never shall know joy more. But she will not, I
am sure she will not; and then I shall be happy again, even after the
sad death of my little William.”
“She is innocent, my Elizabeth,” said I, “and that shall be proved;
fear nothing, but let your spirits be cheered by the assurance of her
acquittal.”
“How kind you are! every one else believes in her guilt, and that
made me wretched; for I knew that it was impossible: and to see
every one else prejudiced in so deadly a manner, rendered me
hopeless and despairing.” She wept.
“Sweet niece,” said my father, “dry your tears. If she is, as you
believe, innocent, rely on the justice of our judges, and the activity
with which I shall prevent the slightest shadow of partiality.”

CHAPTER VII.
We passed a few sad hours, until eleven o’clock, when the trial was
to commence. My father and the rest of the family being obliged to
attend as witnesses, I accompanied them to the court. During the
whole of this wretched mockery of justice, I suffered living torture. It
was to be decided, whether the result of my curiosity and lawless
devices would cause the death of two of my fellow-beings: one a
smiling babe, full of innocence and joy; the other far more dreadfully
murdered, with every aggravation of infamy that could make the
murder memorable in horror. Justine also was a girl of merit, and
possessed qualities which promised to render her life happy: now all
was to be obliterated in an ignominious grave; and I the cause! A
thousand times rather would I have confessed myself guilty of the
crime ascribed to Justine; but I was absent when it was committed,
and such a declaration would have been considered as the ravings of
a madman, and would not have exculpated her who suffered
through me.
The appearance of Justine was calm. She was dressed in mourning;
and her countenance, always engaging, was rendered, by the
solemnity of her feelings, exquisitely beautiful. Yet she appeared
confident in innocence, and did not tremble, although gazed on and
execrated by thousands; for all the kindness which her beauty might
otherwise have excited, was obliterated in the minds of the
spectators by the imagination of the enormity she was supposed to
have committed. She was tranquil, yet her tranquillity was evidently
constrained; and as her confusion had before been adduced as a
proof of her guilt, she worked up her mind to an appearance of
courage. When she entered the court, she threw her eyes round it,
and quickly discovered where we were seated. A tear seemed to dim
her eye when she saw us; but she quickly recovered herself, and a
look of sorrowful affection seemed to attest her utter guiltlessness.

The trial began; and after the advocate against her had stated the
charge, several witnesses were called. Several strange facts
combined against her, which might have staggered any one who had
not such proof of her innocence as I had. She had been out the
whole of the night on which the murder had been committed, and
towards morning had been perceived by a market-woman not far
from the spot where the body of the murdered child had been
afterwards found. The woman asked her what she did there; but she
looked very strangely, and only returned a confused and
unintelligible answer. She returned to the house about eight o’clock;
and when one inquired where she had passed the night, she replied,
that she had been looking for the child, and demanded earnestly, if
any thing had been heard concerning him. When shewn the body,
she fell into violent hysterics, and kept her bed for several days. The
picture was then produced, which the servant had found in her
pocket; and when Elizabeth, in a faltering voice, proved that it was
the same which, an hour before the child had been missed, she had
placed round his neck, a murmur of horror and indignation filled the
court.
Justine was called on for her defence. As the trial had proceeded,
her countenance had altered. Surprise, horror, and misery, were
strongly expressed. Sometimes she struggled with her tears; but
when she was desired to plead, she collected her powers, and spoke
in an audible although variable voice:—
“God knows,” she said, “how entirely I am innocent. But I do not
pretend that my protestations should acquit me: I rest my innocence
on a plain and simple explanation of the facts which have been
adduced against me; and I hope the character I have always borne
will incline my judges to a favourable interpretation, where any
circumstance appears doubtful or suspicious.”
She then related that, by the permission of Elizabeth, she had
passed the evening of the night on which the murder had been
committed, at the house of an aunt at Chêne, a village situated at
about a league from Geneva. On her return, at about nine o’clock,

she met a man, who asked her if she had seen any thing of the child
who was lost. She was alarmed by this account, and passed several
hours in looking for him, when the gates of Geneva were shut, and
she was forced to remain several hours of the night in a barn
belonging to a cottage, being unwilling to call up the inhabitants, to
whom she was well known. Unable to rest or sleep, she quitted her
asylum early, that she might again endeavour to find my brother. If
she had gone near the spot where his body lay, it was without her
knowledge. That she had been bewildered when questioned by the
market-woman, was not surprising, since she had passed a sleepless
night, and the fate of poor William was yet uncertain. Concerning
the picture she could give no account.
“I know,” continued the unhappy victim, “how heavily and fatally this
one circumstance weighs against me, but I have no power of
explaining it; and when I have expressed my utter ignorance, I am
only left to conjecture concerning the probabilities by which it might
have been placed in my pocket. But here also I am checked. I
believe that I have no enemy on earth, and none surely would have
been so wicked as to destroy me wantonly. Did the murderer place it
there? I know of no opportunity afforded him for so doing; or if I
had, why should he have stolen the jewel, to part with it again so
soon?
“I commit my cause to the justice of my judges, yet I see no room
for hope. I beg permission to have a few witnesses examined
concerning my character; and if their testimony shall not overweigh
my supposed guilt, I must be condemned, although I would pledge
my salvation on my innocence.”
Several witnesses were called, who had known her for many years,
and they spoke well of her; but fear, and hatred of the crime of
which they supposed her guilty, rendered them timorous, and
unwilling to come forward. Elizabeth saw even this last resource, her
excellent dispositions and irreproachable conduct, about to fail the
accused, when, although violently agitated, she desired permission
to address the court.

“I am,” said she, “the cousin of the unhappy child who was
murdered, or rather his sister, for I was educated by and have lived
with his parents ever since and even long before his birth. It may
therefore be judged indecent in me to come forward on this
occasion; but when I see a fellow-creature about to perish through
the cowardice of her pretended friends, I wish to be allowed to
speak, that I may say what I know of her character. I am well
acquainted with the accused. I have lived in the same house with
her, at one time for five, and at another for nearly two years. During
all that period she appeared to me the most amiable and benevolent
of human creatures. She nursed Madame Frankenstein, my aunt, in
her last illness with the greatest affection and care; and afterwards
attended her own mother during a tedious illness, in a manner that
excited the admiration of all who knew her. After which she again
lived in my uncle’s house, where she was beloved by all the family.
She was warmly attached to the child who is now dead, and acted
towards him like a most affectionate mother. For my own part, I do
not hesitate to say, that, notwithstanding all the evidence produced
against her, I believe and rely on her perfect innocence. She had no
temptation for such an action: as to the bauble on which the chief
proof rests, if she had earnestly desired it, I should have willingly
given it to her; so much do I esteem and value her.”
Excellent Elizabeth! A murmur of approbation was heard; but it was
excited by her generous interference, and not in favour of poor
Justine, on whom the public indignation was turned with renewed
violence, charging her with the blackest ingratitude. She herself
wept as Elizabeth spoke, but she did not answer. My own agitation
and anguish was extreme during the whole trial. I believed in her
innocence; I knew it. Could the dæmon, who had (I did not for a
minute doubt) murdered my brother, also in his hellish sport have
betrayed the innocent to death and ignominy. I could not sustain the
horror of my situation; and when I perceived that the popular voice,
and the countenances of the judges, had already condemned my
unhappy victim, I rushed out of the court in agony. The tortures of
the accused did not equal mine; she was sustained by innocence,

but the fangs of remorse tore my bosom, and would not forego their
hold.
I passed a night of unmingled wretchedness. In the morning I went
to the court; my lips and throat were parched. I dared not ask the
fatal question; but I was known, and the officer guessed the cause
of my visit. The ballots had been thrown; they were all black, and
Justine was condemned.
I cannot pretend to describe what I then felt. I had before
experienced sensations of horror; and I have endeavoured to bestow
upon them adequate expressions, but words cannot convey an idea
of the heart-sickening despair that I then endured. The person to
whom I addressed myself added, that Justine had already confessed
her guilt. “That evidence,” he observed, “was hardly required in so
glaring a case, but I am glad of it; and, indeed, none of our judges
like to condemn a criminal upon circumstantial evidence, be it ever
so decisive.”
When I returned home, Elizabeth eagerly demanded the result.
“My cousin,” replied I, “it is decided as you may have expected; all
judges had rather that ten innocent should suffer, than that one
guilty should escape. But she has confessed.”
This was a dire blow to poor Elizabeth, who had relied with firmness
upon Justine’s innocence. “Alas!” said she, “how shall I ever again
believe in human benevolence? Justine, whom I loved and esteemed
as my sister, how could she put on those smiles of innocence only to
betray; her mild eyes seemed incapable of any severity or ill-humour,
and yet she has committed a murder.”
Soon after we heard that the poor victim had expressed a wish to
see my cousin. My father wished her not to go; but said, that he left
it to her own judgment and feelings to decide. “Yes,” said Elizabeth,
“I will go, although she is guilty; and you, Victor, shall accompany
me: I cannot go alone.” The idea of this visit was torture to me, yet I
could not refuse.

We entered the gloomy prison-chamber, and beheld Justine sitting
on some straw at the further end; her hands were manacled, and
her head rested on her knees. She rose on seeing us enter; and
when we were left alone with her, she threw herself at the feet of
Elizabeth, weeping bitterly. My cousin wept also.
“Oh, Justine!” said she, “why did you rob me of my last consolation.
I relied on your innocence; and although I was then very wretched, I
was not so miserable as I am now.”
“And do you also believe that I am so very, very wicked? Do you also
join with my enemies to crush me?” Her voice was suffocated with
sobs.
“Rise, my poor girl,” said Elizabeth, “why do you kneel, if you are
innocent? I am not one of your enemies; I believed you guiltless,
notwithstanding every evidence, until I heard that you had yourself
declared your guilt. That report, you say, is false; and be assured,
dear Justine, that nothing can shake my confidence in you for a
moment, but your own confession.”
“I did confess; but I confessed a lie. I confessed, that I might obtain
absolution; but now that falsehood lies heavier at my heart than all
my other sins. The God of heaven forgive me! Ever since I was
condemned, my confessor has besieged me; he threatened and
menaced, until I almost began to think that I was the monster that
he said I was. He threatened excommunication and hell fire in my
last moments, if I continued obdurate. Dear lady, I had none to
support me; all looked on me as a wretch doomed to ignominy and
perdition. What could I do? In an evil hour I subscribed to a lie; and
now only am I truly miserable.”
She paused, weeping, and then continued—“I thought with horror,
my sweet lady, that you should believe your Justine, whom your
blessed aunt had so highly honoured, and whom you loved, was a
creature capable of a crime which none but the devil himself could
have perpetrated. Dear William! dearest blessed child! I soon shall

see you again in heaven, where we shall all be happy; and that
consoles me, going as I am to suffer ignominy and death.”
“Oh, Justine! forgive me for having for one moment distrusted you.
Why did you confess? But do not mourn, my dear girl; I will every
where proclaim your innocence, and force belief. Yet you must die;
you, my playfellow, my companion, my more than sister. I never can
survive so horrible a misfortune.”
“Dear, sweet Elizabeth, do not weep. You ought to raise me with
thoughts of a better life, and elevate me from the petty cares of this
world of injustice and strife. Do not you, excellent friend, drive me to
despair.”
“I will try to comfort you; but this, I fear, is an evil too deep and
poignant to admit of consolation, for there is no hope. Yet heaven
bless thee, my dearest Justine, with resignation, and a confidence
elevated beyond this world. Oh! how I hate its shews and mockeries!
when one creature is murdered, another is immediately deprived of
life in a slow torturing manner; then the executioners, their hands
yet reeking with the blood of innocence, believe that they have done
a great deed. They call this retribution. Hateful name! When that
word is pronounced, I know greater and more horrid punishments
are going to be inflicted than the gloomiest tyrant has ever invented
to satiate his utmost revenge. Yet this is not consolation for you, my
Justine, unless indeed that you may glory in escaping from so
miserable a den. Alas! I would I were in peace with my aunt and my
lovely William, escaped from a world which is hateful to me, and the
visages of men which I abhor.”
Justine smiled languidly. “This, dear lady, is despair, and not
resignation. I must not learn the lesson that you would teach me.
Talk of something else, something that will bring peace, and not
increase of misery.”
During this conversation I had retired to a corner of the prison-room,
where I could conceal the horrid anguish that possessed me.

Despair! Who dared talk of that? The poor victim, who on the
morrow was to pass the dreary boundary between life and death,
felt not as I did, such deep and bitter agony. I gnashed my teeth,
and ground them together, uttering a groan that came from my
inmost soul. Justine started. When she saw who it was, she
approached me, and said, “Dear Sir, you are very kind to visit me;
you, I hope, do not believe that I am guilty.”
I could not answer. “No, Justine,” said Elizabeth; “he is more
convinced of your innocence than I was; for even when he heard
that you had confessed, he did not credit it.”
“I truly thank him. In these last moments I feel the sincerest
gratitude towards those who think of me with kindness. How sweet
is the affection of others to such a wretch as I am! It removes more
than half my misfortune; and I feel as if I could die in peace, now
that my innocence is acknowledged by you, dear lady, and your
cousin.”
Thus the poor sufferer tried to comfort others and herself. She
indeed gained the resignation she desired. But I, the true murderer,
felt the never-dying worm alive in my bosom, which allowed of no
hope or consolation. Elizabeth also wept, and was unhappy; but
her’s also was the misery of innocence, which, like a cloud that
passes over the fair moon, for a while hides, but cannot tarnish its
brightness. Anguish and despair had penetrated into the core of my
heart; I bore a hell within me, which nothing could extinguish. We
staid several hours with Justine; and it was with great difficulty that
Elizabeth could tear herself away. “I wish,” cried she, “that I were to
die with you; I cannot live in this world of misery.”
Justine assumed an air of cheerfulness, while she with difficulty
repressed her bitter tears. She embraced Elizabeth, and said, in a
voice of half-suppressed emotion, “Farewell, sweet lady, dearest
Elizabeth, my beloved and only friend; may heaven in its bounty
bless and preserve you; may this be the last misfortune that you will
ever suffer. Live, and be happy, and make others so.”

As we returned, Elizabeth said, “You know not, my dear Victor, how
much I am relieved, now that I trust in the innocence of this
unfortunate girl. I never could again have known peace, if I had
been deceived in my reliance on her. For the moment that I did
believe her guilty, I felt an anguish that I could not have long
sustained. Now my heart is lightened. The innocent suffers; but she
whom I thought amiable and good has not betrayed the trust I
reposed in her, and I am consoled.”
Amiable cousin! such were your thoughts, mild and gentle as your
own dear eyes and voice. But I—I was a wretch, and none ever
conceived of the misery that I then endured.
END OF VOL. I.

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