Mini courses 

Many phenomena in the real world can be phrased in terms of networks. Examples include the World-Wide Web, social interactions and Internet, but also the interaction patterns between proteins, food webs and citation networks. Many large-scale networks have, despite their diversity in backgrounds, surprisingly much in common. Many of these networks are small worlds, in the sense that one requires few links to hop between pairs of vertices. Also the variability of the number of connections between elements tends to be enormous, which is related to the scale-free phenomenon.

Percolation is a model for random damage to a network. It is one of the simplest models that displays a phase transition: when the network is severely damaged, it falls apart in many small connected components, while if the damage is light, connectivity is hardly affected. We study the location and nature of the phase transition on random graphs. In particular, we focus on the connectivity structure close to, or below, criticality, where components display intricate scaling behaviour.

We review the recent progress that has been made in two important settings: random graphs whose expected adjacency matrix is close to being rank-1, the most prominent examples being the configuration model and rank-1 inhomogeneous random graphs, and dynamic random graphs, i.e., random graphs that grow with time, such as uniform attachment models. Remarkably, these two settings behave rather differently. In all cases, the inhomogeneity of the underlying random graph on which we perform percolation is of crucial importance.

We also describe some of the key techniques used to study percolation on random graphs. Local convergence is essential to study the local structure of random graphs, and predicts the limit of the percolation critical value, since the giant in percolated random graphs is 'almost local'; To study near-critical connected component sizes, we rely on graph exploration processes, and their scaling limits obtained through weak convergence of stochastic processes and martingale central limit theorems. For dynamic random graphs, we rely on stochastic approximation and martingale techniques.

In more detail, the mini-course consists of the following lectures:

Lecture 1: Random Graphs, Networks and Percolation.

Lecture 2: Giants and Percolation Phase Transition on Random Graphs.

Lecture 3: Critical Percolation on Rank-1 Random Graphs.

Lecture 4: Percolation on Dynamic Random Graphs.

We assume no prior knowledge in random graph theory.

In my lectures I will discuss the basic theory of the Brownian web, some of its applications, and more recent developments. The plan for the lectures is:

Lecture 1: The Brownian web; Convergence of systems of nearest-neighbour coalescing random walks to the Brownian web, structure of the Brownian web including special points, and an application to the Arratia flow.

Lecture 2: Applications of the Brownian web; The true self-repelling motion, the directed spanning forest, the Brownian castle, and the Brownian marble.

Lecture 3: The Brownian net; Equivalent definitions and applications to voter model perturbations and Howitt-Warren flows.

Lecture 4: Universality of the Brownian web; Convergence of non-nearest neighbour coalescing random walks to the Brownian web, the augmented Brownian web, and multi-scale webs.