### Prè-requis

Probability, Linear Algebra, Basic Statistics, Programming (*eg *C, Python, R).

Some training in Statistical Physics may be a bonus but it is absolutely not a prerequisite.

### Objectif du cours

Interactions are an intergal part of many systems, be they natural, social, artificial and/or virtual. Their modelling and analysis has long posed challenges to the sciences – one may think of the three-body problem in classical mechanics, chaos theory in dynamical systems, feedback loops in climate science, particle systems in statistical physics, confounding variables in statistics, cities and transport systems in quantitative geography, markets in economics, opinion dynamics in social and political science…

With the unprecedented availability of data and the advent of new theoretical and computational methods, several approaches in the contemporary mathematical sciences are relevant to tackle such systems: stochastic modelling, simulations, machine learning, to name but a few. This course will provide an introduction to the general question of modelling systems of interactions, covering notions such as complexity, emergence and self-organized criticality, as well as simulation tools known as agent-based models (ABMs). These models provide ways of designing and running *in silico *controlled experiments of complex real systems. They also provide means of testing statistical and analytical methods: in particular, we will examine methods to learn the structure and content of interactions and causal relationships using ABMs. We will also look at the occurrence, in stochastic processes, ABMs and other algorithms, of extreme and rare events arising from interactions.

### Organisation des séances

Twelve 2-hour lectures (no exercice classes) covering the following points:

**Interacting Particles Systems (IPS):**(i) Basics (Markov processes, semigroups, gen- erators) (ii) Contact Process, Voter Model, Exclusion Process (iii) Stochastic Ising Models (Metropolis, Gibbs, Kawasaki, Glauber dynamics).*References*: Liggett 2005, Aldous 2013

**Agent-Based Models (ABMs):**(i) Generative Models & Generative Social Science (complex systems, emergence, self-organized criticality) (ii) Schelling Model (iii) Epi- demics Dynamics & Public Health Modelling (iv) Markets & Socio-Technical Systems (v) Avalanches, Timeliness Criticality.*References*: Epstein 2012, Sethna 2021, Adam 2020, Moran 2024

**Analysis & Inference in IPS and ABMs:**(i) Markov Chain Aggregation (ii) Inferring Interaction Structure (iii) Learning Interaction Laws (iv) Causal Inference.*References*: Banisch 2015, Krakauer 2023, Lu et al. 2019, Manzo 2022.

**Extrema,****Rare****Events****and****Heavy-Tail****Processes:**(i) Conspiracy vs Catastrophe Principle (ii) Higher-Dimensional Extrema (convex hulls of Lévy processes, combinatorial lemmas) (iii) Heavy-Tails in Stochastic Gradient Descent.*References*: Nair 2022, Majumdar et al. 2010, Gurbuzbalaban et al. 2022

### Mode de validation

Project based

### Références

Adam, D. *The simulations driving the world’s response to COVID-19*. Nature, 580(7802), 316-319. (2020) Aldous D. *Interacting* *particle* *systems* *as* *stochastic* *social* *dynamics*. Bernoulli 19(4), 1122-1149. (2013) Banisch, S. *Markov* *chain* *aggregation* *for* *agent-based* *models*. Springer. (2015)

de Bruyne, B., Randon-Furling, J., and Redner, S. *Optimization in first-passage resetting*. Physical Review Letters, 125 (5), 050602. (2020)

Epstein, J. M. *Generative* *social* *science.* *Studies* *in* *agent-based* *computational* *modeling*. Princeton University Press. (2012)

Gurbuzbalaban, M., Hu, Y., Simsekli, U., Yuan, K., and Zhu, L. *Heavy-tail phenomenon in decentralized SGD*. preprint arXiv:2205.06689. (2022)

Krakauer, D. C. *Unifying complexity science and machine learning*. Frontiers in Complex Systems 1: 1235202. (2023)

Liggett, T. *Interacting* *particle* *systems*. Springer. (2005)

Lu, F., Zhong, M., Tang, S., and Maggioni, M. *Nonparametric* *inference* *of* *interaction* *laws* *in* *systems* *of* *agents **from trajectory data*. PNAS 116 (29) 14424-14433. (2019)

Majumdar, S. N., Comtet, A., and Randon-Furling, J. *Random convex hulls and extreme value statistics*. Jour- nal of Statistical Physics, 138, 955-1009. (2010)

Manzo, G. *Agent-based models and causal inference*. John Wiley & Sons. (2022) Moran, J., Romeijnders, M., Le Doussal, P., Pijpers, F. P., Weitzel, U., Panja, D., and Bouchaud, J. P. *Timeliness criticality*. preprint arXiv:2309.15070. To appear in *Nature Physics*. (2024)

Sethna, J. P. *Entropy, order parameters, and complexity*. Oxford University Press. (2021)

### Thèmes abordés

Links with other MVA courses :

Several approaches that will be discussed in the course rely on, or are connected to, techniques and methods studied in other MVA courses, in particular of course Markov chains, MCMC and network models that feature in *Introduction* *to* *Probabilistic* *Graphical* *Models* *and* *Deep* *Gen**erative* *Models* (P. Latouche, P.A. Mattei), *Graphical* *Models:* *Discrete* *Inference* *and* *Learning *(K. Alahari, D. Wassermann), and *Graphs in machine learning *(D. Calandriello, M. Valko). However the perspective and the emphasis here is placed on different ways to model interactions, and to simulate systems of interactions through Agent-Based Models in order to provide a basis for decision and policy making (eg during pandemics). Notions of complexity and the statistical physics of disordered and complex systems will connect with parts of what used to be in the course *M**o**d**é**lisation* *en neurosciences et ailleurs *(by J.P. Nadal), but with a broader scope and a more probability-theoretical basis here. The introduction of ABMs and their hybridation with ML techniques and analysis of extrema and rare events will be new additions complementing the course *Apprentissage et génération par échantillonnage aléatoire* (S. Mallat).

#### Julien RANDON-FURLING

(Centre Borelli - ENS Paris-Saclay)