Related papers: Phase transitions for Erdos-Renyi graphs

Consider a critical random multigraph $\mathcal{G}_n$ with $n$ vertices constructed by the configuration model such that its vertex degrees are independent random variables with the same distribution $\nu$ (criticality means that the second…

We identify the scaling limit of random intersection graphs inside their critical windows. The limit graphs vary according to the clustering regimes, and coincide with the continuum Erdos--Renyi graph in two out of the three regimes. Our…

We present an analytic proof of the existence of phase transition in the large $N$ limit of certain random noncommutaitve geometries. These geometries can be expressed as ensembles of Dirac operators. When they reduce to single matrix…

We show that almost surely the rank of the adjacency matrix of the Erd\"os-R\'enyi random graph $G(n,p)$ equals the number of non-isolated vertices for any $c\ln n/n<p<1/2$, where $c$ is an arbitrary positive constant larger than 1/2. In…

We present a large-deviations/thermodynamic approach to the classic problem of percolation on the complete graph. Specifically, we determine the large-deviation rate function for the probability that the giant component occupies a fixed…

In this work we prove that the giant component of the Erd\"os--Renyi random graph $G(n,c/n)$ for c a constant greater than 1 (sparse regime), is not Gromov $\delta$-hyperbolic for any positive $\delta$ with probability tending to one as…

The exponential family of random graphs is one of the most promising class of network models. Dependence between the random edges is defined through certain finite subgraphs, analogous to the use of potential energy to provide dependence…

We provide a sufficient condition on the isoperimetric properties of a regular graph $G$ of growing degree $d$, under which the random subgraph $G_p$ typically undergoes a phase transition around $p=\frac{1}{d}$ which resembles the…

In the classical Erd\"os-R\'enyi random graph G(n,p) there are n vertices and each of the possible edges is independently present with probability p. The random graph G(n,p) is homogeneous in the sense that all vertices have the same…

The following random graph model was introduced for the evolution of protein-protein interaction networks: Let $\mathcal G = (G_n)_{n=n_0, n_0+1,...}$ be a sequence of random graphs, where $G_n = (V_n, E_n)$ is a graph with $|V_n|=n$…

The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate 1, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we…

We introduce perhaps the simplest models of graph evolution with choice that demonstrate discontinuous percolation transitions and can be analyzed via mathematical evolution equations. These models are local, in the sense that at each step…

We examine the order of the phase transition in the Potts model by using the graph representation for the partition function, which allows treating a non-integer number of Potts states. The order of transition is determined by the analysis…

We study a random graph model which combines properties of the edge percolation model on Z^d and a classical random graph G(n,c/n). We show that this model, being a homogeneous random graph, has a natural relation to the so-called "rank 1…

Random directed graphs $D(n,p)$ undergo a phase transition around the point $p = 1/n$, and the width of the transition window has been known since the works of Luczak and Seierstad. They have established that as $n \to \infty$ when $p = (1…

We study phase transitions in the Ising model on random graphs using graph limits. We show that the critical temperatures are determined by the eigenvalues of the kernel operator associated with the graph limit. Bifurcation diagrams for…

Random intersection graphs (RIGs) are an important random structure with applications in social networks, epidemic networks, blog readership, and wireless sensor networks. RIGs can be interpreted as a model for large randomly formed…

We study random walks on Erd\"os-R\'enyi random graphs in which, every time the random walk returns to the starting point, first an edge probability is independently sampled according to a priori measure $\mu$, and then an Erd\"os-R\'enyi…

We show that there is a constant c>0 so that for any fixed r which is at least 3 a.a.s. an r-regular graph on n vertices contains a complete graph on c n^{1/2} vertices as a minor. This confirms a conjecture of Markstrom. Since any minor of…

The Norros-Reittu model is a random graph with $n$ vertices and i.i.d. weights assigned to them. The number of edges between any two vertices follows an independent Poisson distribution whose parameter is increasing in the weights of the…