The phase transition in inhomogeneous random graphs

The “classical” random graph models, in particular G(n,p), are “homogeneous,” in the sense that the degrees (for example) tend to be concentrated around a typical value. Many graphs arising in the real world do not have this property, having, for example, power‐law degree distributions. Thus there has been a lot of recent interest in defining and studying “inhomogeneous” random graph models. One of the most studied properties of these new models is their “robustness”, or, equivalently, the “phase transition” as an edge density parameter is varied. For G(n,p), p = c/n, the phase transition at c = 1 has been a central topic in the study of random graphs for well over 40 years. Many of the new inhomogeneous models are rather complicated; although there are exceptions, in most cases precise questions such as determining exactly the critical point of the phase transition are approachable only when there is independence between the edges. Fortunately, some models studied have this property already, and others can be approximated by models with independence. Here we introduce a very general model of an inhomogeneous random graph with (conditional) 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). © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 31, 3–122, 2007     

Sparse random graphs with clustering

In 2007, we introduced a general model of sparse random graphs with (conditional) independence between the edges. The aim of this article 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 (conditionally) 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. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 269–323, 2011     

The augmented multiplicative coalescent,bounded size rules and critical dynamics of random graphs

Random graph models with limited choice have been studied extensively with the goal of understanding the mechanism of the emergence of the giant component. One of the standard models are the Achlioptas random graph processes on a fixed set of \(n\) vertices. Here at each step, one chooses two edges uniformly at random and then decides which one to add to the existing configuration according to some criterion. An important class of such rules are the bounded-size rules where for a fixed \(K\ge 1\) , all components of size greater than \(K\) are treated equally. While a great deal of work has gone into analyzing the subcritical and supercritical regimes, the nature of the critical scaling window, the size and complexity (deviation from trees) of the components in the critical regime and nature of the merging dynamics has not been well understood. In this work we study such questions for general bounded-size rules. Our first main contribution is the construction of an extension of Aldous’s standard multiplicative coalescent process which describes the asymptotic evolution of the vector of sizes and surplus of all components. We show that this process, referred to as the standard augmented multiplicative coalescent (AMC) is ‘nearly’ Feller with a suitable topology on the state space. Our second main result proves the convergence of suitably scaled component size and surplus vector, for any bounded-size rule, to the standard AMC. This result is new even for the classical Erd?s–Rényi setting. The key ingredients here are a precise analysis of the asymptotic behavior of various susceptibility functions near criticality and certain bounds from Bhamidi et al. (The barely subcritical regime. Arxiv preprint, 2012) on the size of the largest component in the barely subcritical regime.     

The Bohman‐Frieze process near criticality

The Erd?s‐Rényi process begins with an empty graph on n vertices, with edges added randomly one at a time to the graph. A classical result of Erd?s and Rényi states that the Erd?s‐Rényi process undergoes a phase transition, which takes place when the number of edges reaches n/2 (we say at time 1) and a giant component emerges. Since this seminal work of Erd?s and Rényi, various random graph models have been introduced and studied. In this paper we study the Bohman‐Frieze process, a simple modification of the Erd?s‐Rényi process. The Bohman‐Frieze process also begins with an empty graph on n vertices. At each step two random edges are presented, and if the first edge would join two isolated vertices, it is added to a graph; otherwise the second edge is added. We present several new results on the phase transition of the Bohman‐Frieze process. We show that it has a qualitatively similar phase transition to the Erd?s‐Rényi process in terms of the size and structure of the components near the critical point. We prove that all components at time t_c ? ? (that is, when the number of edges are (t_c ? ?)n/2) are trees or unicyclic components and that the largest component is of size Ω(?^‐2log n). Further, at t_c + ?, all components apart from the giant component are trees or unicyclic and the size of the second‐largest component is Θ(?^‐2log n). Each of these results corresponds to an analogous well‐known result for the Erd?s‐Rényi process. Our proof techniques include combinatorial arguments, the differential equation method for random processes, and the singularity analysis of the moment generating function for the susceptibility, which satisfies a quasi‐linear partial differential equation. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Connectivity of a general class of inhomogeneous random digraphs

We study a family of directed random graphs whose arcs are sampled independently of each other, and are present in the graph with a probability that depends on the attributes of the vertices involved. In particular, this family of models includes as special cases the directed versions of the Erd?s‐Rényi model, graphs with given expected degrees, the generalized random graph, and the Poissonian random graph. We establish a phase transition for the existence of a giant strongly connected component and provide some other basic properties, including the limiting joint distribution of the degrees and the mean number of arcs. In particular, we show that by choosing the joint distribution of the vertex attributes according to a multivariate regularly varying distribution, one can obtain scale‐free graphs with arbitrary in‐degree/out‐degree dependence.     

The evolution of subcritical Achlioptas processes

In Achlioptas processes, starting from an empty graph, in each step two potential edges are chosen uniformly at random, and using some rule one of them is selected and added to the evolving graph. AlthouSgh the evolution of such ‘local’ modifications of the Erd?s–Rényi random graph process has received considerable attention during the last decade, so far only rather simple rules are well understood. Indeed, the main focus has been on ‘bounded‐size’ rules, where all component sizes larger than some constant B are treated the same way, and for more complex rules very few rigorous results are known. In this paper we study Achlioptas processes given by (unbounded) size rules such as the sum and product rules. Using a variant of the neighbourhood exploration process and branching process arguments, we show that certain key statistics are tightly concentrated at least until the susceptibility (the expected size of the component containing a randomly chosen vertex) diverges. Our convergence result is most likely best possible for certain generalized Achlioptas processes: in the later evolution the number of vertices in small components may not be concentrated. Furthermore, we believe that for a large class of rules the critical time where the susceptibility ‘blows up’ coincides with the percolation threshold. © 2014 Wiley Periodicals, Inc. Random Struct. Alg., 47, 174–203, 2015     

A probabilistic approach to the leader problem in random graphs

We study the fixation time of the identity of the leader, that is, the most massive component, in the general setting of Aldous's multiplicative coalescent, which in an asymptotic sense describes the evolution of the component sizes of a wide array of near‐critical coalescent processes, including the classical Erd?s‐Rényi process. We show tightness of the fixation time in the “Brownian” regime, explicitly determining the median value of the fixation time to within an optimal O(1) window. This generalizes ?uczak's result for the Erd?s‐Rényi random graph using completely different techniques. In the heavy‐tailed case, in which the limit of the component sizes can be encoded using a thinned pure‐jump Lévy process, we prove that only one‐sided tightness holds. This shows a genuine difference in the possible behavior in the two regimes.     

Critical random graphs and the differential equations technique

Over the last few years a wide array of random graph models have been postulated to understand properties of empirically observed networks. Most of these models come with a parameter t (usually related to edge density) and a (model dependent) critical time t _c that specifies when a giant component emerges. There is evidence to support that for a wide class of models, under moment conditions, the nature of this emergence is universal and looks like the classical Erd?s-Rényi random graph, in the sense of the critical scaling window and (a) the sizes of the components in this window (all maximal component sizes scaling like n^2/3) and (b) the structure of components (rescaled by n^?1/3) converge to random fractals related to the continuum random tree. The aim of this note is to give a non-technical overview of recent breakthroughs in this area, emphasizing a particular tool in proving such results called the differential equations technique first developed and used extensively in probabilistic combinatorics in the work of Wormald [52, 53] and developed in the context of critical random graphs by the authors and their collaborators in [10–12].     

Stochastic kronecker graphs

A random graph model based on Kronecker products of probability matrices has been recently proposed as a generative model for large‐scale real‐world networks such as the web. This model simultaneously captures several well‐known properties of real‐world networks; in particular, it gives rise to a heavy‐tailed degree distribution, has a low diameter, and obeys the densification power law. Most properties of Kronecker products of graphs (such as connectivity and diameter) are only rigorously analyzed in the deterministic case. In this article, we study the basic properties of stochastic Kronecker products based on an initiator matrix of size two (which is the case that is shown to provide the best fit to many real‐world networks). We will show a phase transition for the emergence of the giant component and another phase transition for connectivity, and prove that such graphs have constant diameters beyond the connectivity threshold, but are not searchable using a decentralized algorithm. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 38, 453–466, 2011     

The size of the giant high‐order component in random hypergraphs

The phase transition in the size of the giant component in random graphs is one of the most well‐studied phenomena in random graph theory. For hypergraphs, there are many possible generalizations of the notion of a connected component. We consider the following: two j‐sets (sets of j vertices) are j‐connected if there is a walk of edges between them such that two consecutive edges intersect in at least j vertices. A hypergraph is j‐connected if all j‐sets are pairwise j‐connected. In this paper, we determine the asymptotic size of the unique giant j‐connected component in random k‐uniform hypergraphs for any and .     

Topological properties of the one dimensional exponential random geometric graph

In this article we study the one‐dimensional random geometric (random interval) graph when the location of the nodes are independent and exponentially distributed. We derive exact results and limit theorems for the connectivity and other properties associated with this random graph. We show that the asymptotic properties of a graph with a truncated exponential distribution can be obtained using the exponential random geometric graph. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

The Reverse H‐free Process for Strictly 2‐Balanced Graphs

Consider the random graph process that starts from the complete graph on n vertices. In every step, the process selects an edge uniformly at random from the set of edges that are in a copy of a fixed graph H and removes it from the graph. The process stops when no more copies of H exist. When H is a strictly 2‐balanced graph we give the exact asymptotics on the number of edges remaining in the graph when the process terminates and investigate some basic properties namely the size of the maximal independent set and the presence of subgraphs.     

Critical random graphs and the structure of a minimum spanning tree

We consider the complete graph on n vertices whose edges are weighted by independent and identically distributed edge weights and build the associated minimum weight spanning tree. We show that if the random weights are all distinct, then the expected diameter of such a tree is Θ(n^1/3). This settles a question of Frieze and Mc‐Diarmid (Random Struct Algorithm 10 (1997), 5–42). The proofs are based on a precise analysis of the behavior of random graphs around the critical point. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009     

Sparse graphs: Metrics and random models

Recently, Bollobás, Janson and Riordan introduced a family of random graph models producing inhomogeneous graphs with n vertices and Θ(n) edges whose distribution is characterized by a kernel, i.e., a symmetric measurable function κ: [0, 1]² → [0, ∞). To understand these models, we should like to know when different kernels κ give rise to “similar” graphs, and, given a real‐world network, how “similar” is it to a typical graph G(n, κ) derived from a given kernel κ. The analogous questions for dense graphs, with Θ(n²) edges, are answered by recent results of Borgs, Chayes, Lovász, Sós, Szegedy and Vesztergombi, who showed that several natural metrics on graphs are equivalent, and moreover that any sequence of graphs converges in each metric to a graphon, i.e., a kernel taking values in [0, 1]. Possible generalizations of these results to graphs with o(n²) but ω(n) edges are discussed in a companion article [Bollobás and Riordan, London Math Soc Lecture Note Series 365 (2009), 211–287]; here we focus only on graphs with Θ(n) edges, which turn out to be much harder to handle. Many new phenomena occur, and there are a host of plausible metrics to consider; many of these metrics suggest new random graph models and vice versa. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 1‐38, 2011     

Rainbow hamilton cycles in random graphs

One of the most famous results in the theory of random graphs establishes that the threshold for Hamiltonicity in the Erd?s‐Rényi random graph G_n,p is around . Much research has been done to extend this to increasingly challenging random structures. In particular, a recent result by Frieze determined the asymptotic threshold for a loose Hamilton cycle in the random 3‐uniform hypergraph by connecting 3‐uniform hypergraphs to edge‐colored graphs. In this work, we consider that setting of edge‐colored graphs, and prove a result which achieves the best possible first order constant. Specifically, when the edges of G_n,p are randomly colored from a set of (1 + o(1))n colors, with , we show that one can almost always find a Hamilton cycle which has the additional property that all edges are distinctly colored (rainbow).Copyright © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 44, 328‐354, 2014     

On rigidity,orientability, and cores of random graphs with sliders

Suppose that you add rigid bars between points in the plane, and suppose that a constant fraction q of the points moves freely in the whole plane; the remaining fraction is constrained to move on fixed lines called sliders. When does a giant rigid cluster emerge? Under a genericity condition, the answer only depends on the graph formed by the points (vertices) and the bars (edges). We find for the random graph the threshold value of c for the appearance of a linear‐sized rigid component as a function of q, generalizing results of Kasiviswanathan, Moore, and Theran. We show that this appearance of a giant component undergoes a continuous transition for and a discontinuous transition for . In our proofs, we introduce a generalized notion of orientability interpolating between 1‐ and 2‐orientability, of cores interpolating between the 2‐core and the 3‐core, and of extended cores interpolating between the 2 + 1‐core and the 3 + 2‐core; we find the precise expressions for the respective thresholds and the sizes of the different cores above the threshold. In particular, this proves a conjecture of Kasiviswanathan, Moore, and Theran about the size of the 3 + 2‐core. We also derive some structural properties of rigidity with sliders (matroid and decomposition into components) which can be of independent interest.     

The phase transition in the cluster‐scaled model of a random graph

For 0 < p < 1 and q > 0 let G_q(n,p) denote the random graph with vertex set [n]={1,…,n} such that, for each graph G on [n] with e(G) edges and c(G) components, the probability that G_q(n,p)=G is proportional to . The first systematic study of G_q(n,p) was undertaken by 6 , who analyzed the phase transition phenomenon corresponding to the emergence of the giant component. In this paper we describe the structure of G_q(n,p) very close the critical threshold. © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2006     

On a random graph with immigrating vertices: Emergence of the giant component

A randomly evolving graph, with vertices immigrating at rate n and each possible edge appearing at rate 1/n, is studied. The detailed picture of emergence of giant components with O(n^2/3) vertices is shown to be the same as in the Erdős–Rényi graph process with the number of vertices fixed at n at the start. A major difference is that now the transition occurs about a time t=π/2, rather than t=1. The proof has three ingredients. The size of the largest component in the subcritical phase is bounded by comparison with a certain multitype branching process. With this bound at hand, the growth of the sum‐of‐squares and sum‐of‐cubes of component sizes is shown, via martingale methods, to follow closely a solution of the Smoluchowsky‐type equations. The approximation allows us to apply results of Aldous [Brownian excursions, critical random graphs and the multiplicative coalescent, Ann Probab 25 (1997), 812–854] on emergence of giant components in the multiplicative coalescent, i.e., a nonuniform random graph process. © 2000 John Wiley & Sons, Inc. Random Struct. Alg., 17: 79–102, 2000     

Distribution of subgraphs of random regular graphs

We obtain the asymptotic distribution of the number of copies of a fixed subgraph H in a random d‐regular graph, provided H is strictly balanced and d = d(n) is chosen so that the expected number of copies of H tends to infinity (but not too quickly), and the expected number of copies sharing edges with two other copies is bounded. The proof of asymptotic normality of the distribution uses a method of factorial moments for variables with unbounded means that was recently derived by the authors. © 2007 Wiley Periodicals, Inc. Random Struct. Alg., 2008     

Saturation in random graphs

A graph H is K_s ‐saturated if it is a maximal K_s ‐free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K_s ‐saturated graph was determined over 50 years ago by Zykov and independently by Erd?s, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal K_s ‐free subgraph of the Erd?s‐Rényi random graph G (n, p ). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 169–181, 2017