Sparse random graphs with clustering

In 2007 we introduced a general model of sparse random graphs with independence between the edges. The aim of this paper is to present an extension of this model in which the edges are far from independent, and to prove several results about this extension. The basic idea is to construct the random graph by adding not only edges but also other small graphs. In other words, we first construct an inhomogeneous random hypergraph with independent hyperedges, and then replace each hyperedge by a (perhaps complete) graph. Although flexible enough to produce graphs with significant dependence between edges, this model is nonetheless mathematically tractable. Indeed, we find the critical point where a giant component emerges in full generality, in terms of the norm of a certain integral operator, and relate the size of the giant component to the survival probability of a certain (non-Poisson) multi-type branching process. While our main focus is the phase transition, we also study the degree distribution and the numbers of small subgraphs. We illustrate the model with a simple special case that produces graphs with power-law degree sequences with a wide range of degree exponents and clustering coefficients.Comment: 62 pages; minor revisio

Stability for large forbidden subgraphs

We extend the classical stability theorem of Erdos and Simonovits for forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference

Spread-out percolation in R^d

Let $X$ be either $Z^d$ or the points of a Poisson process in $R^d$ of intensity 1. Given parameters $r$ and $p$, join each pair of points of $X$ within distance $r$ independently with probability $p$. This is the simplest case of a `spread-out' percolation model studied by Penrose, who showed that, as $r\to\infty$, the average degree of the corresponding random graph at the percolation threshold tends to 1, i.e., the percolation threshold and the threshold for criticality of the naturally associated branching process approach one another. Here we show that this result follows immediately from of a general result of the authors on inhomogeneous random graphs.Comment: 9 pages. Title changed. Minor changes to text, including updated references to [3]. To appear in Random Structures and Algorithm

A note on the Harris-Kesten Theorem

Recently, a short proof of the Harris-Kesten result that the critical probability for bond percolation in the planar square lattice is 1/2 was given, using a sharp threshold result of Friedgut and Kalai. Here we point out that a key part of this proof may be replaced by an argument of Russo from 1982, using his approximate zero-one law in place of the Friedgut-Kalai result. Russo's paper gave a new proof of the Harris-Kesten Theorem that seems to have received little attention.Comment: 4 pages; author list changed, acknowledgement adde

Duality in inhomogeneous random graphs, and the cut metric

The classical random graph model $G(n,\lambda/n)$ satisfies a `duality principle', in that removing the giant component from a supercritical instance of the model leaves (essentially) a subcritical instance. Such principles have been proved for various models; they are useful since it is often much easier to study the subcritical model than to directly study small components in the supercritical model. Here we prove a duality principle of this type for a very general class of random graphs with independence between the edges, defined by convergence of the matrices of edge probabilities in the cut metric.Comment: 13 page

Distances in random graphs with finite variance degrees

In this paper we study a random graph with $N$ nodes, where node $j$ has degree $D_j$ and $\{D_j\}_{j=1}^N$ are i.i.d. with \prob(D_j\leq x)=F(x). We assume that $1-F(x)\leq c x^{-\tau+1}$ for some $\tau>3$ and some constant $c>0$. This graph model is a variant of the so-called configuration model, and includes heavy tail degrees with finite variance. The minimal number of edges between two arbitrary connected nodes, also known as the graph distance or the hopcount, is investigated when $N\to \infty$. We prove that the graph distance grows like $\log_{\nu}N$, when the base of the logarithm equals \nu=\expec[D_j(D_j -1)]/\expec[D_j]>1. This confirms the heuristic argument of Newman, Strogatz and Watts \cite{NSW00}. In addition, the random fluctuations around this asymptotic mean $\log_{\nu}{N}$ are characterized and shown to be uniformly bounded. In particular, we show convergence in distribution of the centered graph distance along exponentially growing subsequences.Comment: 40 pages, 2 figure

Counting connected hypergraphs via the probabilistic method

In 1990 Bender, Canfield and McKay gave an asymptotic formula for the number of connected graphs on $[n]$ with $m$ edges, whenever $n$ and the nullity $m-n+1$ tend to infinity. Asymptotic formulae for the number of connected $r$-uniform hypergraphs on $[n]$ with $m$ edges and so nullity $t=(r-1)m-n+1$ were proved by Karo\'nski and \L uczak for the case $t=o(\log n/\log\log n)$, and Behrisch, Coja-Oghlan and Kang for $t=\Theta(n)$. Here we prove such a formula for any $r\ge 3$ fixed, and any $t=t(n)$ satisfying $t=o(n)$ and $t\to\infty$ as $n\to\infty$. This leaves open only the (much simpler) case $t/n\to\infty$, which we will consider in future work. ( arXiv:1511.04739 ) Our approach is probabilistic. Let $H^r_{n,p}$ denote the random $r$-uniform hypergraph on $[n]$ in which each edge is present independently with probability $p$. Let $L_1$ and $M_1$ be the numbers of vertices and edges in the largest component of $H^r_{n,p}$. We prove a local limit theorem giving an asymptotic formula for the probability that $L_1$ and $M_1$ take any given pair of values within the `typical' range, for any $p=p(n)$ in the supercritical regime, i.e., when $p=p(n)=(1+\epsilon(n))(r-2)!n^{-r+1}$ where $\epsilon^3n\to\infty$ and $\epsilon\to 0$; our enumerative result then follows easily. Taking as a starting point the recent joint central limit theorem for $L_1$ and $M_1$, we use smoothing techniques to show that `nearby' pairs of values arise with about the same probability, leading to the local limit theorem. Behrisch et al used similar ideas in a very different way, that does not seem to work in our setting. Independently, Sato and Wormald have recently proved the special case $r=3$, with an additional restriction on $t$. They use complementary, more enumerative methods, which seem to have a more limited scope, but to give additional information when they do work.Comment: Expanded; asymptotics clarified - no significant mathematical changes. 67 pages (including appendix

The phase transition in inhomogeneous random graphs

We introduce a very general model of an inhomogenous random graph with independence between the edges, which scales so that the number of edges is linear in the number of vertices. This scaling corresponds to the p=c/n scaling for G(n,p) used to study the phase transition; also, it seems to be a property of many large real-world graphs. Our model includes as special cases many models previously studied. We show that under one very weak assumption (that the expected number of edges is `what it should be'), many properties of the model can be determined, in particular the critical point of the phase transition, and the size of the giant component above the transition. We do this by relating our random graphs to branching processes, which are much easier to analyze. We also consider other properties of the model, showing, for example, that when there is a giant component, it is `stable': for a typical random graph, no matter how we add or delete o(n) edges, the size of the giant component does not change by more than o(n).Comment: 135 pages; revised and expanded slightly. To appear in Random Structures and Algorithm

Hitting time results for Maker-Breaker games

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property \mP. We focus on three natural properties for Maker's graph, namely being $k$-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the $k$-vertex connectivity game exactly at the time the random graph process first reaches minimum degree $2k$; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree $2$; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree $4$. The latter two statements settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page

The minimal density of triangles in tripartite graphs

We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomass\'e and Thomassen characterizing those edge densities guaranteeing the existence of a triangle in a tripartite graph. To be precise we show that a suitably weighted copy of the graph formed by deleting a certain 9-cycle from $K_{3,3,3}$ has minimal triangle density among all weighted tripartite graphs with prescribed edge densities.Comment: 44 pages including 12 page appendix of C++ cod