## Nice summer school on random walks and complex networks

### Nice, Laboratoire Dieudonné, July 8-19, 2019

#### The school is supported by the following institutions:

### Scientific focus

Models of complex networks appear in different fields (telecommunication networks, computer networks, neural networks, the World Wide Web, friendship networks as facebook, epidemiology spreads, protein-protein-interaction networks, semantic networks, interbank payment networks), and we hope to attract participants from other areas as well.
### Idea of the school

The school will focus on two topics: random walks (corresponding lectures will be given by Louigi Addario-Berry, Anna Ben-Hamou and Perla Sousi), and stochastic models for complex networks (the corresponding lectures will be given by Simon Griffiths, Colin McDiarmid and Bruce Reed). The two topics will be relatively independent, but the lectures corresponding to the same topic will be coordinated. Both topics will span both weeks. Exercises will be assigned in the courses and a significant amount of time will be allowed for students to work on them.

The course on complex networks will present a range of topics related to different models of random graphs: models studied are the G(n,p) and G(n,m) model, the configuration model, the uniform (simple) model of graphs with a fixed degree sequence, the preferential attachment model, the triangle-free random graph process, the model of a uniform random graph without having a certain graph as a minor and without having a certain graph as an induced subgraph. Topics covered include basic tools of the analysis, analysis of modularity of the Erdos-Renyi graph, the differential equation method of Wormald, Szemeredi's regularity lemma (for random graphs without having a certain graph as an induced subgraph as well as for the chromatic threshold of graphs).

The course on random walks will present a range of topics related to convergence of Markov chains and sampling with Markov chains, ranging from fundamental results and techniques to recent advances in the field. The topics to be covered include hitting time and cover time bounds, sampling of spanning trees and connections with the algorithmic Lovasz Local lemma, definitions and characterizations of mixing times, the cutoff phenomenon, random walks on random graphs, and non-backtracking random walks.

Students with a basic undergraduate knowledge of probability and discrete mathematics should easily be able to pick up the extra background required after a small amount of reading.