Advanced Topics in Probability

Even years: Random graphs and complex networks.

Random graphs have become one of the most studied models in probability theory. Understanding the behaviour of large networks (eg. social media, financial transactions, communication networks) has become of paramount importance for practicioners and scientists alike. In this course, we aim at providing a general theory to answer the most immediate questions about large random graphs.

We will begin by recalling some tools of discrete probability (moment methods Markov's and Chebyshev's inequality), and learn some new tools: coupling and stochastic dominations. We will then study Erdos Rényi graphs, one of the first examples of random networks in probability. We will then move on to more modern models of complex networks: configuration model, preferential attachment graphs, random geometric graphs.

Odd years: Interacting Particle Systems

This course provides an introduction in the theory of interacting particle systems, a family of interacting Markov processes used to model many real-world phenomena such as the spread of an infection, the evolution of opinions in a population, traffic, the transport phenomena of interacting molecules, temperature-dependent behavior of magnetic systems.

The course treats the following subjects

1. Markov processes in continuous time, generators, semigroups, martingales.

2. Basic techniques of interacting particle systems:

a) Coupling, monotonicity and positive correlations.

b) Duality.

3. These techniques will be applied in the study of the exclusion process, a basic interacting particle system used in non-equilibrium statistical physics, biology and modelling of traffic.Random Graphs part: The following topics will be discussed. Note that this is only a tentative schedule and the pace at which the material will be covered will depend on the class as well.

Real life complex networks and their properties, main modelling approaches.

Probabilistic toolkit: Notions of convergence, coupling, stochastic ordering, Probabilistic bounds (1 lecture)

Branching processes (0.5-1 lecture)

Phase transition in the Erdos-Renyi random graph (1-2 lectures)

Configuration model: (intro 0.5. lecture)

loops and multiple edges (0.5 lecture)

erased configuration model (0.5 lecture)

generating simple graphs

connection to number of simple graphs with prescribed degree sequence (0.5 lecture)

ultra-small world property with power-law degrees. (1 lecture)

Inhomogeneous random graphs (1 lectures)

Preferential attachment model

The model (0.5 lecture)

degree distribution (1.5 lecture)

Lindeberg's Central limit theorem (0.5 lecture)

component sizes with Lindeberg's theorem (1 lecture)

Additional topics if time allows might include small world phenomenon, etc.

New models GIRGs, scale-free percolation, etc.