Strong random correlations in

networks of heterogeneous agents

Imre Kondor Parmenides Foundation

Munich, Germany

Talk given at the Workshop „Modeling Financial Systems”,

a satellite of the 2012 Latsis Symposium „Economics on the Move – Trends and Challenges from the Natural Sciences”

ETH, Zürich, September 11-14, 2012

This work forms part of the EU Collaborative project FOC – Forecasting Financial Crises, grant No. 255987, and

the INET project Correlations in Complex Heterogeneous Networks, grant No. 5343

Abstract

Correlations and other collective phenomena in a schematic model of binary agents (voting yes or no, trading or inactive, etc.) are

considered. The agents are placed at the nodes of a network and they collaborate or compete with each other according to a fixed set of

positive or negative links between the nodes. They may also be subject to some external influence equally impacting each of them, and

some random noise. We study this system by running numerical simulations of its stochastic dynamics.

A microscopic state of the system is a vector with binary components, describing the actual state of each agent. The totality of these

vectors span the „phase space” of the system. Under the dynamics the micro-state vector executes a random walk in phase space. At high

noise levels the system has a single attractor with a broad basin of attraction. As the noise level is lowered, the heterogeneous interactions

between the agents come to the fore and will divide phase space into typically several basins of attraction. For small systems sizes it is

possible to completely map out the attractors and the low lying states belonging to their basins. This map will define a graph in phase

space, and we study the random walk of the system along this graph. At low noise levels the system will typically spend a long period in

the immediate vicinity of one of the attractors until it finds a low saddle point along which it escapes, only to be trapped in the basin of

the next attractor. The dynamics of the system will thus be reminiscent of the punctuated equilibrium type evolution of biosystems – or

human societies.

It is clear that evolution in such a landscape will depend on the initial condition, but the landscape itself will also be extremely sensitive

to details of the concrete distribution of interactions, as well as to small shifts in the values of the noise or the external field.

The evolution is so slow that one can meaningfully speak of some quasi-equilibrium while the system is exploring the vicinity of one or

the other attractor. Performing measurements of correlations in such a quasi-equilibrium state we find that (due to the heterogeneous

nature of the system) these correlations are random both as to their sign and absolute value, but on average they fall off very slowly with

distance. This means that the system is essentially non-local, small changes at one end may have a strong impact at the other, or small

changes in the boundary conditions may influence the agents even deep inside. These strong, random correlations tend to organize a large

fraction of the agents into strongly correlated clusters that act together and behave as if they were occupying a complete graph where

every agent interacts with every other one.

If we think about this model as a distant metaphore of economic agents or bank networks, the systemic risk implications of this tendency

are clear: any impact on even a single agent will spread, in an unforeseeable manner, to the whole system via the strong random

correlations.

This is a report on work in (slow) progress

Recent collaborators from Eötvös University,

Budapest:

István Csabai

Gábor Papp

Mones Enys

Gábor Czimbalmos

Máté Csaba Sándor

Contents

• A schematic heterogeneous agent model (of

the Libor fixing collusion?)

• The interaction network and the stochastic

dynamics

• Long range correlations

The model

This model appears in the theory of a class

of random magnets called spin glasses

There the couplings and external fields are

given, and one tries to determine the averages

(along a simulation trajectory, or over the

canonical ensemble) of some aggregate

quantities like the magnetization, the total

energy, etc., or some local quantities like the

local magnetizations or correlations. This is

the setup of statistical physics.

• In principle, al these averages depend on the

concrete realization of the couplings. In

statistical physics, we are interested in

macroscopic systems (N very large), and it

can be shown that the averages of extensive

quantities self-average, i.e. they are the

same for every realization of the couplings

with probability 1. Therefore, in almost all

works on spin glasses one averages over the

couplings.

• Note, however, that local quantities do not

self-average.

Furthermore:

• In statphys one is mainly interested in

equilibrium. These heterogeneous systems

reach equilibrium extremely slowly, and a

huge effort has been invested in developing

algorithms that ensure that simulations

actually reach equilibrium.

• The large N limit and equilibrium simplify

the treatment tremendously.

Our goal

Our goal is to learn what such a schematic

model can teach us about the stability of

multiagent models: about their sensitivity to

details of the geometry of interactions, to

changes in boundary conditions and in

control parameter values.

In order to get a feeling about

these features

• We regard the MC dynamics as the history

of this mini-society of agents, and do not

necessarily try to reach thermal equilibrium.

However, the slow dynamics allows us to

do time averaging over reasonably long-

lived quasi-equilibrium states.

• As we renounce the simplifying effects of

the thermodynamic limit and thermal

equilibrium, we have to face a wild variety

of behaviours. Yet, some typical features do

emerge.

What we find

• At low temperatures ergodicity is severely

broken, there are several attractors whose

number, basins and properties are determined

by the whole arrangement of couplings and

fields. The random walk on such a landscape

leads to „punctuated equilibrium”.

• Even relatively large rearrangements of the

couplings can leave the phase space structure

unchanged, while sometimes small changes

reorganize the structure completely.

• Initial conditions have a strong effect on

evolution in a given phase space landscape,

but this landscape depends sensitively on

boundary conditions and control parameters

(external fields, temperature); there is a

chaotic response to small changes in all these

factors.

• As a rule, long range random correlations are

generated between the agents, large, strongly

correlated clusters emerge that act as if

agents were placed on a complete graph.

Think of the Libor fixing collusion

• The agents are the officials charged with

submitting their bank’s estimate of the rate at

which they could finance themselves.

• The binary choice they make is whether to

join the collusion or opt out.

• The couplings are their relationships to their

colleagues at the other banks.

• The external field can be the percieved

pressure from their central banks.

SOME ILLUSTRATIONS

Phase space structure for a given

interaction matrix • N=6, complete graph

– Number of positive coupling: 7

– Number of negative coupling : 8

– All triangles in the system: 20

– Frustrated triangles: 10

– J:

Phasespace Landscape (with the lowest 4 energy states)

Same structure for very different

interaction matrix • N=6, complete graph

– Number of positive coupling: 6

– Number of negative coupling : 9

– All triangles in the system: 20

– Frustrated triangles: 10

– J:

Phasespace Landscape (with the lowest 4 energy states)

Different structure for almost

identical interaction matrix • N=6, complete graph

– Number of positive coupling: 8

– Number of negative coupling : 7

– All triangles in the system: 20

– Frustrated triangles: 12

– J:

Phasespace Landscape (with the lowest 4 energy states)

Random walk in phase space

Tav=1

Tav=104 Tav=1000

Tav=100

Tav=10

Tav=107

Slow dynamics, degree of order depends on

observation time

Same, with signs

Same, with absolute values of correlations

measured from an other point

Distribution of correlations in 3d (log scale!)

Sorted distribution of correlations, two samples on a

complete graph of size N= 2048, at T=0.4, averaging

over all microstates

Dependence on boundary conditions (free,

periodic, random) for two reference sites, 2d

Dependence on boundary conditions (free,

periodic, random) for two reference sites, 3d

Absolute values of correlations averaged over 500

samples, free, periodic and random boundary conditions

Absolute values of correlations in 2d, resp. 3d, averaged

over 500 samples, free, periodic and random boundary

conditions

Effect of fixing a single agent out

of 10 000, T=0.5

Effect of fixing a single agent out

of 10 000, T=1.25

Conclusions

• Long range correlations make the system

non-local, „more than the sum of its parts”.

• Non-locality also implies irreducibility: the

system depends on many small details.

• Aggregate quantities may behave more

regularly

• A large part of the system tends to organize

itself as if on the complete graph.

• Implications for systemic risk.

THANK YOU!