07 Whole Network Descriptive Statistics

WHOLE NETWORK
DESCRIPTIVES
Molly Copeland
PhD Candidate
Duke Sociology
Social Networks & Health, 2019
Duke Network Analysis Center

“Whole” Network Descriptives
• Sociocentric
• Sampling and network boundary

Overview
• Centrality
• Two-mode Networks
• Connectivity & Cohesion
• Roles

Centrality: Individual Nodes
How can we distinguish “important” actors?
• Centrality:
• Who is at the ‘center’ of the network?
…but what is meant by ‘center’ gets complicated

Centrality

Centrality
• Freeman’s (1979) criteria:
1. Calculated on individuals
2. Normalized by network size to compare across
networks
3. Derive network-level centralization counterpart

Centrality
• Most useful measurement depends in part on type
of network flow
Borgatti 2005

Centrality
• Centrality measurement approaches:
• Degrees
• Closeness
• Betweenness
• Information & Power
• Eigenvector Centrality & PageRank

Centrality
• Degree Centrality – number of ties
• Undirected
 
j
ijiiD XXndC )(

“Add Health” Example: Degree

Centrality
• Degree Centrality:
• Directed
• In-degrees
• Out-degrees
• Isolates

Centrality
Kornienko et al. 2013:
Test cortisol (stress)
on in-degrees, out-
degrees, and ego-
network density.
Find significantly
higher cortisol for low
out-degree, low and
high popularity

Centrality
But…
• Degree Centrality – local measure only
• Can be deceiving
• Less appropriate
for non-local
questions

“Add Health” Example: In-degree

Centrality
Closeness: Actors considered important if less
distance to all other actors in the network
• inverse distance of each actor to every other
• Geodesic distance – shortest path between
2 actors
• Measurable within a connected graph or
component where each node is reachable
1
1
),()(








 
g
j
jiic nndnC

Centrality
• Closeness
• Inverse of the sum of distances from actor to
all other actors
• Normalized by graph size to range 0-1
1
1
),()(








 
g
j
jiic nndnC
)1))((()('
 gnCnC iCiC

Distance Closeness normalized
0 1 1 1 1 1 1 1 .143 1.00
1 0 2 2 2 2 2 2 .077 .538
1 2 0 2 2 2 2 2 .077 .538
1 2 2 0 2 2 2 2 .077 .538
1 2 2 2 0 2 2 2 .077 .538
1 2 2 2 2 0 2 2 .077 .538
1 2 2 2 2 2 0 2 .077 .538
1 2 2 2 2 2 2 0 .077 .538
Closeness Centrality in the examples
Distance Closeness normalized
0 1 2 3 4 4 3 2 1 .050 .400
1 0 1 2 3 4 4 3 2 .050 .400
2 1 0 1 2 3 4 4 3 .050 .400
3 2 1 0 1 2 3 4 4 .050 .400
4 3 2 1 0 1 2 3 4 .050 .400
4 4 3 2 1 0 1 2 3 .050 .400
3 4 4 3 2 1 0 1 2 .050 .400
2 3 4 4 3 2 1 0 1 .050 .400
1 2 3 4 4 3 2 1 0 .050 .400

Centrality
• Closeness
• Can be directed – (in-closeness and out-
closeness)
…But:
• Non-linear distortion from taking inverse
• Different non-infinity solutions for
disconnected nodes
• Often not calculated as inverse distance

Centrality
• Betweenness
• Actor considered important if controls
information flow or bridges relatively
disconnected portions of the network
• Counts number of paths for actor j where j is
on the shortest path between actors i and k


kj
jkijkiB gngnC /)()(

Centrality
• Betweenness - number of paths for actor j where j is
on the shortest path between actors i and k
• Landon et al. 2012:
Network contexts of
physicians sharing
patients (bipartite) find
specialists have greater
betweenness in rural
regions than urban

“Add Health” Example: Betweenness

Centrality
• Information Centrality
• Like betweenness, but not restricted to
geodesics; information can probably flow
through paths other than geodesics
Betweenness Information Centrality

Low Degree/High Betweenness High Degree/Low Betweenness
(few ties crucial for network flow) (many redundant ties)
Centrality: Comparing Measures

“Add Health” Example: Comparing Centrality
ID Degree Indegree Betw. Close.
1 6 4 0.50 0.28
2 4 3 28.00 0.20
3 6 1 22.47 0.35
4 6 6 0 0
5 11 3 38.17 0.37
6 11 2 51.50 0.40
7 13 7 78.95 0.39
8 9 2 120.7 0.47
9 5 2 14.67 0.35
10 5 2 15.33 0.34
Degree Indegree Betw. Close.
Degree -- 0.77 0.71 0.65
Indegree 0.77 -- 0.53 0.25
Betw. 0.71 0.53 -- 0.54
Close. 0.65 0.25 0.54 --

Low correlations probably tell you something interesting:
Low
Degree
Low
Closeness
Low
Betweenness
High Degree Embedded in cluster
that is far from the
rest of the network
Ego's connections
are redundant -
communication
bypasses him/her
High Closeness Key player tied to
important
important/active
alters
Probably multiple
paths in the
network, ego is near
many actors, but so
are many others
High
Betweenness
Ego's few ties are
crucial for network
flow
Very rare. Would
mean ego
monopolizes the ties
from a small # of
actors to many others.
Centrality: Individual Nodes – Comparing Measures

Centrality
• Power- actors are important if tied to other
important actors
• Bonacich Power Centrality (prestige) –
actors tied to other important actors
1)(),( 1
RRIC 
 
Haas et al. 2010 find adolescents with worse self-rated health
have significantly lower Bonacich centrality in their overall school
network.

“Add Health” Example: Bonacich Power Centrality
 = .5  = -.5

Centrality
• Eigenvector centrality – similar to Bonacich, but
without beta parameter; ego’s eigenvector centrality
is equal to the sum of the centrality of its alters
• Symmetric/undirected data only
• PageRank – eigenvector centrality for directed data
Google describes this as: “PageRank relies on the uniquely democratic nature of
the web by using its vast link structure as an indicator of an individual page’s
value. In essence, Google interprets a link from page A to page B as a vote, by
page A, for page B. But, Google looks at more than the sheer volume of votes, or
links a page receives; it also analyzes the page that casts the vote. Votes cast by
pages that are themselves “important” weigh more heavily and help to make other
pages ‘important.’”

Many More Measures
• Peer Influence based measures (Friedkin and others). Based
on the assumed network autocorrelation model of peer
influence; variant of the eigenvector centrality measures
• Fragmentation centrality – Borgatti’s Key Player - nodes are
central if they can easily break up a network
• Removal Centrality – effect on the rest of the network (for any
given statistic) with the removal of a given node; system-
contribution of a particular actor

Many More Measures
• Often multiple measures in analysis
• Ennett et al. 2006: higher betweenness, indegree, reach, Bonacich
Power significantly associated with higher alcohol use for 15 year olds
But…
Multicollinearity (don’t forget theory!)

• Rather than considering important individual actors,
describing characteristics of the overall network
• To what extent are the links or is the ‘power’ of a
network concentrated in a few nodes, versus spread
throughout the network?
• Degree Distributions
• Centralization
• Density
Centrality in the Network

Whole Network Centrality
• Degree Distribution – frequency distribution of
degree values of actors
• A simple random graph will have a Poisson
degree distribution, so variation from that
suggest non-random processes

• Degree Distribution – frequency distribution of
degree values of actors

• Typical social network degree distribution

• Centralization – extent to which centrality is
concentrated in one/few actors; dispersion of
centrality in graph as a whole (Freeman
centralization)
 
)]2)(1[(
)()(1
*




gg
nCnC
C
g
i iDD
D

• Density – volume of relations in network - number
of ties relative to the number of possible ties
Density = .09

• Density – volume of relations in network - number
of ties relative to the number of possible ties
Guan & Kamo 2016: contagion of friends’ depression
varies with density of high school network (Add Health)

Describing Networks
• Beyond centrality, consider structural arrangements
in combination with attributes:
• Similarity of attributes: typically Homophily –
tendency for actors with similar attributes to be more
likely to be connected
• Assortativity - assortative/disassortative mixing
• Individual attributes: gender, same firm
• Structural attributes: same degree
• Descriptive, not distinguishing selection/influence

Connecting Measures to Mechanisms: Structural Holes
• Structural holes: absence of ties between alters
• Bridging structural holes: connecting people who
otherwise would not be connected; social capital,
access to resources
• Redundancy (ties that connect ego to alters already
connected to) introduces constraint
• Power, brokerage by controlling info or resources by
bridging structural holes (Simmel’s tertius gaudens)
(Burt 1992)

Connecting Measures to Mechanisms: Structural Holes
• 4 related network features:
• Effective Size – (size – redundancy) – average degree of
ego network without counting alters’ ties to ego
• Efficiency – (effective size / observed size)
• Constraint – room to exploit structural holes or negotiate;
extent to which network alters are connected with each
other (direct/indirect, proportion of network ‘time & energy)
• Hierarchy – for Burt/structural holes, many measures of
hierarchy generally – extent to which constraint is
concentrated in one actor

What does centrality look like
in 2-mode networks?

Two-Mode Network Data
• Affiliation/Bipartite Network: 2 sets of nodes, with
between-set ties and no within-set ties
Davis’ 1941 ‘Southern
women” dataset (that
we’ll use in lab) shows
women attending social
events
• “Duality of persons and groups”
(Brieger 1974)

Two-Mode to One-Mode Network
• One-Mode Projection: Joint connection through opposite
set of nodes becomes the connecting edges

Centrality in One-Mode Projections
• Potential bias:
• Projection inflates edges
• Tendency toward cliques, which leads to high density,
clustering coefficients, betweenness, and other measures
• Opaque interpretation
• Solutions:
• Consider order or relative, not absolute, values
• Re-normalize centrality measures (e.g., divide the centrality
measure by the number of degrees in opposite set)
For more information: (Everett 2016; Faust 1997; Jasny 2012; Latapy et al. 2008; Opsahl 2013)

Centrality in One-Mode Projections
Pettey et al. 2016 analyze Veterans’ medical visit codes to examine co-
incidence of homelessness and health co-morbidities.

Pettey et al. 2016: Co-morbid diagnoses among Veterans

Overview
• Centrality
• Individual nodes
• Centralization: Whole Networks
• Structural Holes
• Two-mode Networks
• Roles

Describing Networks
What are the structural arrangements or
characteristics of networks underlying actors’ power,
influence, or centrality?

Overview
• Centrality
• Triads & Transitivity
• Clustering
• Structural Cohesion
• Roles

Connectivity & Cohesion: Local Processes
• Dyadic – Reciprocity
• Dyad Census: MAN
• Mutual
• Asymmetric
• Null
Proportion Symmetric:
Proportion reciprocal of non-null
𝑴 + 𝑵
𝑴 + 𝑨 + 𝑵
𝑴
𝑴 + 𝑨

Network Sub-Structure: Triad Census & UMAN
003
(0)
012
(1)
102
021D
021U
021C
(2)
111D
111U
030T
030C
(3)
201
120D
120U
120C
(4)
210
(5)
300
(6)

• Characterize non-random social patterns in triad
connections with the triad census – counting
observed triads of each possible type
• Transitivity
• If i  j and j  k, then i  k
• With directed ties, observe transitive,
intransitive, vacuous triads

Network Sub-Structure: Triad Census
003
(0)
012
(1)
102
021D
021U
021C
(2)
111D
111U
030T
030C
(3)
201
120D
120U
120C
(4)
210
(5)
300
(6)
Intransitive
Transitive
Mixed

• Extending to a network level:
• Comparing triad census to expectation by chance
• Clustering coefficient:
– Transitivity ratio - # closed triads/total # triads
Verdery et al. 2017 – comparing two Respondent Driven Sampling
networks of persons who inject drugs in Philippines; see higher
clustering is associated with higher and faster spread of HIV/AIDS

• Social Balance Theory: like nodes, edges can have
attributes, too:
my friend’s friend is my friend,
my friend’s enemy is my enemy,
my enemy’s friend is my enemy,
my enemy’s enemy is my friend Heider (1958)
+
+
+ - -
-
- -
-

Dyad census:
M A N
22 67 506
Triad Census:
003 012 102 021D 021U 021C
4084 1536 519 82 43 88
111D 111U 030T 030C
41 69 9 0
201 120D 120U 120C
13 13 15 6
210 300
14 3
Transitive Triads: 141
Transitive Clustering Coefficient: 0.36

Connectivity & Cohesion: Clustering
Getting less local:
How can we describe the connective or cohesive
nature of a network overall?

• Scale-free networks:
• Locally: high-degree nodes act as hubs
– Preferential Attachment
• Globally: networks with highly skewed degree
distributions
• Scale-free implications for diffusion
– Ex: disease transmission via high-degree ‘hubs’
• Often social constraints and patterns mean we
don’t usually see the full scale-free structure

• Small world graphs:
generally large, sparse,
decentralized, highly
clustered
(Watts & Strogatz, 1998)
• ‘Small World’ phenomenon:
What’s the probability two nodes are connected?

• Small local changes can have
big effects on the global
network
(Watts & Strogatz, 1998)
• A ‘small world graph’ has relatively small average
path lengths and relatively large clusters

• Small world graphs:
• ‘shortcuts’ between
clusters dramatically
reduce average path
length
• Can dramatically affect
capacity for disease
transmission or other
network features

• How else can we measure the ‘small wordliness’ or
other cohesive characteristics of a network?
• Clustering coefficient:
• Average local density (ego-network density/n)
• Transitivity

Connectivity & Cohesion: Structural Cohesion
• Structural Cohesion – extent to which networks or
sub-groups within networks are ‘sticky’,
interconnected, or resistant to disruption
• Challenging to measure
• Connectedness maintained through one or a few
actors
• More paths linking network that don’t rely on
one actor = more cohesive

• Reachability – actors i and j are reachable if any path in
the network connects them; more paths linking (and re-linking
actors in the group) increases the ability of the group to ‘hold
together’
• Pattern of ties, not just density
D = . 25 D = . 25

Node Connectivity
0 1 2 3
Same volume of ties, but graph on right has more independent
paths connecting network, making it more cohesive

Connectivity & Cohesion: Components
• Component – maximal connected sub-graph -
connected graph where there is a path between
every node
• Cut-point – node whose removal would
disconnect the graph
• Cut-set – set of nodes necessary for keeping
graph connected
1
2
5
4 3
6
8
7

• Formally defining structural cohesion:
1. Minimum number of actors, who if removed,
would disconnect the group
2. Minimum number of independent paths linking
each pair of actors in the group
1
2
5
4 3
6
8
7

“Add Health” Example: Cutpoints

• Features of components:
• k-components – maximal subset of actors linked
by at least k node-independent paths
• Every member must have at least k ties (but having k
ties doesn’t necessarily make a component)
• 2 k-components can only overlap by k-1 members (or
would be same component)
• Can be nested

• Embeddedness – identify cohesive groups
(blocks) in a network, then remove k-cutsets identify
successively deeper embedded groups in graph
Moody & White 2003

Connectivity & Cohesion: Groups
• Different types of sub-structures in networks:
• Cliques – all members connected to all other
members
• n-clique – where n is number of steps greater than
direct tie, so can consider 2-clique, defined by 2-step
(friend of a friend) ties
• k-plex – every member to connected to at least n-k
others in the graph (relaxed from connected to all but
self, n-1, of the clique)
• Mostly intractable in large networks

Connectivity & Cohesion: Groups
• Different types of sub-structures in networks:
• n-clans – members connected at distance n or
less, only through other members
• k-cores – members joined to at least k other
members, even if not connected to all other
members

Overview
• Centrality
• Roles
• Structural Equivalence
• Regular Equivalence

Roles & Positions: Overall
• Measures that describe subsets
of actors/nodes who have
similarly structured relations
• Might expect different risks or
behaviors for actors occupying
similar positions or roles

Roles & Positions: Structural Equivalence
• Structural Equivalence:
• Actors are equivalent if they have the same ties to
the exact same people in the network
• Rare, maybe more restrictive than you want for
thinking about roles and positions in a network,
so can relax to:
• Regular Equivalence:
• Actors are equivalent if have ties to same types
(but not necessarily the exact same) of alters

Roles & Positions: Structural Equivalence & Blockmodeling
• Blockmodeling – process of identifying similar positions
(groups of similar actors in a ‘block’ of the adjacency matrix)
• Based on attributes
• Based on patterns of ties
• Can be increasingly generalized and abstracted
• Ex: Core/periphery
1 2
3

Roles & Positions: Structural Equivalence
• Structural (Regular) Equivalence: Actors are
equivalent if have ties to same types (but not
necessarily the exact same) of alters
Fujimoto & Valente 2012: Exposure to substance use
through structural equivalence is a better predictor of
drinking and smoking (being connected to the same types
of peers with the same types of behaviors matters more)
than traditional measures of cohesion.

In Summary
• Many ways of describing networks or characterizing
nodes of interest within them
• Here: individual node properties, entire network
counterparts, then structures and sub-groups:
• Centrality
• Roles
• Frame as micro/meso/macro
• Micro: individual nodes
• Meso: sub-groups, sub-graph structures, roles
• Macro: features of entire networks

Describing Networks: Summary
• Not considered here:
• Dynamics – churn or stability over time, effects of
changes, etc.
• More 2-mode network methods
• Many more challenging concepts and additions in
describing networks:
• Centrality or structural measures specific to
certain topics or processes
• Groups & Community Detection

Thank you!
Questions?
molly.copeland@duke.edu

Resources
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Resources, Cont.
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07 Whole Network Descriptive Statistics

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