From Randomness to Order George Barmpalias

Victoria University of Wellington Joint with Elwes (Leeds) and Lewis-Pye (LSE)

Thanks to the Chinese Acad. Sci. and the NFSC

Populations

• Humans, other species, economies, neurones

• Particles in space, under pressure, temperature

• Electrons with positive/negative spin interacting

• Spread of viruses, technology, innovation

• Social networks, racial segregation, opinions

Conformism

Homogeneity

Segregation

Convention

Conformism

Homogeneity

Segregation

Convention

Εμπεδοκλ�ς

People are like liquids

Water to wine more nearly is allied, but will not mix with oil.

Ising model

• Models ferromagnetism

• Phase transitions with temperature

• 1D case no phase transitions

• same not true in higher dimensions

• 800 papers on the Ising model published annually

• Statistical mechanics, neural networks

• Protein folding, biological membranes

• Social behaviour, networks

Show them a simulation

Meet Thomas Schelling

Arms and influence Strategy of conflict

Micromotives and macrobehaviour

2005 Nobel memorial prize !

“Schelling analyzed how racially mixed societies and neighborhoods can suddenly become segregated... A modest preference for not forming part of a minority in one’s neighborhood, but not necessarily favoring dominance of one’s own race, can cause small microshocks to have drastic consequences at the macro level.” !Royal Swedish Academy of Sciences, 2005

Marshall plan, White house, Exec. office of the president 1948-53

Foreign affairs, national security, nuclear strategy

Schelling models

Problem !

Given the parameters, predict: !

Extent of segregation

Expected time of process

Analyse the process

Parameters !

Population n Neighbourhood radius w

Intolerance τ Distribution ρ

!Dimensions

Open/Closed dynamics Perturbations or not

Archetype of agent based modelling in Economics

“The absurd simplicity of the model is also its main fascination” - Brian Hayes, American Scientist

What have we done?

An Analysis of One-Dimensional Schelling Segregation Brandt, Immorlica, Kamath, Kleinberg, STOC 2012

Digital morphogenesis via Schelling segregation Barmpalias, Elwes, Lewis, 2013

Minority population in the Schelling model Barmpalias, Elwes, Lewis, 2014

Tipping points in Schelling segregation Barmpalias, Elwes, Lewis, 2014

Unperturbed Schelling segregation in two or three dimensions Barmpalias, Elwes, Lewis, 2014

Swapping process often leads to segregation

As a dynamical system Irregular Markov chain Not reversible, doesn’t satisfy “detailed balance” It has many stationary distributions (state explosion) Most introduce noise to overcome these problems Decisions detrimental to their utility (with small probability)

We study the unperturbed model in all dimensions

Reveal phase transitions (intolerance)

Increased tolerance leads to increased segregation11

Complete characterization of 1D

Brandt, Immorlica, Kamath, and Kleinberg dealt with τ=ρ=0.5

We dealt with the other values of τ, concluding that

Population 1M, w = 3K and τ = 0.485, 0.49, 0.495, 0.5. About 130K swaps

Increased tolerance leads to increased segregation when τ is in (κ0,0.5)

Population 1M, w = 3K and τ = 0.485, 0.49, 0.495, 0.5. About 130K swaps

Increased tolerance leads to increased segregation when τ is in (κ0,0.5)

What is κ0 ?

Population 1M, w = 3K and τ = 0.485, 0.49, 0.495, 0.5. About 130K swaps

Increased tolerance leads to increased segregation when τ is in (κ0,0.5)

What is κ0 ?

Stable intervals / unhappy nodes

B(w, 2wτ) and B(2w, 2w(1-τ))

Tails of Binomial distributions

Methods and Tools

Combinatorial arguments

Discrete geometry

Martingale arguments

Fluid limits: discrete to continuous

Wormald Diff. Equation method

Concentration inequalities

Examples…

Threshold for the 2D model:

The case when ρ is not 0.5

Infected area (when ρ is not 0.5)

If τ>0.5, with probability one the process arrives to dormant state or complete segregation.

Infected area (when ρ is not 0.5)

If τ>0.5, with probability one the process arrives to dormant state or complete segregation.

Infected area dynamics (when ρ is not 0.5)

1. An Analysis of One-Dimensional Schelling Segregation Brandt, Immorlica, Kamath, Kleinberg, STOC 2012

2. Digital morphogenesis via Schelling segregation Barmpalias, Elwes, Lewis, 2013

3. Minority population in the Schelling model of segregation Barmpalias, Elwes, Lewis, 2014

4. Tipping points in Schelling segregation Barmpalias, Elwes, Lewis, 2014

5. Unperturbed Schelling segregation in two or three dimensions Barmpalias, Elwes, Lewis, 2014

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