2.1

1. .
Introduction to the Theory of Complex Systems
Stefan Thurner
Thurner Intro to the theory of CS KCL london mar 20, 2023 1 / 55

2. About this course
This course is an introduction to the theory of complex systems. We
learn about principles and technical tools to understand, deal with, and
manage CS. What are CS? The answer is not unique, we will try give a
practical and useful one.
Science of CS is about 30 years old. It’s state of development is maybe
best comparable to the state of Quantum Mechanics in 1920, when no
comprehensive literature existed on QM, and it was just a collection of
bits and pieces. There has not been a ’Copenhagen’ yet for CS.
However, there is a pattern emerging what a theory of CS must include.
This lecture is an attempt to show this pattern. It is a bit like discovering
a Greek mosaic that is not yet fully excavated, parts are missing, but the
main characters are already clearly visible and one understands the
picture.
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3. Course outline
Introduction
Scaling and driven systems
Statistical mechanics and information theory for CS
Systemic risk in financial networks and supply chains
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4. Literature
This course will follow
S. Thurner, P. Klimek, R. Hanel
Introduction to the Theory of Complex Systems
Oxford University Press, 2018
graphics, if not mentioned otherwise, were produced by the authors
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5. What are CS?
What are CS ?
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6. What are CS?
To provide a feeling for what complex systems are let us first remind
ourselves on three other questions
• what is physics?
• what is biology and life sciences?
• what are social sciences?
By describing what these sciences are – and what they are not – we should
develop a feeling of what complex systems are
The aim is to make clear that the science of CS is a non-trivial combination of
three disciplines and that this mix becomes something like a discipline by
itself
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7. What are CS?
What is physics?
Physics is the experimental, quantitative and predictive science of matter and
their interactions
quantitative statements are made with numbers: less ambiguous than
words
predictive means that statements are given in form of predictions which
can be experimentally tested
Basically one asks specific questions to nature in form of experiments – and
in fact, one gets answers. This methodology is unique in world history - the
scientific method
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8. What are CS?
What is physics?
Matter Relevant interaction types Length scale
Macroscopic matter gravity, electromagnetism all ranges
Molecules electromagnetism all ranges
Atoms electromagnetism, weak force ∼ 10−18
m
Hadrons & leptons electromagnetism, weak/strong 10−18
− 10−15
m
Quarks & gauge bosons electromagnetism, weak/strong 10−18
− 10−15
m
All interactions in the physical world are mediated by the exchange of gauge
bosons. For gravity the situation is not yet experimentally founded.
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9. What are CS?
What is physics?
The nature of fundamental forces
Typically the four fundamental forces act homogeneously and
isotropically in space and time. There are famous exceptions, however:
the strong force or interaction acts in a way as if the interaction is limited
to a ‘string’ – similar to type II superconductivity.
These interactions work on different scales, from light years to femto
meters: this means that one typically has to consider only one force for a
given phenomenon of interest. The other one can be 100, 106
, or up to
1039
times stronger than the other.
Traditionally physics does not specify which particles interact with each
other. Usually they all interact equally, the interaction strength depends
on interaction type and the form of the potential.
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10. What are CS?
What is physics?
What does predictive mean?
Assume you can do an experiment over and over again, e.g. drop a stone.
The theoretical task is to predict the reproducible results.
Since Newton physics follows the following recipe
Find the equations of motion to code your understanding of a dynamical
system
dp
dt
= F(x) , p = m
dx
dt
Predictive means: once F is specified the problem is solved, if the initial
and or boundary conditions are known. The result is x(t)
Compare the result with your experiments
Note: fixing initial conditions and boundary conditions means taking the
system out of the context – out of the rest of the universe. This is why physics
works!
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11. What are CS?
What is physics?
The same philosophy holds for arbitrarily complicated systems. Assume a
vector X(t) represents the state of a system (for example all positions and
momenta), then we get a set of equations of motion of this form
dX(t)
dt
= G(X(t))
Predictive means that in principle you can solve these equations. However,
these can be hard to solve.
Already for three bodies, this becomes a hard task, the famous three-body
problem (sun, earth, moon)
Laplace: if a daemon knowing all the initial conditions and able to solve all
equations – we could predict everything
The problem is that this daemon is hard to find. In fact the Newton-Laplace
program becomes completely useless for most systems. So are these
systems not predictable?
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12. What are CS?
What is physics?
Consider water and make the following experiment over and over again
cool it to 0o
C – it freezes
heat it to 100o
C it boils – almost certainly (under standard conditions)
One can maybe measure the velocity of a single gas molecule at a point in
time, but not of all, O(1023
) at the same time. But one can compute the
probability that a gas molecule has a velocity v,
p(v) ∝ exp(−
m
2kT
v2
)
For many non-interacting particles these probabilities become extremely
precise and one can make predictions about aggregates of particles.
Necessary: interactions are weak and the number of particles is large.
Note that to compute the freezing temperature of water was impossible
before simulations.
Note: The word prediction now has a much weaker meaning than in the
Laplace-Newton sense. The concept of determinism is diluted.
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13. What are CS?
What is physics?
The triumph of statistical mechanics
The idea of statistical mechanics is to understand the macroscopic
properties of a system from its microscopic components: relate the
micro- with the macro world
Typically in physics the macroscopic description is often simple and
corresponds to the state of the phase in which the system is (solid,
gaseous, liquid).
Physical systems have often very few phases. A system is often
prepared in one macro state (e.g. temperature and pressure is given).
There are usually many possible microstates that are related to that
macro state.
In statistical mechanics the main task is to compute the probabilities for
the many microstates that lead to that single macro state
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14. What are CS?
What is physics?
Traditional physics works fine for a few particles (Newton-Laplace) and for
many non-interacting particles (Boltzmann-Gibbs).
In other words, the class of systems that can be understood by physics is not
so big.
There were more severe shifts to the concept of predictability
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15. What are CS?
What is physics?
What does prediction mean? The crises of physics
Prediction in the 18th century is quite different from the concept of prediction
in the 21st
classical physics: exact prediction of trajectories
→crisis 1900: too many particles →
statistical physics: laws of probability allow stochastic predictions of the
macro (collective) behavior of gases, assuming trajectories are
predictable in principle
→crisis 1920s: concept of determinism evaporates completely →
QM and non-linear dynamics: unpredictable components – collective
phenomena remain predictable
→crisis 1990s: can not deal with strong interactions in stat. systems →
Complex Systems: situation can be worse than QM: unpredictable
components and complicated interactions – hope that collective is still
predictable
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16. What are CS?
What is physics?
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 in physics do not change over time.
This is radically different for complex systems: interactions themselves can
change over time. In that sense, complex systems change their internal
interaction structure as they evolve–co-evolution.
Systems that change their internal structure dynamically can be viewed as
machines and are best described as algorithms—a list of rules how the
system updates its states and future interactions.
Many complex systems work like this: states of components and interactions
between them are simultaneously updated, which leads to tremendous
mathematical difficulties. Whenever it is possible to ignore the changes in the
interactions in a dynamical system, analytic descriptions become meaningful.
Physics is analytic; complex systems are algorithmic. Experimentally testable
quantitative predictions can be made with analytic or algorithmic descriptions
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17. What are CS?
What is physics?
What are complex systems from a physics point of view?
many particle systems (as in statistical physics)
may have stochastic components and elements (as e.g. in QM)
interactions may be specific between two objects (networks)
interactions may be complicated and co-evolving: states and interactions
they are often chaotic and driven systems
interacting bodies are not limited to matter
interactions are not limited to the 4 fundamental forces
they can show very rich phase structure
they can have many macro states, even simultaneously realized
Most physicists will not have a problem to call these extensions to traditional
physics CS. For them it is still physics. The prototype model of CS in physics
are the so-called spin glasses.
Thurner Intro to the theory of CS KCL london mar 20, 2023 17 / 55

18. What are CS?
What is physics?
This is not what we call Complex Systems yet—there are crucial concepts
missing
However, due to more specific interactions and the increased variety of
types of interactions, the variety of macroscopic states changes
drastically. These emerge from the properties of the system’s
components and the interactions. The phenomenon that a priori
unexpected properties may arise as a consequence of the generalized
interactions is sometimes called emergence. Such CSs can have an
extremely rich phase structure.
In the case when there is a plurality of macro-states in a system, this
leads to entirely new questions one can ask to the system:
I What is the number of macro states?
I What are their co-occurrence rates?
I What are the typical sequences of occurrence?
I What is their life-times?
I What are the transition probabilities? etc.
Thurner Intro to the theory of CS KCL london mar 20, 2023 18 / 55

19. What are CS?
What is physics?
What we definitely want to keep from physics
CS is the experimental, quantitative and predictive science of generalized mat-
ter with generalized interactions
Generalized interactions are described by the interaction type α, and who
interacts with whom. If there are more than 2 objects involved, interactions
are conveniently indicated by networks
Mα
ij (t)
Interactions themselves remain based on concept of exchange. For example
think of communication, where messages are exchanged, trade where goods
and services are exchanged, friendships where wine bottles are exchanged,
etc.
For many CS the framework of physics is incomplete: What are the missing
concepts? Equilibrium, co-evolution, adjacent possible, ...
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20. What are CS?
A note on chemistry
A note on chemistry – the science of equilibria
In chemistry interactions between atoms and molecules can be already
quite specific. So why is chemistry usually not a CS?
Classically, chemistry is based on the law of mass action – many
particles interact in a way to reach equilibrium.
αA + βB σS + τT
α, β, σ, τ are the stoichiometric constants, k+, k− are reaction rates
forward reaction: k+{A}α
{B}β
, ({.} reactive mass)
backward reaction: k−{S}σ
{T}τ
they are the same in equilibrium
K =
k+
k−
=
{S}σ
{T}τ
{A}α{B}β
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21. What are CS?
A note on chemistry
Many CS are characterized by the fact that they are out of equilibrium
(driven). This means that there are no fixed point type equations that can
be used to solve the problem.
In this situation the help from statistical mechanics becomes very
limited. It is hard to handle systems that are out of equilibrium.
Even the so-called stationary non-equilibrium is hard to understand –
even computationally (thermostats)
On the positive side: many CS are self-organized critical
let’s keep from chemistry
Many CS are out of equilibrium
Many CS are non-ergodic
Note: as soon as one focuses on e.g. cyclical catalytic reactions on networks
chemistry becomes a CS very soon
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22. What are CS?
What is Biology?
What are CS ?
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23. What are CS?
What is Biology?
What is biology?
Life Science is the experimental science of living matter.
What is living matter?
What are the minimal conditions for living matter to exist?
According to S.A. Kauffman
living matter has to be self-replicating
has to run through at least one Carnot cycle
has to be localized
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24. What are CS?
What is Biology?
Living matter is a self-sustained sequence of genetic activity over life.
It uses energy and performs work. It is constantly out of equilibrium.
Genetic activity has to do with chemical reactions which take place e.g.
within cells (compartments). Unfortunately, this is only partly true.
Chemical reactions usually involve billions of atoms or molecules.
What happens in the cell is chemistry with few molecules. If you have a
few molecules only, there arise problems:
I the law of large numbers becomes inappropriate
I the laws of diffusion become inadequate
I the concept of equilibrium becomes shaky
I without equilibrium what is the law of mass action?
If we do not have a law of mass action, how is chemistry to be done?
Consequently, traditional chemistry is often rather inadequate for living matter
Thurner Intro to the theory of CS KCL london mar 20, 2023 24 / 55

25. What are CS?
What is Biology?
More complications in the cell:
Molecules may be transported from site of production to where they are
needed. This changes law of diffusion even more, it becomes
anomalous diffusion, d
dt p(x, t) = D d2+ν
dx2+ν p(x, t)µ
.
Chemical binding depends on 3D structure of molecules.
Chemical binding depends on ’state’ of molecules, e.g. if they are
phosphorylated or not.
’Reaction rates’ – if one still wants to use this term – depend on the
statistical mechanics of small systems, i.e. fluctuation theorems might
become important.
Thurner Intro to the theory of CS KCL london mar 20, 2023 25 / 55

26. What are CS?
What is Biology?
Biological interactions happen on networks – almost exclusively
Genetic regulation governs the temporal sequence of abundance of
proteins, nucleic material and metabolites within a living organism
Genetic regulation can be viewed as a discrete interaction
Protein-protein binding is discrete, for example complex formation
Discrete interactions are described by networks:
gene-regulatory network (e.g. Boolean)
metabolic network
protein-protein network
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27. What are CS?
What is Biology?
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28. What is Biology?
Evolution
Nothing in biology makes sense except in the light of evolution. Dobzhansky
Genetic material and the process of replication involve several stochastic
components, and lead to variations in copies. Replication and variation are
two of the three main ingredients for all evolutionary processes.
What is evolution? Remember the Darwinian story
Consider a population of some kind. The offspring of this population has some
random variations (e.g. mutations). Individuals with the optimal variations (gi-
ven a certain surrounding) have a selection advantage, i.e. fitness. This fitness
manifests itself such that these individuals have the more offspring, and thus
pass on the particular variation on to a new generation. In this way ’optimal’
variations get selected over time.
Is this definition predictive science? Or is it just a convincing story?
Is the Darwinian story falsifiable? How can we measure fitness? It is an a
posteriori concept. Survival of the fittest ≡ – survival of those who survive.
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29. What is Biology?
Evolution
Evolution is a three-step process.
1 new thing comes into being in a given environment.
2 new thing has the chance to interact with environment. The result of this
interaction: possibility to get selected or destroyed.
3 if the new thing gets selected (survives) in this environment it becomes
part of this environment – it becomes part of the new environment for all
future, new and arriving things.
Evolution happens simultaneously on various scales (time & space): cells –
organisms – populations
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30. What is Biology?
Evolution
Evolution is not physics.
If you think of this three step process in terms of equations of motion
1 Write down the dynamics of the system in the form of equations of
motion.
2 Boundary conditions depend on these equations – you can not fix them.
3 Consequently, you can not solve the equations.
4 Newtonian recipe breaks down, you fail – program becomes
mathematical monster if you think of dynamically coupled boundary
conditions with the dynamical system.
The task is: try to solve it nevertheless. We will see that multi-scale methods
can be used to address this type of problems.
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31. What is Biology?
Evolution
The concept of evolution is fundamentally different from physics.
It is immediately evident that we are confronted with two huge problems:
Boundary conditions can not be fixed.
Phase space is not well defined – it changes over time. New elements
may emerge that change the environment substantially.
The evolutionary aspect is essential for many CS, it can not be neglected.
Thurner Intro to the theory of CS KCL london mar 20, 2023 31 / 55

32. What is Biology?
Evolution
Evolution is a natural phenomenon.
Evolution is a process that increases diversity, it looks as if it is a
’creative’ process.
Evolution has many forms of appearance – it is ’universal’: biological
evolution, technological innovation, economy, financial markets, history,
etc.
Evolution follows patterns, regardless in which form of appearance.
These patterns are surprisingly robust. For example: wherever you look,
there is life. Ecological niches that can get filled will get filled. Power
laws.
As a natural phenomenon it deserves a scientific explanation
(quantitative & predictive).
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33. What is Biology?
Evolution
A side note on evolution as a natural phenomenon.
Natural phenomena are to certain degree predictable, i.e. one can trust
patterns: drop stone it will fall down, if water gets cold it freezes, etc.
Credo in natural sciences: one can trust in the fact that natural
phenomena work. If this is the case there must be basic laws on
scientific grounds
With almost certainty: no ecological niche is empty, life is creative and
keeps on going regardless of catastrophes etc. Certain statistical
patterns in the evolutionary timeseries are repeated over and over again
– regardless of the specific system.
It will not be possible to predict what animals live on earth in 500.000
years. But it should be possible to make falsifiable predictions on
systemic quantities such as diversity, diversification rates, robustness,
resilience, adaptability. See our discussion on predictability.
Missing: ways to predict these systemic features quantitatively.
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34. What is Biology?
Evolution
A language for evolutionary processes – the adjacent possible
algorithmic view
This ’science’ is at its very beginning. Maybe not even the right language has
been established so far.
The adjacent possible is the set of all possible worlds that could
potentially exist in the next timestep. The adjacent possible depends
strongly on the present state of the world.
With this definition evolution is a process that continuously fills the AP.
Different from physics: a given state determines the next state. In
physics all potential states are known → nature is algorithmic
Evolution: given state determines the realization of a (huge) set of
possible states. The future states may not even be known at all –
’creative process’.
Filling of the adjacent possible determines the next adjacent possible.
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35. What is Biology?
Evolution
We have learned for evolutionary processes
One can not fix boundary conditions of evolutionary systems. This
means that it is impossible to take the system apart without possibly
losing critical properties. Here the triumphal concept of reductionism
starts to become inadequate.
Evolutionary CS feel their boundary conditions.
Evolutionary CS change their boundary conditions.
In physics the adjacent possible is very small. For example imagine a
falling stone. The AP is that it is on the floor in two seconds. There are
practically no other options.
Lagrangian trajectories are completely specified by initial & end point. In
evolution the AP evolves: AP(t) → AP(t + 1) → AP(t + 2) + · · ·
In physics: realization of the AP does (almost) not influence the next AP.
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36. What are CS?
What is Biology?
Biological systems are adaptive and robust – the concept of the edge of chaos
The possibility to adapt and robustness seem to exclude each other.
However, living systems are clearly adaptive and robust at the same time. To
explain how this is possible one can take the picturesque view:
Every dynamical system has a maximal Lyapunov exponent. It measures
how – 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(0) is the initial separation
If the exponent is positive the system is called chaotic, or strongly mixing
If the exponent λ is negative the system approaches an attractor – two
initially infinitesimally adjacent trajectories converge. The system is
periodic
If the exponent is zero: the system is called quasi-periodic, or ’at the
edge of chaos’.
Thurner Intro to the theory of CS KCL london mar 20, 2023 36 / 55

37. What are CS?
What is Biology?
How does nature find the edge of chaos?
The set of points where the Lyapunov exponents are zero is usually of
measure zero. However, evolution seems to find and select these points.
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38. What are CS?
What is Biology?
How does nature find the edge of chaos?
How can a mechanism of evolution detect something of measure zero?
One explanation is self-organized criticality, where systems organize
themselves to operate at a critical point between order and randomness.
SOC can emerge from interactions in different systems, including sand
piles, precipitation, heartbeat, avalanches, forest fires, earthquakes, etc.
Or is this set simply not of measure zero?
Learn more about these questions in chapter ‘Evolution’ in the book.
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39. What are CS?
What is Biology?
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40. What are CS?
What is Biology?
Biological systems are self-organized and critical
Self-organized systems are dynamical systems that have a critical point
as an attractor. Very often these systems – at a macroscopic scale – are
characterized by scale invariance. Scale invariance means the absence
of a characteristic (length) scale.
A critical point of a system is reached at conditions (temperature,
pressure, slope in sandpile, etc.) where the characteristic length-scale
(e.g. correlation length) becomes divergent.
It is typical to slowly driven systems, where driven means that they are
driven softly away from equilibrium.
Applications cover all the sciences
Physics: particle-, geo-, plasma-, solar physics, cosmology, quantum gravity,...
Biology: evolutionary biology, ecology, neurobiology, ...
Social sciences: economics, sociology, ...
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41. What are CS?
What is Biology?
An intuition for self-organized critical systems—sandpile models
Imagine a pile of sand.
If the slope is too steep avalanches go off → slope becomes flatter
If the slope is too flat sand gets deposited → slope becomes steeper
The pile self-organizes toward a critical slope. The system is robust and
adaptive.
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42. What are CS?
What is Biology?
Let us collect the components for CS that we get from the life sciences
Interactions take place on networks
Out of equilibrium – they are driven
Evolutionary dynamics
Core components are discrete – e.g. Boolean networks
Adaptive and robust: edge of chaos – self-organized critical
Most evolutionary complex systems are path-dependent and have
memory (non-ergodic or non-Markovian). The adjacent possible is a way
of conceptualizing the evolution of the ‘reachable’ phasespace.
We learn how to do statistics of driven out of equilibrium systems in courses:
Scaling and Statistical Mechanics
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43. What are CS?
What is Biology?
We conclude this section with listing a few aims of current Life Science
Understand basic mechanisms (molecular, geno-phenotype relations,
epigenetics)
Manipulate and manage living matter (medicine, smart matter)
Create living matter from scratch (artificial life)
This sounds too optimistic? It is to a certain extent, but several prerequisites
for these tasks are looking certainly good at the moment. The components
are understood and basically under experimental control.
Genetic sequences of hundreds of organisms are available.
Have proteins of thousands of organisms available.
Can synthesize proteins that nature does not produce.
Can synthesize arbitrary RNA sequences on demand.
Have all metabolites of organisms available.
Can count molecules, understand basic transport mechanisms, etc.
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44. Lecture I What are CS?
What is social science?
What are CS ?
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45. What are CS?
What is social science?
Social science is the science of dynamical social interactions and their impli-
cations to society.
Traditionally it is neither quantitative nor predictive, nor does not produce
experimentally testable predictions. Why is that so?
Lack of detailed data
Lack of reproducibility / repeatability
The queen of the social sciences is economics, i.e. the science of the
invention, production, distribution, consumption, and disposal of goods and
services.
All of these components happen on networks, i.e. the associated interactions
are very much directed and interconnected. Maybe network aspects are most
illustrative when studied in the social sciences, even more than in the life
sciences.
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46. What are CS?
What is social science?
What are societies?
People, goods or institutions are represented by nodes of a network.
Interactions are represented by links in networks of various types.
One node may be engaged in several types of interaction.
Nodes are characterized by ‘states’: wealth, opinion, age, etc.
Nodes and links change over time.
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47. What are CS?
What is social science?
What are societies?
Societies are co-evolving multiplex networks, Mα
ij (t)
A multiplex network is a collection of networks on the same set of nodes
Links. α = 1: communication: full line; α = 2: trading: dashed line; α = 3: friendship: dotted line.
States. black – votes for Hillary; grey – votes for Trump.
Nodes i (humans or institutions) are characterized by states σi (t)
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48. What are CS?
What is social science?
Let us collect the components for CS that we get from the social sciences
Interactions happen on a collection of networks (Multiplex networks).
Networks may interact with themselves.
Networks show a rich variety in growth and re-structuring.
Networks are evolutionary and co-evolutionary objects.
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49. What are CS?
What is co-evolution?
The concept of co-evolution.
What is an interaction? In general an interaction can change the state of
the interacting objects, or the environment. For example, the collision of
two particles changes their momentum. The magnetic interaction of two
spins may change their orientations. An economic interaction changes
the portfolios of the participants involved.
The interaction partners (network or multiplex) of a node can be seen as
the ‘environment’ (space) of that node. The environment determines the
future state of the node.
Interactions can change over time. For example people establish new
friendships or economical links, countries terminate diplomatic relations.
The state of nodes determine the future state of the link, meaning if it
exists in the future or not.
The state (topology) of the network determines the future states of the nodes.
The state of the nodes determines the future state of the links of the network.
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50. What are CS?
A pictorial view
Co-evolving multiplex networks – more formally.
d
dt
σα
i (t) ∼ F
Mα
ij (t), σβ
j (t)
and
d
dt
Mα
ij (t) ∼ G
Mα
ij (t), σβ
j (t)
This is maybe the simplest way to illustrate what a CS is, or looks like.
Networks are observable (big data)
Collections of networks are manageable
States of individual nodes are (will be) observable
From a practical point of view it is useless, because G and F are not
specified. They can be stochastic. However, it should be possible to express
most CS of the form we discussed in this form.
Thurner Intro to the theory of CS KCL london mar 20, 2023 50 / 55

51. What are CS? A summary
1 Complex systems are composed of many elements (latin indices i).
2 Elements interact through one or more types (greek indices α).
3 Interactions are not static but change over time, Mα
ij (t).
4 Elements are characterized by states, σi (t).
5 States and interactions often evolve together by mutually updating each
other, they co-evolve.
6 Dynamics of co-evolving multilayer networks is usually non-linear.
7 CS are context-dependent. Multilayer networks provide that context and
thus offer the possibility of a self-consistent description of CSs.
8 CS are algorithmic.
9 CS are path-dependent and consequently often non-ergodic.
10 CS often have memory. Information about the past can be stored in
nodes or in the network structure of the various layers.
Thurner Intro to the theory of CS KCL london mar 20, 2023 51 / 55

52. What are CS?
The role of the computer
The computer is the game changer.
CS have never been accessible to control because of computational limits.
The analytic tools available until the 1980’s were simply too limited to address
problems beyond simple physics problems and equilibrium solutions in
(evolutionary) biology and economics. In the later the most famous concept
has been game theory.
Thurner Intro to the theory of CS KCL london mar 20, 2023 52 / 55

53. What are CS?
The role of the computer
The computer is the game changer.
The computer changed the path of science and opened the way to CS.
Mimic systems through simulation. Agent based models model
interactions by brute force and study the consequences for the collective.
In many systems there is only a single history, especially in the social
sciences, or in evolution. The computer allows to create artificial
histories of statistically equivalent copies – ensemble pictures help to
understand systemic properties that would otherwise not be handleable.
This solves the repeatability problem of CS.
Develop intuition by building the models. It forces you to systematically
think problems through. Maybe one can condense the such gained
intuition into traditional formula-language.
Computation itself profited immensely from CS in the past 3 decades.
Computational limits are very often not the limit for understanding anymore –
i.e. controlling and managing – CS. Also data issues become less and less
relevant in present times. Science experiences an unheardof revolution.
Thurner Intro to the theory of CS KCL london mar 20, 2023 53 / 55

54. What are CS?
Some triumphs of the science of CS:
Network theory
Genetic regulatory networks and Boolean networks
Self-organized criticality
Genetic algorithms
Auto-catalytic networks
Theory of increasing returns
Origin and statistics of power laws
Mosaic vaccines
Statistical mechanics of complex systems
Network models in epidemiology
Complexity economics and systemic risk
Allometric scaling in biology
Science of cities
Thurner Intro to the theory of CS KCL london mar 20, 2023 54 / 55

55. What are CS?
Aim of the course
Clarify the origin of power laws, especially in the context of driven
non-equilibrium systems.
Derive a framework for the statistics of driven systems.
Categorize probabilistic complex systems into equivalence classes that
characterize their statistical properties.
Present a generalization of statistical mechanics, and information theory,
so that they become useful for CS. In particular, we derive an entropy for
complex systems.
The overarching theme is understand co-evolutionary dynamics of states
and interactions.
Thurner Intro to the theory of CS KCL london mar 20, 2023 55 / 55