Random Networks Workshop
20th May, 2026

Title: Into space - and back again!

Abstract: The configuration model is a standard model for generating a random graph with a prescribed degree distribution. I will describe attempts to analyze a spatial version of the model, and ongoing work to revert the spatial model to the non-spatial setting.

Title: The transference principle for random hypergraphs

Abstract: Results in extremal combinatorics typically only tell us something about the structure of rather dense structures with certain properties. This is why the last few decades have seen considerable interest in the question when and how such results can be "transferred" to a sparser random setting. In the talk I will survey some important results and techniques in this direction, and introduce some new results (including a so-called counting lemma for sparse random hypergraphs) that I obtained in joint work with Peter Allen, Joanna Lada, and Domenico Mergoni Cecchelli.

Title: Barak-Erdos directed random graphs and related models: limit theorems, perfect simulation and around.

Abstract: We consider directed random graphs, the prototype of which being the Barak-Erdős graph on the integers, and study the way that long (or heavy, if weights are present) paths grow. This is done by relating the graphs to certain particle systems that we call Infinite Bin Models (IBM). We formulate a number of limit theorems. Regenerative techniques are used where possible, exhibiting random sets of vertices over which the graphs regenerate. When edges have random weights we show how the last passage percolation constants behave and when central limit theorems exist. When the underlying vertex set is partially ordered, new phenomena occur, e.g., there are relations with last passage Brownian percolation. We also look at weights that may possibly take negative values and study in detail some special cases that require combinatorial/graph theoretic techniques that exhibit some interesting non-differentiability properties of the last passage percolation constant. We also explain how to approach the problem of estimation of last passage percolation constants by means of perfect simulation.

The talk is based on a series of joint papers with Takis Konstantopoulos, Bastien Mallein, Sanjay Ramassami, James Martin, Denis Denisov, and several other colleagues, and in particular on the survey paper https://arxiv.org/abs/2312.02884

Title: The Augmented Brownian Web

Abstract: We consider coalescing random walks in 1+1 dimensional space-time, with a jump kernel that has finite moments up to order alpha. We view this system as a random set of coalescing paths, with one path starting from each point of space-time. When alpha>3, the diffusive scaling limit is known to be the Brownian web. We study the regime in which alpha is between 2 and 3. In this regime tightness fails (in the sense of continuous paths) due to erratic behaviour near the start times of some of the paths. We show that a surprising transition in behaviour occurs at alpha=9/4; when alpha>9/4 a diffusive scaling limit exists in which paths are essentially Brownian but some paths possess jumps at their initial times, whilst when alpha<9/4 tightness "truly" fails.