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Introduction to the Theory
of Complex Systems
Stefan Thurner, Rudolf Hanel, and Peter Klimek
Medical University of Vienna, Austria
1

3
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First Edition published in 2018
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Preface
This book is for people who are interested in the science of complex adaptive systems
and wish to have more than just a casual understanding of it. As with all the sciences,
understanding of complex adaptive systems is reached solely in a quantitative, predictive,
and ultimately experimentally testable manner. Complex adaptive systems are dynamical
systems that are able to change their structure, their interactions, and, consequently,
their dynamics as they evolve in time. This is not a book about complicated systems,
even though most complex systems are complicated. Indeed, over the last 300 years,
scientists have usually dealt with complicated systems that are neither complex nor
adaptive.
The theory of complex systems is the theory of generalized time-varying interactions
between elements that are characterized by states. Interactions typically take place on
networks that connect those elements. The interactions involved may cause the states
of the elements themselves to alter over time. The essence of a complex system is that
the interaction networks may change and rearrange as a consequence of changes in
the states of the elements. Thus, complex systems are systems whose states change
as a result of interactions and whose interactions change concurrently as a result of
states. Due to this chicken–egg-type problem, complex systems show an extremely
rich spectrum of behaviour: they are adaptive and co-evolutionary; they show path-
dependence, emergence, power laws; they have rich phase diagrams; they produce
and destroy diversity; they are inherently prone to collapse; they are resilient, and so
on. The theory of complex systems tries to understand these properties based on its
building blocks and on the interactions between those building blocks that take place
on networks. It combines mathematical and physical principles with concepts borrowed
from biology and the social sciences; it uses new computational techniques and, with
the advent of comprehensive large-scale data sets, is becoming experimentally testable.
The goal of the theory of complex systems is to understand the dynamical systemic
outcomes of interconnected systems, and its ultimate goal is to eventually control and
design systemic properties of systems such as the economy, the ﬁnancial system, social
processes, cities, the climate, and ecology. The theory of complex systems builds partly
on previous attempts to understand systems that interact in non-trivial ways, such as
game theory, cybernetics, or systems theory. However, in its current state, the science
of complex systems goes well beyond these earlier developments, in so many ways,
in fact, that it can be regarded as an independent scientiﬁc branch, which—due to its
quantitative, predictive, and testable nature—is a natural science.
Even though it is fair to say that the theory of complex systems is not yet complete,
in recent years, it has become quite clear just what the theory is going to look like.
Its elements and structure are emerging. The current state of the theory of complex

vi Preface
systems is comparable perhaps to the state of quantum mechanics in the 1920s, before
the famous Copenhagen meetings and Werner Heisenberg’s book. At that time, quantum
mechanics was a collection of experimental and theoretical bits and pieces, which
had not yet been seen within a fully comprehensive framework. Nevertheless, it was
clear that, one day soon, such a framework would exist. The present situation can
be compared to an archaeological project, where a mosaic ﬂoor has been discovered
and is being excavated. While the mosaic is only partly visible and the full picture is
still missing, several facts are becoming clear: the mosaic exists; it shows identiﬁable
elements (for instance, people and animals engaged in recognizable activities); there
are large patches missing or still invisible, but experts can already tell that the mosaic
represents a scene from,say,Homer’s Odyssey.Similarly,for dynamical complex adaptive
systems, it is clear that a theory exists that, eventually, can be fully developed. There
are those who say that complex systems will never be understood or that, by their very
nature, they are incomprehensible. This book will demonstrate that such statements are
incorrect. The elements of a theory of complex systems are becoming clear: dynamical
multilayer networks, scaling, statistical mechanics of algorithmic dynamics, evolution
and co-evolution, and information theory. The essence of this book is to focus on these
components, clarify their meaning in the context of complex systems, and enable the
reader with a mathematical skill set to apply them to concrete problems in the world of
complex systems.
The book is written in mathematical language because this is the only way to
express facts in a quantitative and predictive manner and to make statements that are
unambiguous. We aim for consistency. The book should be comprehensible so that no-
one with an understanding of basic calculus, linear algebra, and statistics need refer
to other works. The book is particularly designed for graduate students in physics or
mathematics. We try to avoid ambiguous statements while, at the same time, being as
general as possible. The hope is that this work will serve as a textbook and as a starting
point for journeys into new and unexplored territory.
Many complex systems are often sensitive to details in their internal setup, to initial
and to boundary conditions. Concepts that proved to be extremely robust and effective in
non-complex systems, such as the central limit theorem, classical statistical mechanics, or
information theory, lose their predictive power when confronted with complex systems.
Extreme care is thus needed in any attempt to apply these otherwise distinguished
concepts to complex systems: doing so could end in confusion and nonsensical results.
In several concrete examples, we will demonstrate the importance of understanding what
these methods mean in the context of complex systems and whether they can or cannot
be applied. We will discuss how some of these classical concepts can be generalized to
become useful for understanding complex systems.
The book is also a statement about our belief that the exact sciences may be entering
a phase of transition from a traditional analytical description of nature, as used with
tremendous success since Galileo and Newton, towards an algorithmic description.
Whereas the analytical description of nature is, conceptually, based largely on differential
equations and analytical equations of motion, the algorithmic view takes into account
evolutionary and co-evolutionary aspects of dynamics. It provides a framework for

Preface vii
systems that can endogenously change their internal interaction networks, rules of
functioning, dynamics, and even environment, as they evolve in time. Algorithmic
dynamics, which is characteristic of complex dynamical systems, may be a key to the
quantitative and predictive understanding of many natural and man-made systems. In
contrast to physical systems, which typically evolve analytically, algorithmic dynamics
describe certainly how living, social, environmental, and economic systems unfold.
This algorithmic view is not new but has been advocated by authors like Joseph A.
Schumpeter, Stuart Kauffman, and Brian Arthur. However, it has not, to date, been
picked up by mainstream science, and it has never been presented in the context of the
theory of complex systems.
This book is based on a two-semester course, that has been held at the Medical
University of Vienna since 2011. We are grateful to our students and to Kathryn Platzer
and Anita Wanjek for helping us with the manuscript.
ST Vienna January 2018

Contents
1 Introduction to Complex Systems 1
1.1 Physics, biology, or social science? 1
1.2 Components from physics 1
1.2.1 The nature of the fundamental forces 2
1.2.2 What does predictive mean? 3
1.2.3 Statistical mechanics—predictability on stochastic grounds 5
1.2.4 The evolution of the concept of predictability in physics 5
1.2.5 Physics is analytic, complex systems are algorithmic 6
1.2.6 What are complex systems from a physics point of view? 7
1.2.7 A note on chemistry—the science of equilibria 9
1.3 Components from the life sciences 10
1.3.1 Chemistry of small systems 10
1.3.2 Biological interactions happen on
networks—almost exclusively 12
1.3.3 Evolution 13
1.3.4 Adaptive and robust—the concept of the edge of chaos 16
1.3.5 Components taken from the life sciences 19
1.4 Components from the social sciences 19
1.4.1 Social systems continuously restructuring networks 20
1.5 What are Complex Systems? 21
1.5.1 What is co-evolution? 24
1.5.2 The role of the computer 25
1.6 The structure of the book 26
1.6.1 What has complexity science contributed to the history of science? 27
2 Probability and Random Processes 29
2.1 Overview 29
2.1.1 Basic concepts and notions 31
2.1.2 Probability and information 36
2.2 Probability 39
2.2.1 Basic probability measures and the Kolmogorov axioms 39
2.2.2 Histograms and relative frequencies 41
2.2.3 Mean, variance and higher moments 41
2.2.4 More than one random variable 44
2.2.5 A note on Bayesian reasoning 47
2.2.6 Bayesian and frequentist thinking 52

x Contents
2.3 The law of large numbers—adding random numbers 53
2.3.1 The central limit theorem 55
2.3.2 Generalized limit theorems and α-stable processes 59
2.4 Fat-tailed distribution functions 65
2.4.1 Distribution functions that show power law tails 66
2.4.2 Other distribution functions 69
2.5 Stochastic processes 75
2.5.1 Simple stochastic processes 76
2.5.2 History- or path-dependent processes 84
2.5.3 Reinforcement processes 85
2.5.4 Driven dissipative systems 86
2.6 Summary 89
2.7 Problems 90
3 Scaling 93
3.1 Overview 93
3.1.1 Deﬁnition of scaling 95
3.2 Examples of scaling laws in statistical systems 96
3.2.1 A note on notation for distribution functions 98
3.3 Origins of scaling 100
3.3.1 Criticality 101
3.3.2 Self-organized criticality 105
3.3.3 Multiplicative processes 106
3.3.4 Preferential processes 108
3.3.5 Sample space reducing processes 110
3.3.6 Other mechanisms 119
3.4 Power laws and how to measure them 120
3.4.1 Maximum likelihood estimator for power law exponents λ < −1 120
3.4.2 Maximum likelihood estimator for power laws for all exponents 122
3.5 Scaling in space—symmetry of non-symmetric objects, fractals 124
3.5.1 Self-similarity and scale-invariance 125
3.5.2 Scaling in space: fractals 125
3.5.3 Scaling in time—fractal time series 129
3.6 Example—understanding allometric scaling in biology 131
3.6.1 Understanding the 3/4 power law 133
3.6.2 Consequences and extensions 136
3.7 Summary 137
3.8 Problems 139
4 Networks 141
4.1 Overview 141
4.1.1 Historical origin of network science 143
4.1.2 From random matrix theory to random networks 143
4.1.3 Small worlds and power laws 144
4.1.4 Networks in the big data era 145

Contents xi
4.2 Network basics 145
4.2.1 Networks or graphs? 146
4.2.2 Nodes and links 146
4.2.3 Adjacency matrix of undirected networks 146
4.3 Measures on networks 151
4.3.1 Degree of a node 151
4.3.2 Walking on networks 153
4.3.3 Connectedness and components 154
4.3.4 From distances on networks to centrality 155
4.3.5 Clustering coefficient 156
4.4 Random networks 159
4.4.1 Three sources of randomness 160
4.4.2 Erdős–Rényi networks 161
4.4.3 Phase transitions in Erdős–Rényi networks 163
4.4.4 Eigenvalue spectra of random networks 165
4.5 Beyond Erdős–Rényi—complex networks 167
4.5.1 Generalized Erdős–Rényi networks 168
4.5.2 Network superposition model 170
4.5.3 Small worlds 171
4.5.4 Hubs 173
4.6 Communities 178
4.6.1 Graph partitioning and minimum cuts 179
4.6.2 Hierarchical clustering 180
4.6.3 Divisive clustering in the Girvan–Newman algorithm 181
4.6.4 Modularity optimization 182
4.7 Functional networks—correlation network analysis 184
4.7.1 Construction of correlation networks 186
4.7.2 Filtering the correlation network 190
4.8 Dynamics on and of networks 194
4.8.1 Diffusion on networks 195
4.8.2 Laplacian diffusion on networks 196
4.8.3 Eigenvector centrality 199
4.8.4 Katz prestige 200
4.8.5 PageRank 200
4.8.6 Contagion dynamics and epidemic spreading 201
4.8.7 Co-evolving spreading models—adaptive networks 205
4.8.8 Simple models for social dynamics 206
4.9 Generalized networks 208
4.9.1 Hypergraphs 209
4.9.2 Power graphs 209
4.9.3 Multiplex networks 210
4.9.4 Multilayer networks 211
4.10 Example—systemic risk in financial networks 212
4.10.1 Quantification of systemic risk 213
4.10.2 Management of systemic risk 218

xii Contents
4.11 Summary 219
4.12 Problems 222
5 Evolutionary Processes 224
5.1 Overview 224
5.1.1 Science of evolution 225
5.1.2 Evolution as an algorithmic three-step process 227
5.1.3 What can be expected from a science of evolution? 230
5.2 Evidence for complex dynamics in
evolutionary processes 232
5.2.1 Criticality, punctuated equilibria, and the abundance of
fat-tailed statistics 232
5.2.2 Evidence for combinatorial co-evolution 234
5.3 From simple evolution models to a general
evolution algorithm 236
5.3.1 Traditional approaches to evolution—the
replicator equation 237
5.3.2 Limits to the traditional approach 241
5.3.3 Towards a general evolution algorithm 242
5.3.4 General evolution algorithm 244
5.4 What is fitness? 246
5.4.1 Fitness landscapes? 247
5.4.2 Simple fitness landscape models 247
5.4.3 Evolutionary dynamics on fitness landscapes 249
5.4.4 Co-evolving fitness landscapes—The Bak–Sneppen model 261
5.4.5 The adjacent possible in fitness landscape models 263
5.5 Linear evolution models 264
5.5.1 Emergence of auto-catalytic sets—the
Jain–Krishna model 265
5.5.2 Sequentially linear models and the edge of chaos 271
5.5.3 Systemic risk in evolutionary
systems—modelling collapse 277
5.6 Non-linear evolution models—combinatorial evolution 281
5.6.1 Schumpeter got it right 282
5.6.2 Generic creative phase transition 282
5.6.3 Arthur–Polak model of technological evolution 286
5.6.4 The open-ended co-evolving combinatorial critical
model—CCC model 288
5.6.5 CCC model in relation to other evolutionary models 298
5.7 Examples—evolutionary models for economic predictions 299
5.7.1 Estimation of fitness of countries from economic data 300
5.7.2 Predicting product diversity from data 304

Contents xiii
5.8 Summary 308
5.9 Problems 311
6 Statistical Mechanics and Information Theory for Complex Systems 313
6.1 Overview 313
6.1.1 The three faces of entropy 314
6.2 Classical notions of entropy for simple systems 318
6.2.1 Entropy and physics 321
6.2.2 Entropy and information 328
6.2.3 Entropy and statistical inference 343
6.2.4 Limits of the classical entropy concept 348
6.3 Entropy for complex systems 349
6.3.1 Complex systems violate ergodicity 350
6.3.2 Shannon–Khinchin axioms for complex systems 352
6.3.3 Entropy for complex systems 352
6.3.4 Special cases 356
6.3.5 Classiﬁcation of complex systems based on their entropy 358
6.3.6 Distribution functions from the complex systems entropy 361
6.3.7 Consequences for entropy when giving up ergodicity 363
6.3.8 Systems that violate more than the composition axiom 365
6.4 Entropy and phasespace for physical complex systems 365
6.4.1 Requirement of extensivity 365
6.4.2 Phasespace volume and entropy 366
6.4.3 Some examples 369
6.4.4 What does non-exponential phasespace growth imply? 373
6.5 Maximum entropy principle for complex systems 374
6.5.1 Path-dependent processes and multivariate distributions 374
6.5.2 When does a maximum entropy principle exist for
path-dependent processes? 375
6.5.3 Example—maximum entropy principle for path-dependent
random walks 380
6.6 The three faces of entropy revisited 382
6.6.1 The three entropies of the Pólya process 383
6.6.2 The three entropies of sample space reducing processes 387
6.7 Summary 393
6.8 Problems 395
7 The Future of the Science of Complex Systems? 397
8 Special Functions and Approximations 399
8.1 Special functions 399
8.1.1 Heaviside step function 399
8.1.2 Dirac delta function 399

xiv Contents
8.1.3 Kronecker delta 400
8.1.4 The Lambert-W function 400
8.1.5 Gamma function 401
8.1.6 Incomplete Gamma function 402
8.1.7 Deformed factorial 402
8.1.8 Deformed multinomial 402
8.1.9 Generalized logarithm 402
8.1.10 Pearson correlation coefﬁcient 403
8.1.11 Chi-squared distribution 403
8.2 Approximations 404
8.2.1 Stirling’s formula 404
8.2.2 Expressing the exponential function as a power 404
8.3 Problems 405
References 407
Index 425

1
Introduction to Complex Systems
1.1 Physics, biology, or social science?
The science of complex systems is not an offspring of physics, biology, or the social
sciences, but a unique mix of all three. Before we discuss what the science of complex
systems is or is not, we focus on the sciences from which it has emerged. By recalling
what physics, biology, and the social sciences are, we will develop an intuitive feel for
complex systems and how this science differs from other disciplines. This chapter thus
aims to show that the science of complex systems combines physics, biology, and the
social sciences in a unique blend that is a new discipline in its own right. The chapter
will also clarify the structure of the book.
1.2 Components from physics
Physics makes quantitative statements about natural phenomena. Quantitative state-
ments can be formulated less ambiguously than qualitative descriptions, which are
based on words. Statements can be expressed in the form of predictions in the sense
that the trajectory of a particle or the outcome of a process can be anticipated. If
an experiment can be designed to test this prediction unambiguously, we say that the
statement is experimentally testable. Quantitative statements are validated or falsified
using quantitative measurements and experiments.
Physics is the experimental, quantitative, and predictive science of matter and its
interactions.
Pictorially, physics progresses by putting specific questions to nature in the form of
experiments; surprisingly, if the questions are well posed, they result in concrete answers
that are robust and repeatable for an arbitrary number of times by anyone who can
do the same experiment. This method of generating knowledge about nature, by using
experiments to ask questions of it, is unique in the history of humankind and is called the
scientific method. The scientific method has been at the core of all technological progress
since the time of the Enlightenment.
Introduction to the Theory of Complex Systems. Stefan Thurner, Rudolf Hanel, and Peter Klimek,
Oxford University Press (2018). © Stefan Thurner, Rudolf Hanel, and Peter Klimek.
DOI: 10.1093/oso/9780198821939.001.0001

2 Introduction to Complex Systems
Physics deals with matter at various scales and levels of granularity, ranging from
macroscopic matter like galaxies, stars, planets, stones, and projectiles, to the scale of
molecules, atoms, hadrons, quarks, and gauge bosons. There are four fundamental forces
at the core of all interactions between all forms of matter: gravity, electromagnetism and
two types of nuclear force: the weak and the strong. According to quantum field theory,
all interactions in the physical world are mediated by the exchange of gauge bosons. The
graviton, the boson for gravity, has not yet been confirmed experimentally.
1.2.1 The nature of the fundamental forces
The four fundamental forces are very different in nature and strength. They are
characterized by a number of properties that are crucial for understanding how and why
it was possible to develop physics without computers. These properties are set out here.
Usually, the four fundamental forces are homogeneous and isotropic in space (and
time). Forces that are homogeneous act in the same way everywhere in space; forces
that are isotropic are the same, regardless of the direction in which they act. These two
properties drastically simplify the mathematical treatment of interactions in physics. In
particular, forces can be written as derivatives of potentials, two-body problems can
effectively be treated as one-body problems, and the so-called mean field approach
can be used for many-body systems. The mean field approach is the assumption that
a particle reacts to the single field generated by the many particles around it. Often,
such systems can be fully understood and solved even without computers. There are
important exceptions, however; one being that the strong force acts as if interactions were
limited to a ‘string’, where flux-tubes are formed between interacting quarks, similar to
type II superconductivity.
The physical forces differ greatly in strength. Compared to the strong force, the
electromagnetic force is about a thousand times weaker, the weak force is about 1016
times weaker, and the gravitational force is only 10−41 of the strength of the strong force
[405]. When any physical phenomenon is being dealt with, usually only a single force
has to be considered. All the others are small enough to be safely neglected. Effectively,
the superposition of four forces does not matter; for any phenomenon, only one force
Characteristic
Matter Interaction types length scale
macroscopic matter gravity, electromagnetism all ranges
molecules electromagnetism all ranges
atoms electromagnetism, weak force ∼ 10−18 m
hadrons and leptons electromagnetism, weak and strong force 10−18 − 10−15 m
quarks and gauge bosons electromagnetism, weak and strong force 10−18 − 10−15 m

Components from physics 3
is relevant. We will see that this is drastically different in complex systems, where a
multitude of different interaction types of similar strength often have to be taken into
account simultaneously.
Typically, physics does not specify which particles interact with each other, as
they interact in identical ways. The interaction strength depends only on the relevant
interaction type, the form of the potential, and the relative distance between particles.
In complex systems, interactions are often specific. Not all elements, only certain pairs or
groups of elements, interact with each other. Networks are used to keep track of which
elements interact with others in a complex system.
1.2.2 What does predictive mean?
Physics is an experimental and a predictive science. Let us assume that you perform an
experiment repeatedly; for example, you drop a stone and record its trajectory over time.
The predictive or theoretical task is to predict this trajectory based on an understanding
of the phenomenon. Since Newton’s time, understanding a phenomenon in physics
has often meant being able to describe it with differential equations. A phenomenon
is understood dynamically if its essence can be captured in a differential equation.
Typically, the following three-step process is then followed:
1. Find the differential equations to encode your understanding of a dynamical
system.In the example of our stone-dropping experiment,we would perhaps apply
Newton’s equation,
m
d2x
dt2
= F(x),
where t is time, x(t) is the trajectory, m is mass of the stone, and F is force on the
stone. In our case, we would hope to identify the force with gravity, meaning that
F = gm.
2. Once the equation is specified, try to solve it. The equation can be solved
using elementary calculus, and we get, x(t) = x0 + v0t + 1
2 gt2. To make a testable
prediction we have to fix the boundary or initial conditions; in our case we have
to specify what the initial position x0 and initial velocity v0 are in our experiment.
Once this is done, we have a prediction for the trajectory of the stone, x(t).
3. Compare the result with your experiments. Does the stone really follow this
predicted path x(t)? If it does, you might claim that you have understood
something on a quantitative, predictive, and experimental basis. If the stone
(repeatedly) follows another trajectory, you have to try harder to find a better
prediction.
Fixing initial or boundary conditions means simply taking the system out of its
context, separating it from the rest of the universe. There are no factors, other than
the boundary conditions, that influence the motion of the system from the outside. That

such a separation of systems from their context is indeed possible is one reason why
physics has been so successful, even before computing devices became available. For
many complex systems, it is impossible to separate the dynamics from the context in
a clear way. This means that many outside influences that are not under experimental
control will simultaneously determine their dynamics.
In principle, the same thinking used to describe physical phenomena holds for
arbitrarily complicated systems. Assume that a vector X(t) represents the state of a
system at a given time (e.g. all positions and momenta of its elements), we then get a
set of equations of motion in the form,
d2X(t)
dt2
= G(X(t)),
where G is a high-dimensional function. Predictive means that, in principle, these
equations can be solved. Pierre-Simon Laplace was following this principle when he
introduced a hypothetical daemon familiar with the Newtonian equations of motion and
all the initial conditions of all the elements of a large system (the universe) and thus able to
solve all equations. This daemon could then predict everything. The problem, however,
is that such a daemon is hard to find. In fact, these equations can be difficult, even
impossible,to solve.Already for three bodies that exert a gravitational force on each other,
the famous three-body problem (e.g. Sun, Earth, Moon), there is no general analytical
solution provided by algebraic and transcendental functions. This was first demonstrated
by Henri Poincaré and paved the way for what is today called chaos theory. In fact,
the strict Newton–Laplace program of a predictable world in terms of unambiguously
computable trajectories is completely useless for most systems composed of many
particles. Are these large systems not then predictable? What about systems with an
extremely large number of elements, such as gases, which contain of the order of O(1023)
molecules?
Imagine that we perform the following experiment over and over again: we heat and
cool water. We gain the insight that if we cool water to 0oC and below, it will freeze,
that if we heat it to 100oC it will start to boil and, under standard conditions, ultimately
evaporate. These phase transitions will happen with certainty. In that sense, they are
predictable. We cannot predict from the equations of motion which molecule will be the
first to leave the liquid. Given appropriate instrumentation, we can perhaps measure the
velocity of a few single gas molecules at a point in time, but certainly not all 1023. What
can be measured is the probability distribution that a gas molecule is observed with a
specific velocity v,
p(v) ∼ v2
exp

−
mv2
2kT

,
where T is temperature, and k is Boltzmann’s constant. Given this probability distribu-
tion, it is possible to derive a number of properties of gases that perfectly describe their
macroscopic behaviour and make them predictable on a macroscopic (or systemic) level.

For non-interacting particles,these predictions can be extremely precise.The predictions
immediately start to degenerate as soon as there are strong interactions between the
particles or if the number of particles is not large enough. Note that the term prediction
now has a much weaker meaning than in the Newton–Laplace program. The meaning
has shifted from being a description based on the exact knowledge of each component
of a system to one based on a probabilistic knowledge of the system. Even though one
can still make extremely precise predictions about multiparticle systems in a probabilistic
framework, the concept of determinism is now diluted. The framework for predictions
on a macroscopic level about systems composed of many particles on a probabilistic
basis is called statistical mechanics.
1.2.3 Statistical mechanics—predictability
on stochastic grounds
The aim of statistical mechanics is to understand the macroscopic properties of a system
on the basis of a statistical description of its microscopic components. The idea behind
it is to link the microscopic world of components with the macroscopic properties of the
aggregate system. An essential concept that makes this link possible is Boltzmann–Gibbs
entropy.
A system is often prepared in a macrostate, which means that aggregate properties
like the temperature or pressure of a gas are known. There are typically many pos-
sible microstates that are associated with that macrostate. A microstate is a possible
microscopic configuration of a system. For example, a particular microstate is one for
which all positions and velocities of gas molecules in a container are known. There are
usually many microstates that can lead to one and the same macrostate; for example, the
temperature and pressure in the container. In statistical mechanics, the main task is to
compute the probabilities for the many microstates that lead to that single macrostate. In
physics, the macroscopic description is often relatively simple. Macroscopic properties
are often strongly determined by the phase in which the system is. Physical systems often
have very few phases—typically solid, gaseous, or liquid.
Within the Newton–Laplace framework, traditional physics works with extreme
precision for very few particles or for extremely many non-interacting particles, where
the statistical mechanics of Boltzmann–Gibbs applies. In other words, the class of
systems that can be understood with traditional physics is not that big. Most systems are
composed of many strongly interacting particles. Often, the interactions are of multiple
types, are non-linear, and vary over time. Very often, such systems are complex systems.
1.2.4 The evolution of the concept of predictability in physics
The concept of prediction and predictability has changed in significant ways over the
past three centuries. Prediction in the eighteenth century was quite different from the
concept of prediction in the twenty-first. The concept of determinism has undergone at
least three transitions [300].

In the classical mechanics of the eighteenth and nineteenth centuries, prediction meant
the exact prediction of trajectories. Equations of motion would make exact statements
about the future evolution of simple dynamical systems. The extension to more than
two bodies has been causing problems since the very beginning of Newtonian physics;
see, for example, the famous conflict between Isaac Newton and John Flamsteed on the
predictability of the orbit of the Moon. By about 1900, when interest in understanding
many-body systems arose, the problem became apparent. The theory of Ludwig Boltz-
mann, referred to nowadays as statistical mechanics, was effectively based on the then
speculative existence of atoms and molecules, and it drastically changed the classical
concept of predictability.
In statistical mechanics, based on the assumption that atoms and molecules follow
Newtonian trajectories, the law of large numbers allows stochastic predictions to be
made about the macroscopic behaviour of gases. Statistical mechanics is a theory of
the macroscopic or collective behaviour of non-interacting particles. The concepts of
predictability and determinism were subject to further change in the 1920s with the
emergence of quantum mechanics and non-linear dynamics.
In quantum mechanics, the concept of determinism disappears altogether due to
the fundamental simultaneous unpredictability of the position and momentum of the
(sub-)atomic components of a system. However, quantum mechanics still allows us to
make extremely high-quality predictions on a collective basis. Collective phenomena
remain predictable to a large extent on a macro- or systemic level.
In non-linear systems, it became clear that even in systems for which the equations of
motion can be solved in principle, the sensitivity to initial conditions can be so enormous
that the concept of predictability must,for all practical purposes,be abandoned.A further
crisis in terms of predictability arose in the 1990s, when interest in more general forms
of interactions began to appear.
In complex systems, the situation is even more difficult than in quantum mechanics,
where there is uncertainty about the components, but not about its interactions. For
many complex systems, not only can components be unpredictable, but the interactions
between components can also become specific, time-dependent, non-linear, and unpre-
dictable. However, there is still hope that probabilistic predictions about the dynamics
and the collective properties of complex systems are possible. Progress in the science of
complex systems will, however, be impossible without a detailed understanding of the
dynamics of how elements specifically interact with each other. This is, of course, only
possible with massive computational effort and comprehensive data.
1.2.5 Physics is analytic, complex systems are algorithmic
Physics largely follows an analytical paradigm. Knowledge of phenomena is expressed
in analytical equations that allow us to make predictions. This is possible because
interactions are homogeneous, isotropic, and of a single type. Interactions in physics
typically do not change over time. They are usually given and fixed. The task is to work
out specific solutions regarding the evolution of the system for a given set of initial and
boundary conditions.

This is radically different for complex systems, where interactions themselves can
change over time as a consequence of the dynamics of the system. In that sense, complex
systems change their internal interaction structure as they evolve. Systems that change
their internal structure dynamically can be viewed as machines that change their internal
structure as they operate. However, a description of the operation of a machine using
analytical equations would not be efficient. Indeed, to describe a steam engine by seeking
the corresponding equations of motion for all its parts would be highly inefficient.
Machines are best described as algorithms—a list of rules regarding how the dynamics of
the system updates its states and future interactions, which then lead to new constraints
on the dynamics at the next time step. First, pressure builds up here, then a valve opens
there, vapour pushes this piston, then this valve closes and opens another one, driving
the piston back, and so on.
Algorithmic descriptions describe not only the evolution of the states of the com-
ponents of a system, but also the evolution of its internal states (interactions) that will
determine the next update of the states at the next time step. Many complex systems work
in this way: states of components and the interactions between them are simultaneously
updated, which can lead to the tremendous mathematical difficulties that make complex
systems so hard to understand. These difficulties in their various forms will be addressed
time and again in this book. Whenever it is possible to ignore the changes in the
interactions in a dynamical system, analytic descriptions become meaningful.
Physics is generally analytic, complex systems are algorithmic. Quantitative pre-
dictions that can be tested experimentally can be made within the analytic or the
algorithmic paradigm.
1.2.6 What are complex systems from a physics point of view?
From a physics point of view, one could try to characterize complex systems by the
following extensions to physics.
• Complex systems are composed of many elements, components, or particles.
These elements are typically described by their state, such as velocity, position,
age, spin, colour, wealth, mass, shape, and so on. Elements may have stochastic
components.
• Elements are not limited to physical forms of matter; anything that can interact
and be described by states can be seen as generalized matter.
• Interactions between elements may be specific. Who interacts with whom, when,
in what form, and how strong is described by interaction networks.
• Interactions are not limited to the four fundamental forces, but can be of a more
complicated type. Generalized interactions are not limited to the exchange of
gauge bosons, but can be mediated through exchange of messages, objects, gifts,
information, even bullets, and so on.
continued

• Complex systems may involve superpositions of interactions of similar strengths.
• Complex systems are often chaotic in the sense that they depend strongly on the
initial conditions and details of the system. Update equations that algorithmically
describe the dynamics are often non-linear.
• Complex systems are often driven systems. Some obey conservation laws, some
do not.
• Complex systems can exhibit a rich phase structure and have a huge variety of
macrostates that often cannot be inferred from the properties of the elements. This
is sometimes referred to as emergence. Simple forms of emergence are, of course,
already present in physics. The spectrum of the hydrogen atom or the liquid phase
of water are emergent properties of the involved particles and their interactions.
With these extensions, we can derive a physics-based deﬁnition for what the theory
of complex systems is.
The theory of complex systems is the quantitative, predictive and experimentally
testable science of generalized matter interacting through generalized interactions.
Generalized interactions are described by the interaction type and who interacts with
whom at what time and at what strength. If there are more than two interacting elements
involved, interactions can be conveniently described by time-dependent networks,
Mα
ij (t),
where i and j label the elements in the system, and α denotes the interaction type.
Mα
ij (t) are matrix elements of a structure with three indices. The value Mα
ij (t) indicates
the strength of the interaction of type α between element i and j at time t. Mα
ij (t)=0
means no interaction of that type. Interactions in complex systems remain based on the
concept of exchange; however, they are not limited to the exchange of gauge bosons. In
complex systems, interactions can happen through communication, where messages are
exchanged, through trade where goods and services are exchanged, through friendships,
where bottles of wine are exchanged, and through hostility, where insults and bullets are
exchanged.
Because of more speciﬁc and time-varying interactions and the increased variety of
types of interaction, the variety of macroscopic states and systemic properties increases
drastically in complex systems. This diversity increase of macrostates and phenomena
emerges from the properties both of the system’s components and its interactions.
The phenomenon of collective properties arising that are, a priori, unexpected from
the elements alone is sometimes called emergence. This is mainly a consequence of the
presence of generalized interactions. Systems with time-varying generalized interactions
can exhibit an extremely rich phase structure and may be adaptive. Phases may co-exist
in particular complex systems.The plurality of macrostates in a system leads to new types

of questions that can be addressed, such as: what is the number of macrostates? What are
their co-occurrence rates? What are the typical sequences of occurrence? What are the
life-times of macrostates? What are the probabilities of transition between macrostates?
As yet, there are no general answers to these questions, and they remain a challenge for
the theory of complex systems. For many complex systems, the framework of physics is
incomplete. Some of the missing concepts are those of non-equilibrium, evolution, and
co-evolution. These concepts will be illustrated in the sections that follow.
1.2.7 A note on chemistry—the science of equilibria
In chemistry, interactions between atoms and molecules are specific in the sense that
not every molecule binds to (interacts with) any other molecule. So why is chemistry
usually not considered to be a candidate for a theory of complex systems? To a large
extent, chemistry is based on the law of mass action. Many particles interact in ways that
lead to equilibrium states. For example, consider two substances A and B that undergo
a reaction to form substances S and T,
αA + βB ⇋ σS + τT,
where α,β,σ,τ are the stoichiometric constants, and k+ and k− are the forward and
backward reaction rates, respectively. The forward reaction happens at a rate that is
proportional to k+{A}α{B}β, the backward reaction is proportional to k−{S}σ {T}τ . The
brackets indicate the active (reacting) masses of the substances. Equilibrium is attained
if the ratio of the reaction rates equals a constant K,
K =
k+
k−
=
{S}σ {T}τ
{A}α{B}β
.
Note that the solution to this equation gives the stationary concentrations of the
various substances. Technically, these equations are fixed point equations. In contrast to
chemical reactions and statistical mechanics, many complex systems are characterized
by being out-of-equilibrium. Complex systems are often so-called driven systems, where
the system is (exogenously) driven away from its equilibrium states. If there is no
equilibrium, there is no way of using fixed-point-type equations to solve the problems.
The mathematical difficulties in dealing with out-of-equilibrium or non-equilibrium
systems are tremendous and generally beyond analytical reach. One way that offers
a handle on understanding driven out-of-equilibrium systems is the concept of self-
organized criticality, which allows essential elements of the statistics of complex systems
to be understood; in particular, the omnipresence of power laws.
Many complex systems are driven systems and are out-of-equilibrium.

By comparing the nature of complex systems and basic equilibrium chemistry, we
learn that the mere presence of speciﬁc interactions does not automatically lead us
to complex systems. However, cyclical catalytic chemical reactions [22, 113, 205], are
classic prototypes of complex systems.
1.3 Components from the life sciences
We now present several key features of complex systems that have been adopted from
biology. In particular, we discuss the concepts of evolution, adaptation, self-organization,
and, again, networks.
The life sciences describe the experimental science of living matter. What is living
matter? A reasonable minimal answer has been attempted by the following three
statements [223]:
• Living matter must be self-replicating.
• It must run through at least one Carnot cycle.
• It must be localized.
Life without self-replication is not sustainable. It is, of course, conceivable that non-
self-replicating organisms can be created that live for a time and then vanish and have to
be recreated. However, this is not how we experience life on the planet, which is basically
a single, continuous, living germ line that originated about 3.5 billion years ago, and has
existed ever since. A Carnot cycle is a thermodynamic cyclical process that converts
thermal energy into work, or vice versa. Starting from an initial state, after the cycle is
completed, the system returns to the same initial state. The notion that living matter must
perform at least one Carnot cycle is motivated by the fact that all living organisms use
energy gradients (usually thermal) to perform work of some kind. For example, this work
could be used for moving or copying DNA molecules. This view also pays tribute to the
fact that all living objects are out-of-equilibrium and constantly driven by energy gradi-
ents. If, after performing work, a system were not able to reach its previous states, it would
be hard to call it a living system. Both self-replication and Carnot cycles require some
sort of localization. On this planet, this localization typically happens at the level of cells.
Living matter uses energy and performs work on short timescales without signiﬁ-
cantly transforming itself. It is constantly driven by energy gradients and is out-of-
equilibrium. Self-replication and Carnot cycles require localization.
1.3.1 Chemistry of small systems
Living matter, as we know it on this planet, is a self-sustained sequence of genetic activity
over time. By genetic activity we mean that genes (locations on the DNA) can be turned

Components from the life sciences 11
Figure 1.1 Schematic view of genetic activity and what a link Mki means in a genetic regulatory
network. (a) Gene i activates gene k if something like the following process takes place: the activity of
gene i means that a speciﬁc sub-sequence of the deoxyribonucleic acid (DNA) (gene) is copied into a
complementary structure, an mRNA molecule. This mRNA molecule from gene i, might get ‘translated’
(copied again) into a protein of type i.This protein can bind with other proteins to form a cluster of proteins,
a ‘complex’. Such complexes can bind to other regions of the DNA, say, the region that is associated with
gene k,and thereby cause the activation of gene k.(b) Gene i causes gene j to become active,which activates
genes m and n. (c) The process, where the activity of gene i triggers the activity of other genes, can be
represented as a directed genetic regulatory network. Complexes can also deactivate genes. If gene j is
active, a complex might deactivate it.
‘on’ and ‘off’. If a gene is on, it triggers the production of molecular material, such as
ribonucleic acid (RNA) that can later be translated into proteins. A gene is typically
turned on by a cluster of proteins that bind to each other to form a so-called ‘complex’.
If such a cluster binds to a speciﬁc location on the DNA, this could cause a copying
process to be activated at this position; the gene is then active or ‘on’; see Figure 1.1.
Genetic activity is based on chemical reactions that take place locally, usually within
cells or their nuclei. However, these chemical reactions are special in the sense that only
a few molecules are involved [341]. In traditional chemistry, reactions usually involve
billions of atoms or molecules. What happens within a cell is chemistry with a few
molecules. This immediately leads to a number of problems:
• It can no longer be assumed that molecules meet by chance to react.
• With only a few molecules present that might never meet to react, the concept of
equilibrium becomes useless.
• Without equilibrium, there is no law of mass action.

If there is no law of mass action, how can chemistry be done? Classical equilibrium
chemistry is inadequate for dealing with molecular mechanisms in living matter. In
cells, molecules are often actively transported from the site of production (typically, the
nucleus, for organisms that have one) to where they are needed in the cell. This means
that diffusion of molecules no longer follows the classical diffusion equation. Instead,
molecular transport is often describable by an anomalous diffusion equation of the form,
d
dt
p(x,t) = D
d2+ν
dx2+ν
p(x,t)µ
,
where p(x,t) is the probability of finding a molecule at position x at time t, D is the
diffusion constant, and µ and ν are exponents that make the diffusion equation non-
linear.
Chemical binding often depends on the three-dimensional structure of the molecules
involved. This structure can depend on the ‘state’ of the molecules. For example, a
molecule can be in a normal or a phosphorylated state.Phosphorylation happens through
the addition of a phosphoryl group (PO2−
3 ) to a molecule, which may change its entire
structure. This means that for a particular state of a molecule it binds to others, but
does not bind if it is in the other state. A further complication in the chemistry of a
few particles arises with the reaction rates. By definition, the term reaction rate only
makes sense for sufficiently large systems. The speed of reactions depends crucially on
the statistical mechanics of the underlying small system and fluctuation theorems may
now become important [122].
1.3.2 Biological interactions happen on
networks—almost exclusively
Genetic regulation governs the temporal sequence of the abundance of proteins, nucleic
material, and metabolites within any living organism. To a large extent, genetic regulation
can be viewed as a discrete interaction: a gene is active or inactive; a protein binds to
another or it does not; a molecule is phosphorylated or not. Discrete interactions are
well-described by networks. In the context of the life sciences, three well-known networks
are the metabolic network, the protein–protein binding network, and the Boolean gene-
regulatory network. The metabolic network1 is the set of linked chemical reactions
occurring within a cell that determine the cell’s physiological and biochemical properties.
The metabolic network is often represented in networks of chemical reactions, where
nodes represent substances and directed links (arrows) correspond to reactions or
catalytic influences. The protein–protein networks represent empirical findings about
protein–protein interactions (binding) in network representations [102]. Nodes are
proteins, and links specify the interaction type between them. Different interaction types
include stable, transient, and homo- or hetero-oligomer interactions.
1 For an example of what metabolic networks look like, see http://biochemical-pathways.com/#/map/1

1.3.3 What is evolution?
‘Nothing in biology makes sense except in the light of evolution’. Theodosius
Dobzhansky
Evolution is a natural phenomenon. It is a process that increases and destroys diversity,
and it looks like both a ‘creative’ and a ‘destructive’ process. Evolution appears in bio-
logical, technological, economical, financial, historical, and other contexts. In that sense,
evolutionary dynamics is universal. Evolutionary systems follow characteristic dynamical
and statistical patterns, regardless of the context. These patterns are surprisingly robust
and, as a natural phenomenon, they deserve a quantitative and predictive scientific
explanation.
What is evolution? Genetic material and the process of replication involve several
stochastic components that may lead to variations in the offspring. Replication and
variation are two of the three main ingredients of evolutionary processes. What evolution
means in a biological context is captured by the classic Darwinian narrative. Consider
a population of some kind that is able to produce offspring. This offspring has some
random variations (e.g. mutations). Individuals with the optimal variations with respect
to a given environment have a selection advantage (i.e. higher fitness). Fitness manifests
itself by higher reproductive success. Individuals with optimal variations will have more
offspring and will thus pass their particular variations on to a new generation. In this
way ‘optimal’ variations are selected over time. This is certainly a convincing description
of what is going on; however, in this form it may not be useful for predictive science.
How can we predict the fitness of individuals in future generations, given that life in
future environments will look very different from what it is today? Except over very
short time periods, this is a truly challenging task that is far from understood. There is a
good prospect, however, of the statistics of evolutionary systems being understood. The
Darwinian scenario fails to explain essential features about evolutionary systems, such as
the existence of boom and crash phases, where the diversity of systems radically changes
within short periods of time. An example is the massive diversification (explosion) of
species and genera about 500 million years ago in the Cambrian era. It will almost cer-
tainly never be possible to predict what species will live on Earth even 500 years from now,
but it may be perfectly possible to understand the statistics of evolutionary events and the
factors that determine the statistics. In particular, statistical statements about expected
diversity, diversification rates, robustness, resilience, and adaptability are coming within
reach. In Chapter 5 we will discuss approaches to formulating evolutionary dynamics in
ways that make them accessible both combinatorially and statistically.
The concept of evolution is not limited to biology. In the economy, the equivalent of
biological evolution is innovation, where new goods and services are constantly being
produced by combination of existing goods and services. Some new goods will be
selected in markets, while the majority of novelties will not be viable and will vanish.
The industrial revolution can be seen as one result of evolutionary dynamics, leading,
as it did, to an ongoing explosion of diversification of goods, services, and innovations.

Another example of evolutionary dynamics outside biology is the sequence of invention
and discovery of chemical compounds. The history of humankind itself is an example of
evolutionary dynamics. Evolutionary dynamics can take place simultaneously at various
scales. In biological settings, it works at the level of molecules, cells, organisms, and
populations; in economic settings, it can work at product, firm, corporation, and country
level. A famous application of evolutionary dynamics in computer science are so-
called genetic algorithms [194]. These algorithms mimic natural selection by iteratively
producing copies of computer code with slight variations. Those copies that perform
best for a given problem (usually an optimization task) are iteratively selected and are
passed onto the next ‘generation’ of codes.
1.3.3.1 Evolution is not physics
To illustrate that evolution is not a process that can be described with traditional physics,
we define an evolutionary process as a three-step process:
1. A new thing comes into existence within a given environment.
2. The new thing has the chance to interact with its environment. The result of this
interaction is that it gets ‘selected’ (survives) or is destroyed.
3. If the new thing gets selected in the environment, it becomes part of this environ-
ment (boundary) and thus transforms the old environment into a new one. New
and arriving things in the future will experience the new environment.In that sense,
evolution is an algorithmic process that co-evolves its boundaries.
If we try to interpret this three-step process in terms of physics, we immediately
see that even if we were able to write down the dynamics of the system in the form
of equations of motion, we would not be able to fix the system’s boundary conditions.
Obviously, the environment plays the role of the boundary conditions within which the
interactions happen. The boundary conditions evolve as a consequence of the dynamics
of the system and change at every instant. The dynamics of the boundary conditions
is dynamically coupled with the equations of motion. Consequently, as the boundary
conditions cannot be fixed, this set of equations cannot, in general, be solved and the
Newtonian method breaks down. A system of dynamical equations that are coupled
dynamically to their boundary conditions is a mathematical monster. That is why an
algorithmic process like evolution is hard to solve using analytical approaches.2
The second problem associated with evolutionary dynamics, from a physics point of
view,is that the phasespace is not well-defined.As new elements may arrive at any point in
time, it is impossible to prestate what the phasespace of such systems will be. Obviously,
this poses problems in terms of producing statistics with these systems. The situation
could be compared to trying to produce statistics by rolling a dice, whose number of
faces changes from one throw to the next.
2 Such systems can be treated analytically whenever the characteristic timescales of the processes involved
are different. In our example, this would be the case if the dynamics of the interactions of the ‘new thing’ with
the environment happens on a fast timescale, while changes in the environment happen slowly.

Evolutionary dynamics is radically different from physics for two main reasons:
• In evolutionary systems, boundary conditions cannot be fixed.
• In evolutionary systems, the phasespace is not well defined—it changes over time.
New elements may emerge that change the environment and therefore also the
dynamics for all the existing elements of the system.
Evolutionary aspects are essential for many complex systems and cannot be ignored.
A great challenge in the theory of complex systems is to develop a consistent framework
that is nevertheless able to deal with evolutionary processes in quantitative and predictive
terms. We will see how a number of recently developed mathematical methods can
be used to address and deal with these two fundamental problems. In particular, in
Chapter 5, we will discuss combinatorial evolution models. These models are a good
example of how algorithmic approaches lead to quantitative and testable predictions.
1.3.3.2 The concept of the adjacent possible
A helpful steppingstone in addressing the problem of dynamically changing phasespaces
is the concept of the adjacent possible, proposed by Stuart Kauffman [223]. The adjacent
possible is the set of all possible states of the world that could potentially exist in the
next time step, given the present state of the world. It drastically reduces the size of
phasespace from all possible states to a set of possibilities that are conditional on the
present. Obviously, not everything can be produced within the next time step. There are
many states that are impossible to imagine, as the components required to make them
do not yet exist. In other words, the adjacent possible is the subset of all possible worlds
that are reachable within the next time step and depends strongly on the present state of
the world. In this view, evolution is a process that continuously ‘fills’ its adjacent possible.
The concrete realization of the adjacent possible at one time step determines the adjacent
possible at the next time step.
Thus, in the context of biological evolution or technological innovation, the adjacent
possible is a huge set, in which the present state of the world determines the potential
realization of a vast number of possibilities in the next time step. Typically, the future
states are not known. In contrast, in physics, a given state often determines the next state
with high precision.This means that the adjacent possible is a very small set.For example,
the adjacent possible of a falling stone is given by the next position (point) on its parabolic
trajectory. In comparison, the adjacent possible of an ecosystem consists of all organisms
that can be born within the next time step, with all possible mutations and variations that
can possibly happen—a large set of possibilities indeed. The concept of the adjacent
possible introduces path-dependence in the stochastic dynamics of phasespace. We will
discuss the statistics of path-dependent evolutionary processes in Chapters 5 and 6.
1.3.3.3 Summary evolutionary processes
Evolutionary processes are relevant to the treatment of complex systems for the following
reasons.

• For evolutionary systems, boundary conditions cannot usually be fixed. This
means that it is impossible to take the system apart and separate it from its
context without massively altering and perhaps even destroying it. The concept
of reductionism is inadequate for describing evolutionary processes.
• Evolutionary complex systems change their boundary conditions as they unfold
in time. They co-evolve with their boundary conditions. Frequently, situations are
difficult or impossible to solve analytically.
• For complex systems,the adjacent possible is a large set of possibilities.For physics,
it is typically a very small set.
• The adjacent possible itself evolves.
• In many physical systems, the realization of the adjacent possible does not
influence the next adjacent possible; in evolutionary systems, it does.
1.3.4 Adaptive and robust—the concept of the edge of chaos
Many complex systems are robust and adaptive at the same time. The ability to adapt
to changing environments and to be robust against changing environments seem to be
mutually exclusive. However, most living systems are clearly adaptive and robust at the
same time. As an explanation for how these seemingly contradictory features could co-
exist, the following view of the edge of chaos was proposed [246]. Every dynamical system
has a maximal Lyapunov exponent, which measures how fast two initially infinitesimally
close trajectories diverge over time. The exponential rate of divergence is the Lyapunov
exponent, λ,
|δX(t)| ∼ eλt
|δX(0)|,
where |δX(t)| is the distance between the trajectories at time t and |δX(0)| is the initial
separation. If λ is positive, the system is called chaotic or strongly mixing. If λ is negative,
the system approaches an attractor, meaning that two initially infinitesimally separated
trajectories converge. This attractor can be a trivial point (fixed point), a line (limit
cycle), or a fractal object. An interesting case arises when the exponent λ is exactly zero.
The system is then called quasi-periodic or at the ‘edge of chaos’. There are many low-
dimensional examples where systems exhibit all three possibilities—they can be chaotic,
periodic, or at the edge of chaos, depending on their control parameters. The simplest
of these is the logistic map.
The intuitive understanding of how a system can be adaptive and robust at the same
time if it operates at the edge of chaos,is given by the following.If λ is close to zero,it takes
only tiny changes in the system to move it from a stable and periodic mode (λ slightly
negative) to the chaotic phase (λ positive). In the periodic mode, the system is stable
and robust; it returns to the attractor when perturbed. When it transits into the chaotic
phase, say, through a strong perturbation in the environment, it will sample large regions

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Cover
THE WIDOW IN THE
BYE STREET
BY
JOHN MASEFIELD
LONDON
SIDGWICK JACKSON LTD.
3 ADAM STREET, ADELPHI
MCMXII

Entered at the Library of Congress, Washington, U.S.A.
All rights reserved
Second Thousand
TO
MY WIFE
I
Down Bye Street, in a little Shropshire town,
There lived a widow with her only son:
She had no wealth nor title to renown,
Nor any joyous hours, never one.
She rose from ragged mattress before sun
And stitched all day until her eyes were red,
And had to stitch, because her man was dead.
Sometimes she fell asleep, she stitched so hard,
Letting the linen fall upon the floor;
And hungry cats would steal in from the yard,
And mangy chickens pecked about the door
Craning their necks so ragged and so sore

To search the room for bread-crumbs, or for mouse,
But they got nothing in the widow's house.
Mostly she made her bread by hemming shrouds
For one rich undertaker in the High Street,
Who used to pray that folks might die in crowds
And that their friends might pay to let them lie sweet;
And when one died the widow in the Bye Street
Stitched night and day to give the worm his dole.
The dead were better dressed than that poor soul.
Her little son was all her life's delight,
For in his little features she could find
A glimpse of that dead husband out of sight,
Where out of sight is never out of mind.
And so she stitched till she was nearly blind,
Or till the tallow candle end was done,
To get a living for her little son.
Her love for him being such she would not rest,
It was a want which ate her out and in,
Another hunger in her withered breast
Pressing her woman's bones against the skin.
To make him plump she starved her body thin.
And he, he ate the food, and never knew,
He laughed and played as little children do.
When there was little sickness in the place

She took what God would send, and what God sent
Never brought any colour to her face
Nor life into her footsteps when she went
Going, she trembled always withered and bent
For all went to her son, always the same,
He was first served whatever blessing came.
Sometimes she wandered out to gather sticks,
For it was bitter cold there when it snowed.
And she stole hay out of the farmer's ricks
For bands to wrap her feet in while she sewed,
And when her feet were warm and the grate glowed
She hugged her little son, her heart's desire,
With 'Jimmy, ain't it snug beside the fire?'
So years went on till Jimmy was a lad
And went to work as poor lads have to do,
And then the widow's loving heart was glad
To know that all the pains she had gone through
And all the years of putting on the screw,
Down to the sharpest turn a mortal can,
Had borne their fruit, and made her child a man.
He got a job at working on the line
Tipping the earth down, trolly after truck,
From daylight till the evening, wet or fine,
With arms all red from wallowing in the muck,
And spitting, as the trolly tipped, for luck,

And singing 'Binger' as he swung the pick
Because the red blood ran in him so quick.
So there was bacon then, at night, for supper
In Bye Street there, where he and mother stay;
And boots they had, not leaky in the upper,
And room rent ready on the settling day;
And beer for poor old mother, worn and grey,
And fire in frost; and in the widow's eyes
It seemed the Lord had made earth paradise.
And there they sat of evenings after dark
Singing their song of 'Binger,' he and she,
Her poor old cackle made the mongrels bark
And 'You sing Binger, mother,' carols he;
'By crimes, but that's a good song, that her be':
And then they slept there in the room they shared,
And all the time fate had his end prepared.
One thing alone made life not perfect sweet:
The mother's daily fear of what would come
When woman and her lovely boy should meet,
When the new wife would break up the old home.
Fear of that unborn evil struck her dumb,
And when her darling and a woman met,
She shook and prayed, 'Not her, O God; not yet.'
'Not yet, dear God, my Jimmy go from me.'

Then she would subtly question with her son.
'Not very handsome, I don't think her be?'
'God help the man who marries such an one.'
Her red eyes peered to spy the mischief done.
She took great care to keep the girls away,
And all her trouble made him easier prey.
There was a woman out at Plaister's End,
Light of her body, fifty to the pound,
A copper coin for any man to spend,
Lovely to look on when the wits were drowned.
Her husband's skeleton was never found,
It lay among the rocks at Glydyr Mor
Where he drank poison finding her a whore.
She was not native there, for she belonged
Out Milford way, or Swansea; no one knew.
She had the piteous look of someone wronged,
'Anna,' her name, a widow, last of Triw.
She had lived at Plaister's End a year or two;
At Callow's cottage, renting half an acre;
She was a hen-wife and a perfume-maker.
Secret she was; she lived in reputation;
But secret unseen threads went floating out:
Her smile, her voice, her face, were all temptation,
All subtle flies to trouble man the trout;
Man to entice, entrap, entangle, flout...

To take and spoil, and then to cast aside:
Gain without giving was the craft she plied.
And she complained, poor lonely widowed soul,
How no one cared, and men were rutters all;
While true love is an ever-burning goal
Burning the brighter as the shadows fall.
And all love's dogs went hunting at the call,
Married or not she took them by the brain,
Sucked at their hearts and tossed them back again.
Like the straw fires lit on Saint John's Eve,
She burned and dwindled in her fickle heart;
For if she wept when Harry took his leave,
Her tears were lures to beckon Bob to start.
And if, while loving Bob, a tinker's cart
Came by, she opened window with a smile
And gave the tinker hints to wait a while.
She passed for pure; but, years before, in Wales,
Living at Mountain Ash with different men,
Her less discretion had inspired tales
Of certain things she did, and how, and when.
Those seven years of youth; we are frantic then.
She had been frantic in her years of youth,
The tales were not more evil than the truth.
She had two children as the fruits of trade

Though she drank bitter herbs to kill the curse,
Both of them sons, and one she overlaid,
The other one the parish had to nurse.
Now she grew plump with money in her purse,
Passing for pure a hundred miles, I guess,
From where her little son wore workhouse dress.
There with the Union boys he came and went,
A parish bastard fed on bread and tea,
Wearing a bright tin badge in furthest Gwent,
And no one knowing who his folk could be.
His mother never knew his new name: she,--
She touched the lust of those who served her turn,
And chief among her men was Shepherd Ern.
A moody, treacherous man of bawdy mind,
Married to that mild girl from Ercall Hill,
Whose gentle goodness made him more inclined
To hotter sauces sharper on the bill.
The new lust gives the lecher the new thrill,
The new wine scratches as it slips the throat,
The new flag is so bright by the old boat.
Ern was her man to buy her bread and meat,
Half of his weekly wage was hers to spend,
She used to mock 'How is your wife, my sweet?'
Or wail, 'O, Ernie, how is this to end?'
Or coo, 'My Ernie is without a friend,

She cannot understand my precious life,'
And Ernie would go home and beat his wife.
So the four souls are ranged, the chess-board set,
The dark, invisible hand of secret Fate
Brought it to come to being that they met
After so many years of lying in wait.
While we least think it he prepares his Mate.
Mate, and the King's pawn played, it never ceases
Though all the earth is dust of taken pieces.
II
October Fair-time is the time for fun,
For all the street is hurdled into rows
Of pens of heifers blinking at the sun,
And Lemster sheep which pant and seem to doze,
And stalls of hardbake and galanty shows,
And cheapjacks smashing crocks, and trumpets blowing,
And the loud organ of the horses going.
There you can buy blue ribbons for your girl
Or take her in a swing-boat tossing high,
Or hold her fast when all the horses whirl

Round to the steam pipe whanging at the sky,
Or stand her cockshies at the cocoa-shy,
Or buy her brooches with her name in red,
Or Queen Victoria done in gingerbread.
Then there are rifle shots at tossing balls,
'And if you hit you get a good cigar.'
And strength-whackers for lads to lamm with mauls,
And Cheshire cheeses on a greasy spar.
The country folk flock in from near and far,
Women and men, like blow-flies to the roast,
All love the fair; but Anna loved it most.
Anna was all agog to see the fair;
She made Ern promise to be there to meet her,
To arm her round to all the pleasures there,
And buy her ribbons for her neck, and treat her,
So that no woman at the fair should beat her
In having pleasure at a man's expense.
She planned to meet him at the chapel fence.
So Ernie went; and Jimmy took his mother,
Dressed in her finest with a Monmouth shawl,
And there was such a crowd she thought she'd smother,
And O, she loved a pep'mint above all.
Clash go the crockeries where the cheapjacks bawl,
Baa go the sheep, thud goes the waxwork's drum,
And Ernie cursed for Anna hadn't come.

He hunted for her up and down the place,
Raging and snapping like a working brew.
'If you're with someone else I'll smash his face,
And when I've done for him I'll go for you.'
He bought no fairings as he'd vowed to do
For his poor little children back at home
Stuck at the glass 'to see till father come.'
Not finding her, he went into an inn,
Busy with ringing till and scratching matches.
Where thirsty drovers mingled stout with gin
And three or four Welsh herds were singing catches.
The swing-doors clattered, letting in in snatches
The noises of the fair, now low, now loud.
Ern called for beer and glowered at the crowd.
While he was glowering at his drinking there
In came the gipsy Bessie, hawking toys;
A bold-eyed strapping harlot with black hair,
One of the tribe which camped at Shepherd's Bois.
She lured him out of inn into the noise
Of the steam-organ where the horses spun,
And so the end of all things was begun.
Newness in lust, always the old in love.
'Put up your toys,' he said, 'and come along,
We'll have a turn of swing-boats up above,
And see the murder when they strike the gong.'

'Don't 'ee,' she giggled. 'My, but ain't you strong.
And where's your proper girl? You don't know me.'
'I do.' 'You don't.' 'Why, then, I will,' said he.
Anna was late because the cart which drove her
Called for her late (the horse had broke a trace),
She was all dressed and scented for her lover,
Her bright blue blouse had imitation lace,
The paint was red as roses on her face,
She hummed a song, because she thought to see
How envious all the other girls would be.
When she arrived and found her Ernie gone,
Her bitter heart thought, 'This is how it is.
Keeping me waiting while the sports are on:
Promising faithful, too, and then to miss.
O, Ernie, won't I give it you for this.'
And looking up she saw a couple cling,
Ern with his arm round Bessie in the swing.
Ern caught her eye and spat, and cut her dead,
Bessie laughed hardly, in the gipsy way.
Anna, though blind with fury, tossed her head,
Biting her lips until the red was grey,
For bitter moments given, bitter pay,
The time for payment comes, early or late,
No earthly debtor but accounts to Fate.

She turned aside, telling with bitter oaths
What Ern should suffer if he turned agen,
And there was Jimmy stripping off his clothes
Within a little ring of farming men.
'Now, Jimmy, put the old tup into pen.'
His mother, watching, thought her heart would curdle,
To see Jim drag the old ram to the hurdle.
Then the ram butted and the game began,
Till Jimmy's muscles cracked and the ram grunted.
The good old wrestling game of Ram and Man,
At which none knows the hunter from the hunted.
'Come and see Jimmy have his belly bunted.'
'Good tup. Good Jim. Good Jimmy. Sick him, Rover,
By dang, but Jimmy's got him fairly over.'
Then there was clap of hands and Jimmy grinned
And took five silver shillings from his backers,
And said th'old tup had put him out of wind
Or else he'd take all comers at the Whackers.
And some made rude remarks of rams and knackers,
And mother shook to get her son alone,
So's to be sure he hadn't broke a bone.
None but the lucky man deserves the fair,
For lucky men have money and success,
Things that a whore is very glad to share,
Or dip, at least, a finger in the mess.

Anne, with her raddled cheeks and Sunday dress,
Smiled upon Jimmy, seeing him succeed,
As though to say, 'You are a man, indeed.'
All the great things of life are swiftly done,
Creation, death, and love the double gate.
However much we dawdle in the sun
We have to hurry at the touch of Fate;
When Life knocks at the door no one can wait,
When Death makes his arrest we have to go.
And so with love, and Jimmy found it so.
Love, the sharp spear, went pricking to the bone,
In that one look, desire and bitter aching,
Longing to have that woman all alone
For her dear beauty's sake all else forsaking;
And sudden agony that set him shaking
Lest she, whose beauty made his heart's blood cruddle,
Should be another man's to kiss and cuddle.
She was beside him when he left the ring,
Her soft dress brushed against him as he passed her;
He thought her penny scent a sweeter thing
Than precious ointment out of alabaster;
Love, the mild servant, makes a drunken master.
She smiled, half sadly, out of thoughtful eyes,
And all the strong young man was easy prize.

She spoke, to take him, seeing him a sheep,
'How beautiful you wrastled with the ram,
It made me all go tremble just to peep,
I am that fond of wrastling, that I am.
Why, here's your mother, too. Good-evening, ma'am.
I was just telling Jim how well he done,
How proud you must be of so fine a son.'
Old mother blinked, while Jimmy hardly knew
Whether he knew the woman there or not;
But well he knew, if not, he wanted to,
Joy of her beauty ran in him so hot,
Old trembling mother by him was forgot,
While Anna searched the mother's face, to know
Whether she took her for a whore or no.
The woman's maxim, 'Win the woman first,'
Made her be gracious to the withered thing.
'This being in crowds do give one such a thirst,
I wonder if they've tea going at The King?
My throat's that dry my very tongue do cling,
Perhaps you'd take my arm, we'd wander up
(If you'd agree) and try and get a cup.
Come, ma'am, a cup of tea would do you good;
There's nothing like a nice hot cup of tea
After the crowd and all the time you've stood;
And The King's strict, it isn't like The Key,

Now, take my arm, my dear, and lean on me.'
And Jimmy's mother, being nearly blind,
Took Anna's arm, and only thought her kind.
So off they set, with Anna talking to her,
How nice the tea would be after the crowd,
And mother thinking half the time she knew her,
And Jimmy's heart's blood ticking quick and loud,
And Death beside him knitting at his shroud,
And all the High Street babbling with the fair,
And white October clouds in the blue air.
So tea was made, and down they sat to drink;
O the pale beauty sitting at the board!
There is more death in women than we think,
There is much danger in the soul adored,
The white hands bring the poison and the cord;
Death has a lodge in lips as red as cherries,
Death has a mansion in the yew-tree berries.
They sat there talking after tea was done,
And Jimmy blushed at Anna's sparkling looks,
And Anna flattered mother on her son,
Catching both fishes on her subtle hooks.
With twilight, tea and talk in ingle-nooks,
And music coming up from the dim street,
Mother had never known a fair so sweet.

Now cow-bells clink, for milking-time is come,
The drovers stack the hurdles into carts,
New masters drive the straying cattle home,
Many a young calf from his mother parts,
Hogs straggle back to sty by fits and starts;
The farmers take a last glass at the inns,
And now the frolic of the fair begins.
All of the side shows of the fair are lighted,
Flares and bright lights, and brassy cymbals clanging,
'Beginning now' and 'Everyone's invited,'
Shatter the pauses of the organ's whanging,
The Oldest Show on Earth and the Last Hanging,
'The Murder in the Red Barn,' with real blood,
The rifles crack, the Sally shy-sticks thud.
Anna walked slowly homewards with her prey,
Holding old tottering mother's weight upon her,
And pouring in sweet poison on the way
Of 'Such a pleasure, ma'am, and such an honour,'
And 'One's so safe with such a son to con her
Through all the noises and through all the press,
Boys daredn't squirt tormenters on her dress.'
At mother's door they stop to say 'Good-night.'
And mother must go in to set the table.
Anna pretended that she felt a fright
To go alone through all the merry babel:

'My friends are waiting at The Cain and Abel,
Just down the other side of Market Square,
It'd be a mercy if you'd set me there.'
So Jimmy came, while mother went inside;
Anna has got her victim in her clutch.
Jimmy, all blushing, glad to be her guide,
Thrilled by her scent, and trembling at her touch.
She was all white and dark, and said not much;
She sighed, to hint that pleasure's grave was dug,
And smiled within to see him such a mug.
They passed the doctor's house among the trees,
She sighed so deep that Jimmy asked her why.
'I'm too unhappy upon nights like these,
When everyone has happiness but I!'
'Then, aren't you happy?' She appeared to cry,
Blinked with her eyes, and turned away her head:
'Not much; but some men understand,' she said.
Her voice caught lightly on a broken note,
Jimmy half-dared but dared not touch her hand,
Yet all his blood went pumping in his throat
Beside the beauty he could understand,
And Death stopped knitting at the muffling band.
'The shroud is done,' he muttered, 'toe to chin.'
He snapped the ends, and tucked his needles in.

Jimmy, half stammering, choked, 'Has any man----'
He stopped, she shook her head to answer 'No.'
'Then tell me.' 'No. P'raps some day, if I can.
It hurts to talk of some things ever so.
But you're so different. There, come, we must go
None but unhappy women know how good
It is to meet a soul who's understood.'
'No. Wait a moment. May I call you Anna?'
'Perhaps. There must be nearness 'twixt us two.'
Love in her face hung out his bloody banner,
And all love's clanging trumpets shocked and blew.
'When we got up to-day we never knew.'
'I'm sure I didn't think, nor you did.' 'Never.'
'And now this friendship's come to us for ever.'
'Now, Anna, take my arm, dear.' 'Not to-night,
That must come later when we know our minds,
We must agree to keep this evening white,
We'll eat the fruit to-night and save the rinds.'
And all the folk whose shadows darked the blinds,
And all the dancers whirling in the fair,
Were wretched worms to Jim and Anna there.
'How wonderful life is,' said Anna, lowly.
'But it begins again with you for friend.'
In the dim lamplight Jimmy thought her holy,
A lovely fragile thing for him to tend,

Grace beyond measure, beauty without end.
'Anna,' he said; 'Good-night. This is the door.
I never knew what people meant before.'
'Good-night, my friend. Good-bye.' 'But, O my sweet,
The night's quite early yet, don't say good-bye,
Come just another short turn down the street,
The whole life's bubbling up for you and I.
Somehow I feel to-morrow we may die.
Come just as far as to the blacksmith's light.'
But 'No' said Anna; 'Not to-night. Good-night.'
All the tides triumph when the white moon fills.
Down in the race the toppling waters shout,
The breakers shake the bases of the hills,
There is a thundering where the streams go out,
And the wise shipman puts his ship about
Seeing the gathering of those waters wan,
But what when love makes high tide in a man?
Jimmy walked home with all his mind on fire,
One lovely face for ever set in flame.
He shivered as he went, like tautened wire,
Surge after surge of shuddering in him came
And then swept out repeating one sweet name,
'Anna, O Anna,' to the evening star.
Anna was sipping whiskey in the bar.

So back to home and mother Jimmy wandered,
Thinking of Plaister's End and Anna's lips.
He ate no supper worth the name, but pondered
On Plaister's End hedge, scarlet with ripe hips,
And of the lovely moon there in eclipse,
And how she must be shining in the house
Behind the hedge of those old dog-rose boughs.
Old mother cleared away. The clock struck eight.
'Why, boy, you've left your bacon, lawks a me,
So that's what comes of having tea so late,
Another time you'll go without your tea.
Your father liked his cup, too, didn't he,
Always another cup he used to say,
He never went without on any day.
How nice the lady was and how she talked,
I've never had a nicer fair, not ever.'
'She said she'd like to see us if we walked
To Plaister's End, beyond by Watersever.
Nice-looking woman, too, and that, and clever;
We might go round one evening, p'raps, we two;
Or I might go, if it's too far for you.'
'No,' said the mother, 'we're not folk for that;
Meet at the fair and that, and there an end.
Rake out the fire and put out the cat,
These fairs are sinful, tempting folk to spend.

Of course she spoke polite and like a friend;
Of course she had to do, and so I let her,
But now it's done and past, so I forget her.'
'I don't see why forget her. Why forget her?
She treat us kind. She weren't like everyone.
I never saw a woman I liked better,
And he's not easy pleased, my father's son.
So I'll go round some night when work is done.'
'Now, Jim, my dear, trust mother, there's a dear.'
'Well, so I do, but sometimes you're so queer.'
She blinked at him out of her withered eyes
Below her lashless eyelids red and bleared.
Her months of sacrifice had won the prize,
Her Jim had come to what she always feared.
And yet she doubted, so she shook and peered
And begged her God not let a woman take
The lovely son whom she had starved to make.
Doubting, she stood the dishes in the rack,
'We'll ask her in some evening, then,' she said,
'How nice her hair looked in the bit of black.'
And still she peered from eyes all dim and red
To note at once if Jimmy drooped his head,
Or if his ears blushed when he heard her praised,
And Jimmy blushed and hung his head and gazed.

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