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

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.

Course outline
Introduction
Scaling and driven systems
Statistical mechanics and information theory for CS
Systemic risk in financial networks and supply chains

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

What are CS?
What are CS ?

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

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

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.

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.

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!

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?

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.

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

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

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

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

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.

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.

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, ...

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

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

What are CS?
What is Biology?
What are CS ?

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

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

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.

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

What are CS?
What is Biology?

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.

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

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.

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.

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).

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.

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.

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.

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’.

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.

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.

What are CS?
What is Biology?

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, ...

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.

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

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.

Lecture I What are CS?
What is social science?
What are CS ?

What are CS?
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.

What are CS?
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.

What are CS?
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)

What are CS?
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.

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.

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.

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.

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.

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.

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

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.

Where do scaling laws come from?
stefan thurner
KCL london mar 21 2023

work done together with
Bernat Corominas-Murtra, Rudolf Hanel and Murray
Gell-Mann
RH, ST, MGM, PNAS 111 (2014) 6905-6910
BCM, RH, ST, PNAS 112 (2015) 5348-5353
BCM, RH, ST, New J Physics 18 (2016) 093010
BCM, RH, ST, J Roy Soc Interface 12 (2016) 20150330
ST, BCM, RH, Phys Rev E 96 (2017) 032124
BCM, RH, ST, Sci Rep (2017) 11223
BCM, RH, LZ, ST, Sci Rep 8 (2018) 10837
RH, ST, Entropy 20 (2018) 838
KCL london mar 21 2023 1

1809
Gauss’ theory of errors

1810
1810 Laplace: “central limit theorem”
add independent random numbers that are from identical
source → sum is a random number from a normal distribution
• simple systems: constituents don’t interact (weakly)
• entered physics through Maxwell half a century later
• handle for localized variables around average

e−x2
x = 1 → e−x2
= 1/e ∼ 0.3679...
x = 10 → e−x2
∼ 0.000000000000000000000000000000000000000000037201...

x−α
x = 1 → x−α
= 1 (for α = 1)
x = 10 → x−α
= 0.1

complex systems
CS are networked, dynamical, (co)evolutionary
• elements are not independent
• networks link dynamic and stochastic components
• often many sources of randomness
variables are neither independent nor errors come from identical
sources
→ no reason to believe that Gaussian statistics holds for CS

statistics of CS is
statistics of power laws

statistics of complex systems
CS variables are not localized around mean → tremendous
difficulties
• that is expressed by power laws
• often not exact power laws – but “fat tailed”
→ statistics of CS is the statistics of outliers
→ outliers are the norm rather than the exception

examples for (approximate)
power laws

city size
MEJ Newman (2005)
multiplicative

rainfall
SOC

landslides
SOC

hurrican damages
secondary (multiplicative) ???

financial interbank loans
multiplicative / preferential

forrest fires in various regions
SOC

moon crater diameters
MEJ Newman (2005)
fragmentation

Gamma rays from solar wind
MEJ Newman (2005)

movie sales
SOC

healthcare costs
multiplicative ???

particle physics
???

words in books
MEJ Newman (2005)
preferential / random / optimization

citations of scientific articles
MEJ Newman (2005)
preferential

website hits
MEJ Newman (2005)
preferential

book sales
MEJ Newman (2005)
preferential

telephone calls
MEJ Newman (2005)
preferential

earth quake magnitude
MEJ Newman (2005)
SOC

seismic events
SOC

war intensity
MEJ Newman (2005)
???

killings in wars
???

size of wars
???

wealth distribution
MEJ Newman (2005)
multiplicative

family names
MEJ Newman (2005)
multiplicative, ???

systemic risk in supply chain

more power laws ...
• networks: literally thousands of scale-free networks
• allometric scaling in biology
• dynamics in cities
• fragmentation processes
• random walks
• crackling noise
• growth with random times of observation
• blackouts
• fossil record
• bird sightings
• terrorist attacks
• fluvial discharge, contact processes
• anomalous diffusion ...

how do power laws arise?

basic routes to power laws
• criticality
• self-organized criticality
• multiplicative processes with constraints
• self-reinforcing / preferential processes

I criticality: power laws at phase transitions
• statistical physics: power laws emerge at phase transition
• this happens at the critical point
• power laws in various quantities: critical exponents
• various materials have same critical exponents →
→ behave identical → universality

II power laws through self-organised criticality
• systems find critical points themselves – no tuning necessary
• since they are at a critical point → power laws
• examples: sandpiles, earthquakes, collapse, economy, ...
(Wiesenfeld)

III power laws through multiplicative processes
• Gaussian distribution: add random numbers (same source)
• power law: multiply random numbers and impose constraints
e.g. minimum number can not be smaller than X
• examples: wealth distribution, city size, double Pareto,
language, ...

IV power laws through preferential processes
• events repeat proportional to how often they occurred before
• examples: network growth models, language formation, ...

V other mechanisms
• Levy stable processes
• constraint optimisation
• extreme value statistics
• return times
• generalized entropies

one phenomenon—many explanations
Zipf law in word frequencies
• Simon: preferential attachment
• Mandelbrot: constraint optimization
• Miller: monkeys produce random texts
• Sole: information theoretic: sender and receiver

motivation I

is there a unique principle?

motivation II

complex systems are driven systems

driven system = driving + relaxing process

would be nice to have:
statistics for driven, (stationary) non-equilibrium
systems that explains the abundance of power laws
and variations

many complex systems are path-dependent
• future events depend on history of past events
• often past events constrain possibilities for the future
→ sample-space of processes reduces as they unfold

sample-space reducing processes (SSR)

example: history-dependent SSR processes
9
13
7
5
1)
4)
2) 3)
5) 6)
3 1

phase spaces are nested
Ω1 ⊂ Ω2 ⊂ Ω3 · · · ⊂ ΩN

restart means
Ω1 ≡ ΩN

sentence-formation is SSR
funny
wolf
word
runs
funny
howls
bites
house
can
rucksack
room
light
moon
night
green
eats
door
open
blue
sun
wind
cloud
buy
high
short
grey
jacket
computer
algebra
write
strong
letter
table
fill
information
article
punch
hold
line
brown
trash
tree
deer
rabbit
space
fly
mosquito
bee
window
think
red
building
hospital
glass
wine
take
sell
plane
envelope
word
runs
funny
howls
bites
house
can
rucksack
room
light
moon
night
green
eats
door
open
blue
sun
wind
cloud
buy
high short
grey
jacket
computer
algebra
write
strong
letter
table
fill
information
article
punch
hold
line
brown
trash
tree
deer
rabbit
space
fly
mosquito
bee
window
think
red
building
hospital
glass
wine
take
sell
plane
envelope
wolf wolf
word
house
can
rucksack
room
light
moon
night
green
door
open
blue
sun
wind
cloud
buy
high
short
jacket
computer
algebra
write
letter
table
fill
information
article
punch
hold
line
brown
trash
tree
deer
rabbit
space
fly
mosquito
bee
window
think
red
building
hospital
glass
wine
take
sell
plane
envelope
runs
howls
bites
eats
grey
strong
wolf
word
runs
funny
howls
bites
house
can
rucksack
room
light
green
eats
door
open
blue
sun
wind
cloud
buy
high
short
grey
jacket
computer
algebra
write
strong
letter
table
fill
information
article
punch
hold
line
brown
trash
tree
deer
rabbit
space
fly
mosquito
bee
window
think
red
building
hospital
glass
wine
take
sell
plane
envelope
moon
night
a) b) c) d)

0 5 10
0
0.45
n
0 5 10
0
0.1
0.2
0.3
0.4
n
p(n)
x x
1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9
p(i)
1 5 10 1 5 10
10
0
0.1
0.2
0.3
0.4
Site i Site i

SSR leads to exact Zipf’s law
p(i) =
1
i
p(i) is probability to visit site i

why?
Clearly, p(i) =
PN
j=1 P(i|j) p(j) holds, with
P(i|j) =
(
1
j−1 for i j (SSR)
1
N for i ≥ j = 1 (restart)
We get p(i) = 1
N p(1) +
PN
j=i+1
1
j−1 p(j)
→ recursive relation p(i + 1) − p(i) = −1
i p(i + 1)
p(i) = ... = p(1)i−1

alternative proof by induction
Let N = 2. There are two sequences φ: either φ directly
generates a 1 with p = 1/2, or first generates 2 with p = 1/2,
and then a 1 with certainty. Both sequences visit 1 but only
one visits 2. As a consequence, P2(2) = 1/2 and P2(1) = 1.
Now suppose PN−1(i) = 1/i holds. Process starts with dice
N, and probability to hit i in the first step is 1/N. Also,
any other j, N ≥ j i, is reached with probability 1/N.
If we get j i, we get i in the next step with probability
Pj−1(i), which leads to a recursive scheme for i N, PN(i) =
1
N

1 +
P
ij≤N Pj−1(i)

. Since by assumption Pj−1(i) =
1/i, with i j ≤ N holds, some algebra yields PN(i) = 1/i.

true for all systems with shrinking
sample-space over time

note: process is slowly driven

what if restart SSR before it is fully relaxed?
restarting or driving rate is (1 − λ) →
p(i) = i−λ

arbitrary driving
p(λ)
(i) =
PN
j=1 P(i|j) p(λ)
(j) , with
P(i|j) =





λ
j−1 + 1−λ
N for i j (SSR)
1−λ
N for i ≥ j 1 (RW)
1
N for i ≥ j = 1 (restart)
We get p(λ)
(i) = 1−λ
N + 1
N p(λ)
(1) +
PN
j=i+1
λ
j−1 p(λ)
(j)
to recursive relation p(λ)
(i + 1) − p(λ)
(i) = −λ1
i p(λ)
(i + 1)
p(λ)
(i)
p(λ)(1)
=
Qi−1
j=1

1 + λ
j
−1
= exp
h
−
Pi−1
j=1 log

1 + λ
j
i
∼ exp

−
Pi−1
j=1
λ
j

∼ exp (−λ log(i)) = i−λ

side note: valid also for continuous case
if p(x) ∼
Z x
0
p(y)
y
dy −→ p(x) is a power

true for all systems with driving rate 1 − λ
(shrinking sample space with probability λ)

SSR processes with fast driving
same convergence speed as CLT for iid processes

Berry-Esseen convergence speed
10
0
10
2
10
4
10
−3
10
−2
10
−1
10
0
Number of jumps T
Distance
10
0
10
1
10
2
10
3
10
4
10
1
10
2
10
3
10
4
10
5
10
6
Rank
Number
of
site
visits
0 0.5 1
0
0.2
0.4
0.6
0.8
1
sim
=0.0
=1.0
=1.0
=0.5
=0.7
-0.5
a) b)
-0.5
-0.7
-1.0

SSR-based Zipf law is extremely robust

proof: same as before
p(i|j) =
(
λ q(i)
g(j−1) + (1 − λ)q(i) if i j
(1 − λ)q(i) otherwise.
...
→ pλ(i) ∼
pλ(1)
q(1)1−λ

q(i)
g(i)λ

where g(j) =
P
ij+1 q(i)
polynomial priors: q(i) ∼ iα
, (α −1) → p(i) ∼ iα(1−λ)−λ
polynomial priors: q(i) ∼ iα
, (α −1)→ p(i) ∼ q(i)
exponential priors: q(i) ∼ eβi
→ uniform distribution

prior probabilities are practically irrelevant!

driven system = driving + relaxation part

relaxation = sample space reducing process

details of relaxing process do NOT matter

Zipf-law is extremely robust—accelerated SSR

not all transitions p(i|j) must exist—diffusion on
networks

SSR and diffusion on networks

SSR is a random walk on directed ordered NW
1
2
3
4
5
1 2 3 4 5
1/2
1/4
1/8
Node rank
Node
occupation
probability
10
0
10
2
10
0
10
−2
10
−4
Path rank
Path
probability
acyclic
p
exit
=0.3
a)
b)
-0.65
-1
pexit=0.3
Start
Stop
1/5
1/4
1/3
1/2
(1-pexit)/5
1
pexit
pexit=1)
pexit=0) 1
note fully connected

SSR = targeted random walk on networks
simple example Directed Acyclic Graph (no cycles)

simple routing algorithm
• take directed acyclic network fix it
• pick start-node
• perform a random walk from start-node to end-node (1)
• repeat many times from other start-nodes
• prediction visiting frequency of nodes follows Zipf law

all diffusion processes on DAGs are SSR
sample ER graph → direct it → pick start and end → diffuse

Exponential NW HEP Co-authors

what happens if introduce weights on links?
ER Graph
poisson weights power weights

what happens if we allow for cycles?
ER → direct it → change link to random direction with 1 − λ
“driving” λ = 0.8 λ = 0.5

Zipf’s law is an immense attractor!

Zipf’s law is an attractor
• no matter what the network topology is → Zipf
• no matter what the link weights are → Zipf
• if have cycles → exponent is less than one

all good search is SSR

what is good search?
a search process is good if ...
• ... at every step you eliminate more possibilities than you
actually sample
• ... every step you take eliminates branches of possibilities
if eliminate fast enough → power law in visiting times
if eliminate too little → sample entire space (exhaustive search)
if no cycles: expect Zipf’s law

clicking on web page is often result of search process

adamic hubermann 2002

breslau et al 99

what about exponents 1?

1 2 3 4 5 6 7 8 9
10
11
1213141516
17181920
Multiplication factor µ
→ p(i) = i−µ
Ansatz: n(j → i) = µp(i|j) (slow driving limit)

10
0
10
1
10
2
10
3
10
4
10
−10
10
−8
10
−6
10
−4
10
−2
10
0
State
Relative
Frequency
µ=1
µ=1.5
µ=2
µ=2.5
10
0
10
2
10
4
10
−6
10
−4
10
−2
10
0
Energy
Relative
Frequency
µ=2.5
µ=3.5
µ=4.5

what if we introduce
conservation laws?

conservation laws in SSR processes
assume you have duplication at every jump, µ = 2
if you are at i → duplicate → one jumps to j, the other to k
conservation means: i = j + k.
conservation means:
f(i) = f(state1) + f(state2) + · · · + f(stateµ)
→ p(i) = i−2
for all µ
same result was found by E. Fermi for particle cascades

10
0
10
1
10
2
10
3
10
4
10
−7
10
−6
10
−5
10
−4
10
−3
10
−2
10
−1
Energy (arbitrary units)
Relative
frequency
µ=2.5
µ=3.5
µ=4.5
−2

complex systems are driven –
always!

complex systems are driven non-equilibrium
systems
• only driven systems produce non-trivial structures
• without driving: just ground state or equilibrium
• every driven system: relaxing part + driving part
• every relaxing part is a SSR

example: inelastic gas in a box

example: inelastic gas in a box
ρ(E0
) =
Z E∗
E0
dEρ(E0
|E, cr) ((1 − ξ)ρ(E) + ξρcharge(E))

inelastic gas: transition probability
ρ(E0
1|E1, cr) = 1
Z(E1)
R π
0
dζg(ζ)
R π
0
dαf(α)
R π
0
dφr(φ)
×
R ∞
0
dE2ρ(E2)θ(E1 − E2)δ(E0
1 − F(E1, E2, α, ζ, φ; c∗
r))
with
F(E1, E2, α, ζ, φ; c∗
r) = E12
1+c2
r
4 +
1−c2
r
4 q cos ζ
+cr
2
p
1 − (q cos ζ)2 (cos ζ cos 2α − sin ζ sin 2α cos φ)

with q = 2
q
E1
E12
E2
E12
and cr(α)2
= 1 − (1 − c∗
r
2
)| sin α|
use this in eigenvalue equation

inelastic gas: energy distribution

inelastic gas: dependence on driving rate

where do specific distributions come from?

assume that driving rate depends on state λ(i)

→ λ(x) = −x d
dx log p(x)
or p(x) ∼ exp(−
R λ(x)
x dx)
simple proof

proof
transition probabilities from state k to i are
pSSR(i|k) =

λ(k) qi
g(k−1) + (1 − λ(k))qi if i k
(1 − λ(k))qi if i k
g(k) is the cdf of qi, g(k) =
P
i≤k qi. Observing that
pλ,q(i + 1)
qi+1

1 + λ(i + 1)
qi+1
g(i)

=
pλ,q(i)
qi
we get
pλ,q(i) =
qi
Zλ,q
Y
1j≤i

1 + λ(j)
qj
g(j − 1)
−1
∼
q(i)
Zλ,q
e
−
P
j≤i λ(j)
q(j)
g(j−1)
Zλ,q is the normalisation constant. For uniform priors, taking logs and
going to continuous variables gives the result, λ(x) = −x d
dx log pλ(x).

the driving process determines distribution

special cases λ(x) = −x d
dx log p(x)
• Zipf: slow driving (λ = 1) → p(x) = x−1
• Power-law: constant driving λ(x) = α → p(x) = x−α
• Exponential: λ(x) = βx → p(x) = e−β(x−1)
• Power-law + cut-off: λ(x) = α+βx → p(x) = x−α
e−βx
• Gamma: λ(x) = 1 − α + βx → p(x) = xα−1
e−βx

special cases λ(x) = −x d
dx log p(x)
• Normal: λ(x) = 2βx2
→ p(x) = e−β
2(x−1)2
• Stretched exp: λ(x) = αβ|x|α
→ p(x) = e−β
α(x−1)α
• Log-normal: λ(x) = 1 − β
σ2 + log x
σ2 → 1
xe
−
(log x−β)2
2σ2
• Gompertz: λ(x) = (βeαx
− 1)βx → p(x) = eβx−αeβx
• Weibull: λ(x) = β−α
αxα
+ α − 1 →p(x) =

x
β
α−1
e
−

x
β
α
• Tsallis: λ(x) = βx
1−βx(1−Q) → p(x) = (1−(1−Q)βx)
1
1−Q

driving determines statistics of driven systems
slow → Zipf’s law
constant → power law
extreme driving → prior distribution (uniform)
driving state dependent → any distribution (depends on driving)
(all true for well-behaved priors only)

systems of SSR–nature
• any driven system (with stationary distributions)
• self-organized critical systems
• search
• fragmentation
• propagation of information in language: sentence formation
• sequences of human behavior
• games: go, chess, life ...
• record statistics

the 3 entropies of SSR processes
information production rate: SIT 1 + 1
2 log W
extensive entropy: SEXT S1,1 = −
P
pi log pi
Maxent functional (SD limit): SMEP −
PW
i=2
h
pi log pi
p1
+ (p1 − pi) log

1 − pi
p1
expressions valid for large large N (iterations) and W (states)

do we now have a ”CLT” for fat tailed
distributions?
• criticality: SSR picture applies for some
• self-organized criticality: subset of SSR
• multiplicative processes with constraints: partly applies
• self-reinforcing processes: for some SSR applies

conclusions
• complex systems are driven and show fat tailed distributions
• processes of SSR type abound in nature
• SSR has extremely robust attractors – priors don’t matter
• relaxation is usually SSR
• details of relaxation do not matter – note CLT !
• details of driving + SSR → explain statistics

Statistics of complex systems II:
entropy for complex systems
stefan thurner

with R. Hanel and M. Gell-Mann
RH, ST, Europhysics Letters 93 (2011) 20006
RH, ST, Europhysics Letters 96 (2011) 50003
RH, ST, MGM, PNAS 108 (2011) 6390-6394
RH, ST, MGM, PNAS 109 (2012) 19151-19154
RH, ST, MGM, PNAS 111 (2014) 6905-6910

Why Statistics of Complex Systems ?
Understand macroscopic system behavior on basis of micro-
scopic elements and their interactions → entropy
• Hope: ’thermodynamical’ relations of CS → phase diagrams
• Hope: understand distribution functions: power laws,
stretched exp, ...
• Dream: way to reduce number of parameters → handle CS
• Dream: max entropy principle for CS: predict distributions

What is Entropy?
Entropy has to do with ...
(1) ... quantifying information production of source
(2) ... thermodynamics: for example dU = TdS − pdV
(3) ... statistical inference: given data what is most likely
distribution?
For simple systems all three are related
S = −k
W
X
i=1
pi log pi
This is no longer true for complex, non-ergodic systems!

Information production

Information theory
• Receiver wants to know how correct received message is →
add redundancy
• Adding redundancy → send more info through channel →
info transmission rate reduces
• What is the transmission rate of information ?

Adding redundancy vs. error probability
Message encoded → received decoded
”Hi there” 100100111 → 100100101 ”Hi thor”
with redundancy
100100111 → 100100111
”Hi there” 100100111 → 100100101 ”Hi there”
100100111 → 100100111

Shannon’s first theorem
can encode a message such that you can transmit it error-free
trough a noisy channel, if the capacity of the channel is higher
than the information production rate of the source S
what is S?
S is a property of the source (thing that produces a message)
if find code that can correct for more errors than are produced
→ error free

What is the source? Example 1
random source: produces letters A, B, C, D, E with probability
pA = 0.4, pB = 0.1, pC = 0.2, pD = 0.2, pE = 0.1
experiment AAACDCBDCEAADADACEDAEADCABEDADDCECAAAAD
at what average rate is information produced in this source?
S = −pA ln pA − pB ln pB − pC ln pC − pD ln pD − pE ln pE

Example 2

Example 2
produces letters A, B, C with pA = 9
27, pB = 16
27, pC = 2
27
Successive symbols not independent, probabilities depend on
preceding ones
p(i|j) j
A B C
A 0 4
5
1
5
i B 1
2
1
2 0
C 1
2
2
5
1
10
experiment ABBABABABABABABBBABBBBBABABABABABBBACACABBABBBBABBABACBBBABA
S = −
P
i,j pip(j|i) ln p(j|i)

A source must be ...
• ... stationary
• ... ergodic (objective probabilities same as in experiment)

stationary and ergodic

No no no

Why this funny form, S = −
P
i pi ln pi ?
Appendix 2, Theorem 2
C.E. Shannon, The Bell System Technical Journal 27, 379-423, 623-656,
1948.

Entropy
S[p] =
W
X
i=1
g(pi)
pi ... probability for finding (micro) state i,
P
i pi = 1
W ... number of states
g ... some function. What does it look like?

The Shannon-Khinchin axioms
Measure for the amount of uncertainty S
• SK1: S depends continuously on p
• SK2: S maximum for equi-distribution pi = 1/W
• SK3: S(p1, p2, · · · , pW ) = S(p1, p2, · · · , pW , 0)
• SK4: S(AB) = S(A) + S(B|A)
Theorem (uniqueness theorem)
If SK1- SK4 hold, the only possibility is S[p] = −
PW
i=1 pi ln pi
Proof crucially depends on the use of SK4 and SK1

Interested in complex systems

What are Complex Systems?
• CS are made up from many elements
• These elements are in strong contact with each other
• Elements and combinations of them constrain themselves
As a consequence
• CS are intrinsically non-ergodic: not all states reachable

CS are ...
• evolutionary
• path-dependent
• long-memory
• long-range
• non-ergodic
all of this violates

What is the IT entropy of CS?

Remember Shannon-Khinchin axioms
• SK1: S depends continuously on p → g is continuous
• SK2: S maximal for equi-distribution pi = 1/W → g concave
• SK3: S(p1, p2, · · · , pW ) = S(p1, p2, · · · , pW , 0) → g(0) = 0
• SK4: S(AB) = S(A) + S(B|A)
note: S[p] =
PW
i g(pi). If SK1-SK4 → g(x) = −kx ln x

Shannon-Khinchin axiom 4 is non-sense for CS
SK4 corresponds to ergodic sources
→ SK4 violated for non-ergodic systems
→ nuke SK4

The ‘Complex Systems axioms’
• SK1 holds
• SK2 holds
• SK3 holds
• Sg =
PW
i g(pi) , W 1
Theorem: All systems for which these axioms hold
(1) can be uniquely classified by 2 numbers, c and d
(2) have the entropy
Sc,d =
e
1 − c + cd
W
X
i=1
Γ (1 + d , 1 − c ln pi) −
c
e
#
e · · · Eulerconst.

Γ(a, b) =
R ∞
b dt ta−1
exp(−t)

The argument: generic properties of g
• scaling transformation W → λW: how does entropy change?

Mathematical property I: a scaling law!
lim
W →∞
Sg(Wλ)
Sg(W)
= ... = λ1−c
Define f(z) ≡ limx→0
g(zx)
g(x) with (0 z 1)
Theorem 1: For systems satisfying SK1, SK2, SK3: f(z) = zc
,
0 c ≤ 1

Theorem 1
Let g be a continuous, concave function on [0, 1] with g(0) = 0
and let f(z) = limx→0+ g(zx)/g(x) be continuous, then f is
of the form f(z) = zc
with c ∈ (0, 1].
Proof. note
f(ab) = lim
x→0
g(abx)
g(x)
= lim
x→0
g(abx)
g(bx)
g(bx)
g(x)
= f(a)f(b)
c 1 explicitly violates SK2, c 0 explicitly violates SK3.

Mathematical property II: yet another one !!
lim
W →∞
S(W1+a
)
S(W)Wa(1−c)
= ... = (1 + a)d
Theorem 2: Define hc(a) = limx→0
g(xa+1
)
xacg(x)...

Theorem 2
Let g be as before and f(z) = zc
then hc(a) = (1 + a)d
for d
constant.
Proof. We determine hc(a) again by a similar trick as we have
used for f
hc(a) = limx→0
g(xa+1
)
xacg(x) =
g

(xb
)(a+1
b
−1)+1

(xb)(a+1
b
−1)c
g(xb)
g(xb
)
x(b−1)cg(x)
= hc
a+1
b − 1

hc (b − 1)
for some constant b. By a simple transformation of variables,
a = bb0
− 1, one gets hc(bb0
− 1) = hc(b − 1)hc(b0
− 1). Setting
H(x) = hc(x − 1) one again gets H(bb0
) = H(b)H(b0
). So
H(x) = xd
for some constant d and consequently hc(a) =
(1 + a)d

Summary
CS are non-ergodic systems → SK1-SK3 hold
→ lim
W →∞
Sg(Wλ)
Sg(W)
= λ1−c
0 ≤ c 1
→ lim
W →∞
S(W1+a
)
S(W)Wa(1−c)
= (1 + a)d
d real

What functions S fulfil these 2 scaling laws?
Sc,d = re
W
X
i=1
Γ (1 + d , 1 − c ln pi)−rc

Which distribution maximizes Sc,d ?
pc,d(x) = e
− d
1−c

Wk

B(1+x
r)
1
d

−Wk(B)

r = 1
1−c+cd, B = 1−c
cd exp 1−c
cd

Lambert-W: solution to x = W(x)eW (x)

Universality classes
all SK 1-3 systems are characterized by 2 exponents: (c, d)

Examples
• S1,1 =
P
i g1,1(pi) = −
P
i pi ln pi + 1 (BGS entropy)
• Sq,0 =
P
i gq,0(pi) =
1−
P
i p
q
i
q−1 + 1 (Tsallis entropy)
• S1,d0 =
P
i g1,d(pi) = e
d
P
i Γ (1 + d , 1 − ln pi) − 1
d
• ...

Classification of entropies: order in the zoo
entropy c d
SBG =
P
i pi ln(1/pi) 1 1
• Sq1 =
1−
P
p
q
i
q−1 (q 1) c = q 1 0
• Sκ =
P
i pi(pκ
i − p−κ
i )/(−2κ) (0 κ ≤ 1) c = 1 − κ 0
• Sq1 =
1−
P
p
q
i
q−1 (q 1) 1 0
• Sb =
P
i(1 − e−bpi) + e−
b − 1 (b 0) 1 0
• SE =
P
i pi(1 − e
pi−1
pi ) 1 0
• Sη =
P
i Γ(η+1
η , − ln pi) − piΓ(η+1
η ) (η 0) 1 d = 1/η
• Sγ =
P
i pi ln1/γ
(1/pi) 1 d = 1/γ
• Sβ =
P
i pβ
i ln(1/pi) c = β 1
Sc,d =
P
i erΓ(d + 1, 1 − c ln pi) − cr c d

Associated distribution functions
• p(1,1) → exponentials (Boltzmann distribution) p ∼ e−ax
• p(q,0) → power-laws (q-exponentials) p ∼ 1
(a+x)b
• p(1,d0) → stretched exponentials p ∼ e−axb
• p(c,d) all others → Lambert-W exponentials p ∼ eaW (xb
)
NO OTHER POSSIBILITIES if only SK4 is violated

q-exponentials (powers) Lambert-exponentials
10
0
10
5
10
−30
10
−20
10
−10
10
0
x
p(x)
(b) d=0.025, r=0.9/(1−c)
c=0.2
c=0.4
c=0.6
c=0.8
10
0
10
5
10
−20
10
0
x
p(x)
(c) r=exp(−d/2)/(1−c)
(0.3,−4)
(0.3,−2)
(0.3, 2)
(0.3, 4)
(0.7,−4)
(0.7,−2)
(0.7, 2)
(0.7, 4)

The Lambert-W: a reminder
• solves x = W(x)eW (x)
• inverse of p ln p: [W(p)]
−1
= p ln p
• delayed diff equations ẋ(t) = αx(t − τ) → x(t) = e
1
τ W (ατ)t
Ansatz: x(t) = x0 exp[1
τ f(ατ)t] with f some function

−1 0 1 2
0
1
violates K2
violates K2 Stretched exponentials − asymptotically stable
(c,d)−entropy, d0
Lambert W0
exponentials
q−entropy, 0q1
compact support
of distr. function
BG−entropy
violates K3
(1,0)
(c,0)
(0,0)
d
c
(c,d)−entropy, d0
Lambert W
−1
exponentials

Relaxing ergodicity (kill SK4) opens door to...
• ... order the zoo of entropies through universality classes
• ... understand ubiquity of power laws (and extremely similar
functions)
• ... understand where Tsallis entropy comes from

Thermodynamics?

Thermodynamic entropy is extensive quantity
if not: thermodynamic relations are non-sense: e.g. dU = TdS
extensive entropy means: S(WA+B) = S(WA) + S(WB)
don’t confuse with additive: S(WA.WB) = S(WA) + S(WB)

If we require entropy to be extensive
extensive: S(WA+B) = S(WA) + S(WB) = · · ·
can proof: W(N) = exp
h
d
1−cWk

µ(1 − c)N
1
d
i
c = lim
N→∞
1 − 1/N
W0
(N)
W(N)
d = lim
N→∞
log W

1
N
W
W0
+ c − 1

growth of phasespace volume determines entropy

Examples
• W(N) = 2N
→ (c, d) = (1, 1) → system is BGS-type
• W(N) = Nb
→ (c, d) = (1 − 1
b, 0) → system is Tsallis-type
• W(N) = exp(λNγ
) → (c, d) = (1, 1
γ)
• ...

A note on phasespace volume
imagine tossing coins
1 coin: ↑, ↓ 2 states: W(1) = 2
2 coins: ↑↑, ↑↓, ↓↑,↓↓ 4 states: W(2) = 4
N coins: 2N
states: W(N) = 2N
imagine process: generate sequence of N ↑ and ↓ symbols with
equal probability. BUT after the first ↑, all subsequent symbols
must be ↑
1 coin: ↑, ↓ 2 states: W(1) = 2
2 coins: ↑↑, ↓↑, ↓↓ 3 states: W(2) = 3
3 coins: ↑↑↑, ↓↑↑, ↓↓↑, ↓↓↓ 4 states: W(3) = 4
N coins: N − 1 states: W(N) = N + 1

What does this imply?
• Information Theory: complex source (non-ergodic) →
Sc,d = re
W
X
i=1
Γ (1 + d , 1 − c ln pi) − rc
• Themodynamics: if want extensivity c and d follow from
phasespace volume W(N)

Examples

Example I: super-diffusion: accelerating RWs
0 10 20 30 40 50
−40
−20
0
20
40
N
X
β
free decisions
1 2
3
4
5
6
7
8
9
∆ N ∝ Nβ
β=0.5
0 2 4 6 8 10
x 10
5
−1
−0.5
0
0.5
1
x 10
5
N
x
β
(b)
β=0.5
β=0.6
β=0.7
• up-down decision of walker is followed by [Nβ
]+ steps in
same direction
• k(N) number of u-d decisions to step N → k(N) ∼ N1−β
• possible sequences: W(N) ∼ 2N1−β
→ (c, d) = (1, 1
1−β)

Example II: Join-a-club spin system
• NW growth: new node links to αN(t) random neighbors,
α 1
constant connectency network A (e.g. person joining club)
• each node i has 2 states: si = ±1 ; YES / NO (e.g. opinion)
• each node i has initial (’kinetic’) energy i (e.g. free will)
• interaction Hij = −JAijsisj
• spin-flip of node can occur if node has enough energy for it
(microcanonic)
→ Can show extensive entropy is Tsallis entropy (c, d) = (q, 0),
Sc,d = Sq,0

Example III: XY ferromagnet with transverse
H field
H =
N−1
X
i=1
(1 + γ)σx
i σx
i+1 + (1 − γ)σy
i σy
i+1 + 2λσz
i
|γ| = 1 → Ising ferromagnet
γ = 0 → isotropic XY ferromagnet
0 γ 1 → anisotropic XY ferromagnet
L ... length of block in spin chain N → ∞
→ Can show extensive entropy for the block, (c, 0)-entropy
F. Caruso, C. Tsallis, PRE 78, 021101 2011

Example IV: Black hole entropy
log Wblack−hole ∝ area
→ Can show extensive entropy is (c, d) = (0, 3/2)-entropy
C. Tsallis L.J.L. Cirto, arxiv 1202.2154 [cond-mat.stat-mech]

Statistical inference —
Maximum entropy principle

What is the probability to find a histogram?
Sequence x = (x1, x2, · · · , xN) with xn ∈ {1, 2} (coin tosses)
1q0
1q0
2
1q1
1q0
2 1q0
1q1
2
1q2
1q0
2 2q1
1q1
2 1q0
1q2
2
1q3
1q0
2 3q2
1q1
2 3q1
1q2
2 1q1
1q3
2
1q4
1q0
2 4q3
1q1
2 6q2
1q2
2 4q1
1q3
2 1q0
1q4
2
q1 and q2 ... probabilities to toss 1 (heads) or 2 (tails) in a trial
k = (k1, k2) ... histogram of x

Probability to find a histogram? Binomial
The probability to find the histogram k is
P(k; q) =
N!
k1!k2!
qk1
1 qk2
2
• N ... length of sequence x
• 2 states
• xn ∈ {1, 2}

Probability to find a histogram? Multinomial
The probability to find the histogram k is
P(k; q) =
N!
k1!k2! · · · kW !
| {z }
multiplicity M(k)
qk1
1 qk2
2 · · · q
kW
W
| {z }
probability G(k;q)
• N ... length of sequence x
• W states
• xn ∈ {1, 2, · · · , W}

Two observations
Observation 1: The factorization property
• Multiplicity M(k) does not depend on q
• All dependencies on q are captured by G(k; q)
Factorization
P(k; q) = M(k) G(k; q)
1
N log P(k; q) = 1
N log M(k) + 1
N log G(k; q)

Observation 2: Log of multiplicity is Shannon entropy
1
N log M(k) = 1
N log

N!
k1!k2!···kW !

∼ 1
N log

NN
k
k1
1 k
k2
2 ···k
kW
W

· · · Stirling
= log N − 1
N
PW
i=1 ki log ki
= −
PW
i=1
ki
N log ki
N = −
PW
i=1 pi log pi = SShannon[p]
1
N
log P(k; q) = −
W
X
i=1
pi log(pi)
| {z }
constraint independent
− α
W
X
i=1
pi − β
W
X
i=1
pii
| {z }
constraints

What is the most likely histogram k?
0 =
∂
∂ki
log P(k; q)
for fixed N ⇒
0 =
∂
∂pi
−
W
X
i=1
pi log(pi) −α
W
X
i=1
pi − β
W
X
i=1
pii
!

Crazy !
Some non-ergodic CS can be factorized too!
If factorization P(k; q) = M(k)G(k; q) exists then a MEP
exists with
1
φ(N)
log P(k; q)
| {z }
relative entropy
=
1
φ(N)
log M(k)
| {z }
generalized entropy
+
1
φ(N)
log G(k; q)
| {z }
cross entropy
When does factorization exist? → read RH, ST, MGM
(2014)
Theorem: for processes of type p(xN+1|k; q) (explicitly path-
dependent) factorization exists (SK1!) iff multinomial M(k)
→ becomes deformed multinomial Mu,T (k)

Deformed deformed multinomials → Sc,d
Mu,T (k) ≡
N !u
Q
i NT

ki
N

!u
| {z }
deformed multinomial
, N !u ≡
N
Y
n=1
u(n)
| {z }
deformed factorial
T monotonic with T(0) = 0, T(1) = 1
deformed multinomials lead to trace-form (c, d)-entropies
S[p] =
1
φ(N)
log Mu,T (k) = ... = −
W
X
i=1
Z pi
0
dz Λ(z)
Λ(z) =
1
a

T0
(z)

N
φ(N)
log u(NT(z))

− log λ

Λ(z) does not depend on N → separation of variables with
separation constant ν (derivatives w.r.t. z and N)→

General solution for deformed multinomials →
(c,d)-entropy
Λ(z) = T 0
(z)T (z)ν
−T 0
(1)
T 00(1)+νT 0(1)2
u(N) = λ(Nν)
−1
λ−1 ...q − shifted factorials (ν = 1)
φ(N) = N1+ν
simplest case: T(z) = z Λ(z) = T 0
(z)T (z)ν
−T 0
(1)
T 00(1)+νT 0(1)2 = zν
−1
ν →
S[p] =
PW
i=1
R pi
0
dzΛ(z) = K
1−
PW
i=1 p
Q
i
Q−1
Q ≡ 1 + ν and K constants

This is Sc,d-entropy !!!

Is extensive entropy the same as MEP-entropy?
No! they are NOT the same – but related
Both are of (c, d)-entropy type
(c, d)TD 6= (c, d)MEP

Example: MEP for path-dependent process
imagine a history dependent process of the following kind
P(k|θ) =
W
X
i
p(i|k − ei, θ)P(k − ei|θ) with
p(i|k, θ) =
θi
Z(k)
W
Y
j=i+1
f(kj) with f(x) = λ(x)ν
use simplest priors θi = 1
W , we have u(y) = λyν
and T(y) = y
gives entropy S =
1−
PW
i=1 p
Q
i
Q−1
we get as the prediction from the maximum entropy principle
pi = [1 − ν(1 − Q)(α + βi)]
1
1−Q with i = (i − 1)

Compare with simulation!
10
0
10
1
10
−4
10
−3
10
−2
10
−1
εi
=i
p
i
5000 10000
0
1
2
N
α,β
simul N=1000
N=5000
N=10000
theory N=1000
N=5000
N=10000
α
β
ν=0.25, λ=1.1

Which processes have non-Shannon type MEP?
• Answer: processes whose sample-space reduces as they unfold
Remarkable
• Start with microscopic path-dependent update-equations →
compute entropy → maximise it → distribution at any time in
the future
• Process is non-ergodic, path-dependent , non-stationary –
yet MEP works

The message
complex systems (out-of-equilibrium, non-ergodic, non-
multinomial) break the degeneracy of entropy −
P
i pi log pi
→ entropy has 3 faces
• thermodynamic - face
• information theoretic - face
• maxent entropy - face
Formulas look different for different systems and processes

Example1: Pólya process
a) b)

Example2: Sample space reducing process
9
13
7
5
1)
4)
2) 3)
5) 6)
3 1

The 3 entropies of Pólya and SSR processes
SIT SEXT SMEP
Pólya
1−qj
γ
1
N log N S1− γ
1−qj
,0 −
P
i log pi
SSR 1 + 1
2 log W S1,1 −
PW
i=2
h
pi log pi
p1
+ (p1 − pi) log

1 − pi
p1
i
expressions valid for large N (iterations) and W (states)

• every system has its entropy
• every entropy has three faces

Conclusions I
• complex systems are non-ergodic by nature
• all statistical SK 1-3 systems classified by 2 exponents (c, d)
• single entropy covers all systems:
Sc,d = re
P
i Γ (1 + d , 1 − c ln pi) − rc
• all known entropies of SK1-SK3 systems are special cases
• distribution functions of all systems are Lambert-W expo-
nentials. There are no exceptions!
• phasespace determines entropy (c, d)
• maxent principle: exists for path-dependent processes

Conclusions II
• information theoretic approach leads to (c, d)-entropies
• maxent approach leads to (c, d)-entropies
• extensive and maxentropy are both (c, d)-entropies but NOT
the same

A note on Rényi entropy
• relax Khinchin axiom 4:
S(A + B) = S(A) + S(B|A) → S(A + B) = S(A) + S(B) →
Rényi entropy
• SR = 1
α−1 ln
P
i pα
i violates our S =
P
i g(pi)
but: our above argument also holds for Rényi-type entropies
S = G
W
X
i=1
g(pi)
!
lim
W →∞
S(λW)
S(W)
= lim
R→∞
G

fg(z)
z G−1
(R)

R
= [for G ≡ ln] = 1

The Lambert-W: a reminder
• solves x = W(x)eW (x)
• inverse of p ln p: [W(p)]
−1
= p ln p
• delayed differential equations
ẋ(t) = αx(t − τ) → x(t) = e
1
τ W (ατ)t
Ansatz: x(t) = x0 exp[1
τ f(ατ)t] with f some function

Amount of information production in a process
Markov chain with states A1, A2, ... An
transition probabilities pik
probability for being in state l: Pl =
P
k Pkpkl
if system is in state Ai then we have the scheme

A1 A2 ... An
pi1 pi2 ... pin

Then Hi =
P
k pik log pik is the information obtained when
Markov chain is moving from Ai step to the next

average this over all initial states:
H = −
P
i Pi Hi = −
P
i
P
k Pipik log pik
this is the information production if Markov chain moves one
step ahead

Complex systems and systemic risk

Aim
• See systems as networks
• Discuss SR, resilience, robustness as network
properties
• Detect weak spots in systems
• Understand idea to quantify systemic risk
• Discuss applications in a few of examples:
finance, economy, credit, global, trading, food
supply, global spreading

Why are we interested in SR?
• Complex systems collapse
• When do they?
• Where do they break?
• Can we understand tipping points?
• Can we identify weak spots?
Remember: we see world as networks

What is collapse?
• Systems perform functions
• Functions depend on how networks form and
restructure (co-evolving)
• Collapse is rapid restructuring of networks

Collapse
• Nodes can vanish
• Nodes can change
• Links can vanish / appear
• Links can change
• Relinking can get wrong

Systemic Risk / Resilience / Robustness

What is systemic risk?
• Probability that system can no longer perform its
function(s)
• Size of system-wide damage (often huge)
• SR is a network property
• SR can so-far practically not be quantified
• SR can so-far practically not be managed

SR is necessary to …
• … make systems more resilient
• … make complex systems managable
• … understand what transition means

To understand SR need to know …
… networks and their dynamics

Complex systems are unstable
• most complex systems are stochastic
• statistics of complex systems is statistics of power laws
– many large outliers – outliers are the norm
– non-manageability
• details matter

Breaking complexity
if understand co-evolution: we might break complexity
• by controlling every component tame complexity
à cut off power law tails
• if we forget a detail à might lose control

Example 1: Financial SR
Nodes are banks
Links are financial contracts
It is dynamic and co-evolutionary

The three types of financial risk
• economic risk: investment in business idea does
not pay off
• credit-default risk: you don't get back what you
have lent to others
• systemic risk: system stops functioning due to
local defaults and subsequent (global) cascading

Financial systemic risk
• risk that significant fraction of financial network
defaults
• systemic risk is not the same as credit-default risk
• banks care about credit-default risk
• banks have no means to manage systemic risk
• role of regulator: manage systemic risk
• incentivise banks to reduce SR

Systemic risk created on multi-layer networks
• layer 1: lending–borrowing loans
• layer 2: derivative networks
• layer 3: collateral networks
• layer 4: securities networks
• layer 5: cross-holdings
• layer 6: overlapping pfolios
• layer 7: liquidity: over-night loans
• layer 8: FX transactions

Quantification of SR
• Wanted: systemic risk-value for every financial institution
• Input: network
Google has similar problem: value for importance of web-
pages: A page is important if many important pages point to it
• number of importance à PageRank

Web page is important if many important pages point to it

Institution systemically risky if systemical risky institutions lend to it

Systemic risk factor – DebtRank R
• Is a ‘different Google’ – adapted to context of SR
(S. Battiston et al. 2012)
• superior to: eigenvector centrality, page-rank, Katz rank ...
Why?
• economic value in network that is a affected by node's default
• capitalization/leverage of banks taken into account
• cycles taken into account: no multiple defaults

How to compute it?
Basic idea is to kick one node out of the NW and see
what happens:
• How much damage is caused?
• R is percentage of damage wrt total value
• Repeat for all nodes – one after the other, then in
pairs, triples, …

Systemic risk of nodes
• Input: network of contracts between banks
• Output: all banks i get ‘damage value’ Ri
(% of total damage)

Systemic risk spreads by borrowing

DebtRank Austria Sept 2009
size is not proportional to systemic risk
note: core-periphery structure
(a)

Systemic risk profile
Austria
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
BANK
SYST.
RISK
FACTOR
(a)

Systemic risk profile
Mexico
0 10 20 30 40
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
BANK
SYST.
RISK
FACTOR
(b) combined

How big is the next financial crisis?
Expected Systemic Loss = Sum i pdefault(i) . DebtRank (i)
Expected loss(i) = Sum j pdefault(j) . Loss-given-default(j) . Exposure(i,j)

How big is the next financial crisis?
time
EL
syst
[$/year]
2007 2008 2009 2010 2011 2012 2013
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
11
EL
syst
[$/year]
^MXGV5YUSAC
^VIX
Lehman Brothers collapse
Uncertainty about the rescue of Greece
International alarm over Eurozone crisis
Loss on derivatives of Mexican companies
Subprime crisis
Mexican GDP fell by more than 10%

ESL: Quantify quality of policy
• expected losses per year within country in case of
severe default and NO bailout
• rational decision on bailouts
– allows to compare countries
– allows to compare situation of country over time
• are policy measures working in Spain? in Greece?

Systemic risk of links
SR of banks change with every transaction

All interbank loans: Austria
10
−6
10
−4
10
−2
10
0
10
−8
10
−6
10
−4
10
−2
LOAN SIZE
SYST.
RISK
INCREASE

How to make financial systems better?
i.e. more resilient

Systemic risk is an externality

Management of systemic risk
• Systemic risk is a network property
• Manage systemic risk: re-structure financial networks
• s.t. cascading failure becomes unlikely / impossible

Systemic risk management
=
re-structure networks

Systemic risk elimination
• systemic risk spreads by borrowing from risky agents
• how risky is a transaction?
• ergo: restrict transactions with high systemic risk
à tax those transactions that increase systemic risk

Incentive to change node-linking strategy:
Systemic risk tax
• tax transactions according to their systemic risk contribution
• Banks look for deals with agents with low systemic risk
• ! liability networks re-arrange ! eliminate cascading
• no-one should pay the tax – tax serves as incentive to
• re-structure networks
• size of tax / expected systemic loss of transaction (society is
• neutral)
• if system is risk free: no tax
• credit volume MUST not be reduced by tax

Test the eficacy of tax: ABM
Rebuild system in computer: simulator = agent-based model
Banks
Firms
Households
loans
deposits
consumption
deposits
wages / dividends

The agents
Firms: ask bank for loans
– Firms sell products to households: realize profit/loss
– if surplus – deposit it bank accounts
– Firms are bankrupt if insolvent, or capital is below threshold
– if firm is bankrupt, bank writes off outstanding loans
Banks try to provide firm-loans. If they do not have enough
– approach other banks for interbank loan at interest rate rib
– bankrupt if insolvent or equity capital below zero
– bankruptcy may trigger other bank defaults
Households single aggregated agent: receives cash from rms
– (through firm-loans) and re-distributes it randomly in banks
– (household deposits), and among other firms (consumption)

For comparison: implement Tobin tax
• tax on all transactions regardless of their risk
contribution
• 0.2% of transaction (5% of interest rate)

Comparison of three schemes
• No systemic risk management
• Systemic Risk Tax (SRT)
• Tobin-like tax

Model results: Systemic risk profile
Austria Model
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
BANK
SYST.
RISK
FACTOR
(a)
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
BANK
SYST.
RISK
FACTOR
no tax
tobin tax
systemic risk tax
(b)

Model results: SR of individual loans
Austria Model
10
−6
10
−4
10
−2
10
0
10
−8
10
−6
10
−4
10
−2
LOAN SIZE
10
−4
10
−3
10
−2
10
−1
10
0
10
−4
10
−3
10
−2
10
−1
LOAN SIZE
SYST.
RISK
INCREASE
no tax
tobin tax
systemic risk tax

Model results: Distribution of losses
0 200 400 600 800
0
0.05
0.1
0.15
0.2
TOTAL LOSSES TO BANKS
FREQUENCY no tax
tobin tax
systemic risk tax
(a)
SRT eliminates systemic risk
How?

Model results: Cascading is suppressed
0 5 10 15 20
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
cascade sizes of defaulting banks (C
frequency
no tax
tobin tax
systemic risk tax
(b)

SR elimination at no econ. cost
30 40 50 60 70
0
0.02
0.04
0.06
0.08
0.1
0.12
0.14
TRANSACTION VOLUME IB MARKET
FREQUENCY no tax
tobin tax
systemic risk tax
(c)
Tobin tax reduces risk
by reducing credit volume

Policy testing
Once you have a simulator: can implement
policies and see outcomes
Implement current financial regulation: Basel III

Data for relevant EU banks
European stress testing data 2016 (EBA)
• 51 relevant European banks (49 included in analysis)
• 44 sovereign bond investment categories (36
included)

Asset-Bank à bank exposure NW

Exposure networks
original optimal

Combine it with interbank NW
Combine interbank NW with firm-bank network
to get a ‘complete’ credit network of a nation
then compute the DebtRank again for all banks AND firms

Systemic risk Austrian banks and firms
0.0
0.25
0.50
0.70
Companies banks ranked by DebtRank
DebtRank
Companies
Banks
M K K M M F K L K K L L Q L F M M L N H M K G G M G M K L F D H N L H G M K K K I M K I K
0.0
0.25
0.50
0.70
B
R
A
U
U
N
I
O
N
A
G
Ö
B
B
-
I
n
f
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A
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Companies ranked by DebtRank
DebtRank
M Services K Finance Insurance F Construction L Real estate N Other services
H Logistics G Automobile sector D Energy I Gastronomy Q Health

Message
more than half of the total financial SR
comes from companies

NW of international trade: Data
• Trade networks between all countries are
available
• Compute effect if supply or demand in one
country changes. What are the implications in
the networks?
• Compute primary and secondary effects

Shock spreading in trade NWs
Public administration and defence;...
Real estate activities
Human health and social work activ...
Wholesale trade, except of motor v...
Retail trade, except of motor vehi...
Administrative and support service...
Legal and accounting activities; a...
Insurance, reinsurance and pension...
Manufacture of food products, beve...
Accommodation and food service act...
Manufacture of coke and refined pe...
Financial service activities, exce...
Other service activities
Mining and quarrying
Manufacture of chemicals and chemi...
Telecommunications
Electricity, gas, steam and air co...
Activities auxiliary to financial ...
Land transport and transport via p...
Manufacture of fabricated metal pr...
Crop and animal production, huntin...
Manufacture of motor vehicles, tra...
Architectural and engineering acti...
Manufacture of computer, electroni...
Wholesale and retail trade and rep...
Manufacture of basic metals
Education
Computer programming, consultancy ...
Motion picture, video and televisi...
Construction
time, t [years]
0 2 4 6 8 10
Response
function,

Y
k
(t)

-2
-1.5
-1
-0.5
0
USA 2014
(A)
(B)
Shock,
X
i
-1
-0.5
0
(C)
t = 0y t = 1y t = 6y
Agriculture Mining Manufacturing Electricity Water Construction Trade Transport
Accomodation InformationCommunication Finance Research Administration Other

Secondary effects:Trump’s EU Tax on steel and aluminum

Data
• Production networks within countries become
visible through national VAT data
• Assume production functions
• Assume how firms can substitute for lost
suppliers
• Compute SR index: ESRI for every company

Every company has production function

Systemic relevance / resilience of economy

Systemic core of economy (HUN)

Firms in NACE are not input-similar

How to reconstruct supply chains?

Reconstruction for Austria: communication NW

Results from reconstructed supply NW

Supply networks on product level
Example pork
production

Systemic Risk Index – same idea

Supply chains on product level: monitor

Use a international SC network
(highly incomplete)
What happens in country B if
A firm in country A defaults?
Compute a inter-country SR flow index

International flow of systemic risks

The poorer – the more imported SR

Economy as a collection of dynamical netzworks

Scenario: flood destroys capital in Austria

0 1 2 3 4 5 6 7 8 9 10
Direct losses in percentage of capital stock [%]
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
Cumulative
change
in
GDP-growth
rate
[pp]
2014
2015
2016
Inflection point w. r. t.
economic growth
250-year
event
100-year
event
1,500-year
event
Maximum: threshold at
which natural disaster
becomes a systemic event
Flood-resilience of Austrian economy

slide1_merged.pdf

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