This document provides an overview of sociophysics and mathematical modeling approaches to understanding social behavior. It discusses how physical models can be applied to social systems by treating individuals as "agents" that interact. Specific models summarized include Schelling's model of ethnic segregation, a strike phenomena model using an Ising model, and a bottom-up voting model using majority rule. The document also outlines key elements of sociophysical models like agents, groups, interactions, and states. Overall it aims to demonstrate how quantitative laws seen in collective human behavior can provide insights into social dynamics.

1. MATHEMATICAL MODELLING
Sociophysics’ Modelling
Bhavin Tandel
Dept. of Mathematics

2. Outline
•Introduction
•Elements of Society
•Physical approach to social behaviour
•Interaction
•Schelling Model
•Strike Phenomena
•Bottom up voting Scheme

3. What is Sociophysics’ ?
Do human behaves like atom ?
Transferring ideas and concept of Physics to understand social behaviour.
Sum of activities of searching for fundamental laws and principles that characterize human
behaviour.
Modelling an individual is really complex.
Quantitative laws in the collective properties of a large number of people.
Different models of sociophysics: Opinion Dynamics, voting in bottom-up hierarchical system,
Decision making, cultures and language evolution, traffic dynamics, Terrorism spreading.

4. Elements of Society
Agents
Groups
Interactions
Classes
Neighbourhood(von Neumann and Moore)
States: Collective vs Individual
North
East
South
West
North
North-east
East
South-east
South
South-west
West
North-west

5. Physical approach to social behaviour
Social system involve cooperative behaviour of number of people.
Group behave like one collective individual.
Model: individual has to make choice ‘yes’ or ‘no’. (minimum pair individual conflict)
Polarization or compromise.
Rationale behind model: Ising Ferromagnetic model
Spin can be 𝜎 = +1 𝑜𝑟 − 1
E is total energy of the system, J is the nearest spin-spin
interaction energy, H is external magnetic field. σ=+1 or -1.

6. Interaction
Pair of person i and j interacts.
(i) 𝑐𝑖 = 𝑐𝑗 = +1, (ii)𝑐𝑖 = 𝑐𝑗 = −1, (iii) 𝑐𝑖 = −𝑐𝑗 = +1, (iv) 𝑐𝑖 = −𝑐𝑗= −1
They agree in configuration (i) & (ii) and disagree in (iii) & (iv).
𝐺𝑟𝑜𝑢𝑝 𝑐𝑜𝑛𝑓𝑙𝑖𝑐𝑡, 𝐺𝐼 ≡ 𝐼 <𝑖,𝑗> 𝐶𝑖 𝐶𝑗
Majority rule
Starting configuration:
50% random ‘on’
Rule:
‘on’ if 5 or more Moore neighbours
and self are ‘on’,
‘off’ if 5 or more Moore
neighbours and self are ‘off’

7. Schelling Model
•Study of ethnic segregation in USA in 1971.
•Schelling supposed that people had a threshold of tolerance of other ethnic
groups based on the neighboured people.
40% Threshold
HAPPY

8. Strike Phenomena
To Work To strike
Individual Production 𝜇𝑖 = 2( 𝜇𝑖 −
1
2
), 𝜇𝑖 = 1 (working) , 𝜇𝑖 =-1 (not working)
Two agents i& j interacts with coupling 𝐽𝑖,𝑗
Depending upon synchronization, Dissatisfaction: −𝐽𝑖,𝑗 𝜇𝑖 𝜇 𝑗
External field, 𝐻 = 𝑊 − 𝐸, Dissatisfaction: −𝐻𝜇𝑖
(W = actual wage and E = minimum expected salary)
Collective dissatisfaction function, 𝓗 = − 𝒊,𝒋 𝑱𝒊,𝒋 𝝁𝒊 𝝁𝒋 − 𝑯 𝒊=𝟏
𝑵
𝝁𝒊
OR

9. Dissatisfaction function F versus M for K < K
The limit of metastability for H < 0. F has only one minimum.
M versus H for K < Kc. Arrows show the direction
of change for H. Dashed lines denote metastable states.
Dot-dash lines show possible jumps from metastable to stable
states.
A metastable case with H < 0. The dash arrow
indicates the eventual jump into a stable strike state driven
by some external action

10. Insight obtained
(i) Striking versus working state: Amplitude of strike will be identical to that of working.
(ii) Avoiding strike cheaper than to put a firm back to work:
(iii) Disordered state for K > Kc, Average individual production = 0 Then,
If
H > 0
No Strike
If
H < 0
Strike can be avoided if in
metastable state
If
H < 0
StrikeWork TrueFalse

11. Bottom up Voting in Hierarchical systems
Rule: Local Majority Rule
Population: A & B
Group size: r
Outcome of groups: A with P1 = R(P0)
B with (1 – P1)

12. Different group size
• Group of size 3(Odd) : 𝑃𝑛+1 = 𝑝 𝑛
3 + 3𝑝 𝑛
2(1 − 𝑝 𝑛)

13. Including power inertia
Group with size 4 (even): 2A – 2B situation
If tied then outcome favours in B.
For A to be elected at level n+1
𝑃𝑛+1 = 𝑝 𝑛
4
+ 4𝑝 𝑛
3
1 − 𝑝 𝑛
For B to be elected at level n+1
1 − 𝑃𝑛+1 = 𝑝 𝑛
4 + 4𝑝 𝑛
3 1 − 𝑝 𝑛 + 2𝑝 𝑛
2 1 − 𝑝 𝑛
2 Bias in favour of B

14. Insight obtained
Essential and surprising mechanism of majority rule voting.
Empirical difficulty in changing leaderships in well established institution.
Ex. Sudden and quick collapse of eastern European communist parties.( one reason was end of
the Soviet army threat) : can be result of long and punctual phenomena internally.
Successfully predicted the French Referendum case: first proposition of European constitute.
Either parliamentary procedure or national referendum. In France, president Jacques Chirac
decided to hold referendum. Majority ‘yes’

15. References
•Sociophysics: A review of Galam models, Serge Galam
•Rational Group Decision Making A Random field Ising model at T = 0, Serge Galam
•Econophysics and Sociophysics (Edited by B.K. Chakrabarti, A. Chakraborti, A Chatterjee)
Thank You !!!