Scale Invariance in Complex Systems

Geoff Rodgers

School of Information Systems, Computing and Mathematics

Plan

1. Physics – equilibrium statistical mechanics.

2. Self-organised criticality.

3. Networks.

4. Financial systems.

5. Conclusions.

Physics

Consider a classical system, like an Ising model of a magnet. This has a Hamiltonian

H = - J <i,j> Si Sj

Consider it in equilibrium at temperature T in dimension d >2.

Consider the pair correlation function

F(r) = < (Si - <S>) (Sj - <S>) >,

where r = | i - j |. Simple matter to show that

F(r) = r -(d-2+) g(r/)

where the correlation length behaves like

= | T - Tc | -

So at the critical temperature T=Tc, the

correlation length diverges to infinity, and

the pair correlation function is power-law

F(r) ~ r -(d-2+)

i.e. scale invariant.

• In equilibrium statistical mechanics – power laws only occur at the critical point.

• In non-equilibrium statistical mechanics power-laws are also quite rare.

• Other physical quantities for this system have a scaling form, and are power-law at criticality.

• Also, because

F(r) = r -(d-2+) g(r | T - T

c | ),

plots of F(r)r d-2+

against r | T - Tc | ,for

different temperatures, lie on a universal curve g.

• Other systems have the same scaling forms, with different scaling functions g(x) and different exponents.

• Systems with the same exponents have identical physics and are said to be in the same universality class.

• Scale free, scale invariant, power-law behaviour only occurs at criticality.

Self-organised criticality

Simple models of avalanches

Have a square lattice with grains of sand on each site. At each time step, add a grain to the central site.

Then allow toppling and hence avalanches to occur.

• Site (i,j) has z i j grains on it

• For all z i j 4 toppling occurs

z i j z

i j – 4

z i1 j

z i 1 j

+1

z i j 1

z i j 1

+1

Lattice is finite, so grains falls off – this is

the avalanche.

1 1 3

1 2 3

2 2 2

1 1 3

1 3 3

2 2 2

1 1 3

1 4 3

2 2 2

1 2 3

2 0 4

2 3 2

1 2 4

2 1 0

2 3 3

1 3 0

2 1 1

2 3 3

Three time steps; Avalanche sizes 0, 0 and 3.

Time step 1 Time step 2 Time step 3

Time step 4

1 3 0

2 2 1

2 3 3

Measure the avalanches over a long

period of time on a large lattice – the distribution

of avalanches sizes is power-law

N(x) ~ x – 3/2

Such models are called self-organised

critical models – there is no tunable

parameter such as temperature to make

them critical.

• At first models such as this were believed to be robust – with powerful toppling kinetics that made the details of the lattice, toppling rule etc.. irrelevant, and drove the model to a large attractor state with power-law distributions.

• Later became apparent that introducing physical characteristics to the kinetics – for instance, friction, randomness in the toppling, anisotropy, momentum, inhomogeneous grains – destroyed the power-laws.

However, a lot of experimental evidence

suggests that spatial and temporal

correlations in avalanche systems are

power-law. But these real experimental

systems obviously have anisotropies,

inertia, friction etc…

Networks

A lot of networks in economic,

technological and social systems have

been found to have a power-law degree

distribution. That is, the number of

vertices N(m) with m edges is given by

N(m) ~ m -

Examples include

• Web graph

• Telephone call graph

• E-mail graph

• Citation graph

• Co-authorship graph

and many others

Web-graph

• Vertices are web pages

• Edges are html links

• Degree distribution is power law

• These graphs are all grown, i.e. vertices and edges added over time.

• Imagine a model system in which we add a new vertex at each time step.

• Connect the new vertex to an existing vertex of degree k with rate proportional to k.

Total degree of network = 18, 10 verticesConnect new vertex number 11 to

vertex 1 with probability 5/18vertex 2 with probability 3/18vertex 7 with probability 3/18all others, probability 1/18 each

1

2

3

4

5

7

9

8

10

6

This network is completely solvable

analytically – the number of vertices of

degree k at time t, nk(t), obeys the

differential equation

where M(t) = knk(t) is the total degree of the

network.

k1 1)1(

)(

1)(

kkn

knk

tMdt

tk

dn

Simple to show that as t

nk(t) ~ k-3 t

and the behavior is power-law. However, adding other ingredients – for Instance aging, heterogeneities and probabilistic attachment - destroys the power-law.

Financial Systems

Intra-day Returns

Empirical analyses of financial price data show that the price of different assets deviate from Gaussian, which would be expected if the agents were acting independently.

Anomalously large oscillations are observed, which are distributed according to an exponentially truncated power law

x-

exp { -x/x0 }

where takes a value between 1.4 and 1.6 for a number of market indices and stocks.

If the price at time t, is p(t), then normally measure the return at time t defined by

Z(t) = p(t+t)-p(t)

for small t.

S&P 500 price changes, with Gaussian and Levy stable distribution for comparison.

Physically motivated models assume that

this behaviour is caused by co-operative

behaviour of the agents, and in particular

by a phenomenon called herding or

crowding.

This is the traditional explanation provided

by economists.

Models used to explain this phenomena are

based on either

(i) Kinetics of group aggregation and

dissolution

(ii) Processes on social networks

(iii) The kinetics of the order book

(iv) Multiplicative random exchange processes.

All these models give rise to power laws with

varying degrees of generality.

e.g. Kinetics of the order book

• A limit order to sell (or buy) is an instruction to sell (or buy) a share if its price rises above (falls below) the execution price of the limit order.

• A market order is an instruction to immediately sell or buy at the currently available price.

Simple modelAt each time step, with equal probability, place • a limit order to buy• a limit order to sell• a market order to buy• a market order to sellEach new order is placed randomly within ± of the current price.

Kinetics of the Order Book

Price

Current Price

Sell limit ordersBuy limit orders

$

The probability of a price fluctuation of

size x, is given by

x-

with 2.0.

How to make sense of it all?

Traditionally physicists search for the necessary and sufficient ingredients for a phenomena to occur.

Then seek to define universality classes – classes of behaviour as a function of the ingredients.

Of course, systems considered in this

talk may be too diverse to make this

possible.

Have a go anyway.

• Power laws seem to be ubiquitous in a wide range of physical, technological, social and financial systems.

• Traditional models, be they either equilibrium or non equilibrium, only give rise to power laws in rather special circumstances, such as at a critical point.

The models considered in this talk were designed to understand empirical measurements of power-law distributions. The models themselves were • kinetic. They can be characterised or modeled by

a kinetic process.• growing or driven; at each time step something is

added.• random – not deterministic.

Thus a growing or driven random kinetic process might be regarded as necessary for power-laws distribution.

Although these models exhibit power laws for a wide range of some parameters, the power-laws are destroyed by the addition of some ingredients.

• Unfortunately, these ingredients, such as(i) Inertia in sand piles(ii) Randomness in the www or collaboration graph(iii) Strategy scoring in the stock market

are all present in the real systems.

Hence the question of why scale free behaviour is observed is such a wide range of systems still remains open.

Seminar next week

• Networks: Structure and Transport.

• Bosa Tadic, Institute Josef Stefan, Ljubljana.

• Wednesday 4pm, Maths 128.