Related papers: Phase Transitions in Partially Structured Random G…
We study Bernoulli bond percolation on nonunimodular quasi-transitive graphs, and more generally graphs whose automorphism group has a nonunimodular quasi-transitive subgroup. We prove that percolation on any such graph has a non-empty…
In this paper, we study the high-order phase transition in random $r$-uniform hypergraphs. For a positive integer $n$ and a real $p\in [0,1]$, let $H:=H^r(n,p)$ be the random $r$-uniform hypergraph with vertex set $[n]$, where each $r$-set…
We examine the heterogeneous responses of individual nodes in sparse networks to the random removal of a fraction of edges. Using the message-passing formulation of percolation, we discover considerable variation across the network in the…
Limiting distributions are derived for the sparse connected components that are present when a random graph on $n$ vertices has approximately $\half n$ edges. In particular, we show that such a graph consists entirely of trees, unicyclic…
We study the relaxation dynamics of fully clustered networks (maximal number of triangles) to an unclustered state under two different edge dynamics---the double-edge swap, corresponding to degree-preserving randomization of the…
Bootstrap percolation is a well-known activation process in a graph, in which a node becomes active when it has at least $r$ active neighbors. Such process, originally studied on regular structures, has been recently investigated also in…
I consider p-Bernoulli bond percolation on graphs of vertex-transitive tilings of the hyperbolic plane with finite sided faces (or, equivalently, on transitive, nonamenable, planar graphs with one end) and on their duals. It is known…
The Random Language Model, proposed as a simple model of human languages, is defined by the averaged model of a probabilistic context-free grammar. This grammar expresses the process of sentence generation as a tree graph with nodes having…
We consider the problem of coloring the vertices of a large sparse random graph with a given number of colors so that no adjacent vertices have the same color. Using the cavity method, we present a detailed and systematic analytical study…
Percolation is perhaps the simplest example of a process exhibiting a phase transition and one of the most studied phenomena in statistical physics. The percolation transition is continuous if sites/bonds are occupied independently with the…
We consider a version of continuum long-range percolation on finite boxes of $\mathbb{R}^d$ in which the vertex set is given by the points of a Poisson point process and each pair of two vertices at distance $r$ is connected with…
We generalize the random graph evolution process of Bohman, Frieze, and Wormald [T. Bohman, A. Frieze, and N. C. Wormald, Random Struct. Algorithms, 25, 432 (2004)]. Potential edges, sampled uniformly at random from the complete graph, are…
We consider percolation on high-dimensional product graphs, where the base graphs are regular and of bounded order. In the subcritical regime, we show that typically the largest component is of order logarithmic in the number of vertices.…
The $k$-section width and the Max-Cut for the configuration model are shown to exhibit phase transitions according to the values of certain parameters of the asymptotic degree distribution. These transitions mirror those observed on…
Semi-transitive graphs, defined in \cite{hps98} as examples where ``uniform percolation" holds whenever $p>p_c$, are a large class of graphs more general than quasi-transitive graphs. Let $G$ be a semi-transitive graph with one end which…
This is a status report on a companion subject to extremal combinatorics, obtained by replacing extremality properties with emergent structure, `phases'. We discuss phases, and phase transitions, in large graphs and large permutations,…
Random graphs with latent geometric structure are popular models of social and biological networks, with applications ranging from network user profiling to circuit design. These graphs are also of purely theoretical interest within…
We introduce a new oriented evolving graph model inspired by biological networks. A node is added at each time step and is connected to the rest of the graph by random oriented edges emerging from older nodes. This leads to a statistical…
We propose two classes of dynamic versions of the classical Erd\H{o}s-R\'enyi graph: one in which the transition rates are governed by an external regime process, and one in which the transition rates are periodically resampled. For both…
In the Constrained-degree percolation model on a graph $(\mathbb{V},\mathbb{E})$ there are a sequence, $(U_e)_{e\in\mathbb{E}}$, of i.i.d. random variables with distribution $U[0,1]$ and a positive integer $k$. Each bond $e$ tries to open…