Statistical Physics & Complex Systems

I. Introduction [1h30]

   1. What are complex systems?
           → Examples in nature, social sciences, etc.

   2. Modelling complex systems
           → Statistical (descriptive) models & physical (generative) models

   3. Narrative of the syllabus

II.  Interactions, instabilities & collective effects [6h]

   1. Curie-Weiss and the Random Ising Field Model paradigms
           → The Curie-Weiss paradigm: self-fulfilling prophecies, hysteresis
           → The Random Ising Field Model paradigm: crises & sudden opinion shifts
 
   2. Flocking and collective motion 
           → Vicsek model from microscopic to field theory for flocking
           → 1.0.1 statistical methods: maximum likelihood & baysian estimations
           → Generative model vs statistical model 

   3. Wealth exchanges 1h30 JRF
           → Kinetic exchange model 
           → Altruistic vs selfish society 

III.  Agent-based models [4h30]

   1. The Schelling model 
           → Archetypal agent-based model in the social sciences
           → Phase diagram via analogy with spin systems & simulations
           → Markov chain description & analytical results 

   2. Grauwin et al. statphys formulation of the Schelling model
           → Local versus global detailed balance
           → Segregated steady state:  "Maxwell construction" in social science context

   3. “Agentivity” model
           → Mathematical framework
           → Agentic equilibrium
           → Examples & link with other equilibria

IV.  Networks [4h30]

   1. Formal introduction
           → Graph theory
           → Count-based measures: degree, centrality, betweenness, etc.
           → Random-walked based measures: Google PageRank, mixing index, etc.

   2. Networks in practice
           → Preferential attachment for social networks
           → Planar networks for cities
           → Abstract networks: causal/Bayesian, ecosystems, art

  3. Dynamical processes on networks
           → Temporal criticality: synchronization, failures

V.  Heterogeneities [6h]

  1. Inequalities 
           → Thin tails/Fat tails, scale invariance
           → The « problem » with power-laws: concentration/localisation, generalised CLT
           → Example: Wealth condensation model

  2. Multiplicative growth models for population
           → Uncorrelated growth, log-normal fluctuations
           → Sums of log-normals & condensation
           → Growth with redistribution/mixing
           → Continuous limit & the Hamilton-Jacobi-Bellman method 

  3. Measuring heterogeneities 
           → Gini coefficient, segregation indices, etc.
           → Multiscalar measures, Random walk measures
           → Optimal transport measures 

VI. Evolution & transmission [4h30]

  1. Birth, death processes and branching processes 

  2. Tree models in biology and philology
           → Probability models for the tree(s) of life (Lambert)
           → Cultural transmission: the example of manuscripts (Camps, Godreau)
 

• Who?
  
  Julien Randon-Furling, ENS Paris-Saclay (julien.randon-furling@ens-paris-saclay.fr) 
  Camille Scalliet, ENS, CNRS (camille.scalliet@phys.ens.fr)
   
• When?
  
  Wednesdays 9:00 - 12:30.
  Dates : 14, 21, 28 January, 4, 11, 18 February, 4, 11, 18 March 
   
•  Where?
  
  Sorbonne Université Campus, room to be determined. 

Taste for modelling, probabilities and statistics.