DECISION MAKING IN A CONDOMINIUM AN ISING-LIKE SOCIOPHYSICAL - PowerPoint PPT Presentation

DECISION MAKING IN A CONDOMINIUM – AN ISING-LIKE SOCIOPHYSICAL SYSTEM Márta Jávor Tamás Geszti Roland Eötvös University

IN MEMORIAM GEORGE MARX (1927 – 2002) 2

Purpose • to show: how we can use a simple model of statistical physics – the Ising model – and a statistical method – Monte Carlo - to study a complex sociological system 3

Sociophysics adaptation of statistical physics methods to sociological systems • Serge Galam „ father of sociophysics ”: COLLECTIVE DECISION TAKING is like MAGNETIC ORDERING 4

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Two men dissussing with one another are similar to two interacting spins 6

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Simple model of ferromagnetism The game is to learn about human community from the Ising model. Ernst Ising 8

ISING MODEL energy coupling coefficient external magnetic field 𝐹 = − 𝐾 𝑗𝑘 𝑇 𝑗 𝑇 𝑘 − 𝐼 𝑇 𝑗 𝑗 𝑗,𝑘 The coupling is SYMMETRICAL! spin magnetization 9

Only two possibilities 10

Without external magnetic field 𝐼 = 0 𝐹 = − 𝐾 𝑗𝑘 𝑇 𝑗 𝑇 𝑘 𝑗,𝑘 11

Simulation method • MC – Monte Carlo method (using computer-generated random numbers) – using the Metropolis algorithm 12

Metropolis algorithm • run over all spins on a lattice, and for each spin – propose a change from the current state: flip -1 to +1 or +1 to -1, and calculate the acceptance probability – decide to accept or reject this change by using a random number between 0 and 1: • accept the change if random number ≤ acceptance probability • reject the change if random number › acceptance probability 13

Acceptance probability: Evaluate energy change ∆E accompanying the flip; if if 14

Decision • generate a random number: r ⋲ (0,1) Accept change if r‹W, Reject otherwise 15

EXCEL Visual Basic macro 16

2D 25×25 lattice 17

Neighbours i-1;j i;j-1 i,j i;j+1 i+1;j 18

Periodic boundary conditions 19

Periodic boundary conditions 25 × 25 lattice each cell has 4 neighbours 20

MADRID 21

MOSCOW „BUBLIK” 22

TORUS 23

Random initial configuration 1 2 2 1 2 1 1 1 1 2 1 2 2 2 2 1 1 1 1 1 1 1 2 2 1 1 2 2 2 2 1 1 1 1 1 1 2 1 2 2 1 1 2 2 2 2 1 2 2 2 2 1 1 2 1 1 1 1 1 2 2 1 1 1 1 2 2 1 1 2 2 2 1 2 2 1 2 2 2 1 1 2 1 1 2 2 1 1 1 2 2 1 2 1 2 2 2 2 1 1 1 2 2 1 1 1 2 1 2 2 2 1 1 1 1 2 2 2 2 1 1 1 2 1 1 1 2 2 1 1 2 1 1 1 2 2 2 2 2 1 2 1 2 1 2 1 2 2 1 2 1 2 2 2 2 2 1 1 1 2 1 1 2 1 2 2 2 1 2 1 1 2 2 2 1 2 1 2 1 2 1 2 2 1 2 2 1 2 2 2 1 2 1 1 2 2 2 2 1 1 1 2 2 1 2 1 1 1 1 1 2 2 1 2 1 2 2 2 1 2 1 1 2 2 2 1 2 2 2 1 1 1 1 1 2 2 2 1 2 2 2 2 2 1 1 2 2 2 1 1 2 2 2 1 2 2 2 1 2 2 2 1 2 2 1 2 2 1 1 2 1 1 2 2 2 1 1 1 1 2 1 1 1 1 2 1 1 1 2 1 1 2 1 2 1 2 1 2 2 2 2 2 2 2 1 2 1 2 2 2 2 2 2 1 2 2 2 2 1 2 1 1 1 1 1 2 1 1 2 2 2 1 2 2 1 2 2 2 2 2 2 1 2 1 2 2 1 1 1 1 2 1 2 2 2 2 2 2 2 2 1 1 1 1 1 2 2 2 2 1 2 2 2 2 2 2 2 2 1 1 2 1 2 1 2 2 1 2 1 2 2 2 2 2 1 2 1 2 1 1 2 1 1 1 1 2 2 2 2 1 1 1 2 1 1 1 1 1 1 2 1 2 2 1 1 2 2 1 2 1 2 1 2 2 2 2 2 1 1 2 1 2 2 1 1 1 1 1 2 1 1 2 2 1 1 1 2 2 1 1 2 1 2 2 2 1 1 1 2 1 2 1 1 2 1 2 1 1 1 1 1 2 2 1 2 2 1 1 1 1 1 2 1 2 1 1 1 2 2 2 1 1 1 1 2 2 1 2 1 1 1 2 1 2 1 1 1 2 2 1 2 2 2 1 2 2 1 1 1 1 2 1 2 1 2 2 1 2 2 2 1 1 2 1 1 1 2 1 1 2 2 1 1 2 2 2 1 1 1 1 2 2 2 2 2 2 1 1 2 1 1 1 2 1 1 2 2 1 2 1 1 2 1 2 2 2 2 2 1 1 1 1 2 2 1 2 1 1 1 2 1 1 1 1 1 2 1 1 1 1 2 2 1 1 1 2 1 1 1 2 2 2 2 1 1 24

EXCEL worksheet 25

100-step iteration result M/N 1.0000 M/N 0.5000 0.0000 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 kT/J -0.5000 -1.0000 26

Statistics – EXCEL worksheet 27

M/N depending on parameters 28

Decision in the condominium 29

One example • There is a hole on the roof – two possibilities: • to repair • not repair, resolve otherwise 30

Factors influencing decision • psychology – individual background – openness towards other people • Sociology – connections between neighbours – knowledge about the problem to decide • effects from environment unrelated to the decision 31

Everybody having his/her own idea J=0 32

Discussion J≠0 33

Types of coupling symmetrical asymmetrical J›0 J‹0 non-Ising! 34

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Symmetrical coupling ; J›0 36

magnetizations from seven 100-step simulations, as a function of kT/J 37

magnetizations from seven 100-step simulations, as a function of kT/J 2.27 ONSAGER theory 38

Analysis of model ’s results about the decision • main parameter of the model is the ratio 𝑄 = 𝑙𝑈 𝐾 • if this parameter is low (T is low) – the neighbours pay attention to each other • if this parameter is hight (T is high) – the neighbours pay attention to many other things

EXTERNAL MAGNETIC FIELD ma y correspond to: • a strong argument in favour of some decision or a dominant personality • 40

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Magnetization ~ achieving majority („ polarization ”) , even in the presence of strong noise („ high temperature ”)

Beyond the physical model: asymmetric coupling 44

Beyond the physical model: asymmetric coupling i-1;j i;j-1 i,j i;j+1 upper & left i+1;j neighbours J 45

Beyond the physical model: asymmetric coupling i-1;j i;j-1 i,j i;j+1 upper & lower & left right i+1;j neighbours neighbours qJ J 46

Beyond the physical model: asymmetric coupling i-1;j i;j-1 i,j i;j+1 upper & lower & left right i+1;j neighbours neighbours qJ J 47

Beyond the physical model: asymmetric coupling i-1;j i;j-1 i,j i;j+1 upper & lower & left right i+1;j neighbours neighbours qJ J 48

Beyond the physical model: asymmetric coupling i-1;j i;j-1 i,j i;j+1 upper & lower & left right i+1;j neighbours neighbours qJ J 49

q=0.4 kT/J=1.5 50

q=0.2 kT/J=1.0 51

Conclusion: symmetrical coupling: asymmetrical coupling: the decision is final the decision is temporary 52

SUMMARY • Simple physical models may reveal a lot about complex social problems • Some common physical features of the model (e.g. symmetric coupling) should be given up to get closer to social reality • Numerical modeling using simple tools (excel) provide good insight for high-school teaching 53

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Beyond the physical model: asymmetric coupling i-1;j i;j-1 i,j i;j+1 upper & lower & left right i+1;j neighbours neighbours qJ J 55