Asymptotic normality in random graphs with given vertex degrees

We consider random graphs with a given degree sequence and show, under weak technical conditions, asymptotic normality of the number of components isomorphic to a given tree, first for the random multigraph given by the configuration model and then, by a conditioning argument, for the simple uniform random graph with the given degree sequence. Such conditioning is standard for convergence in probability, but much less straightforward for convergence in distribution as here. The proof uses the method of moments, and is based on a new estimate of mixed cumulants in a case of weakly dependent variables. The result on small components is applied to give a new proof of a recent result by Barbour and Röllin on asymptotic normality of the size of the giant component in the random multigraph; moreover, we extend this to the random simple graph.     

Random graphs with given vertex degrees and switchings

Random graphs with a given degree sequence are often constructed using the configuration model, which yields a random multigraph. We may adjust this multigraph by a sequence of switchings, eventually yielding a simple graph. We show that, assuming essentially a bounded second moment of the degree distribution, this construction with the simplest types of switchings yields a simple random graph with an almost uniform distribution, in the sense that the total variation distance is o(1). This construction can be used to transfer results on distributional convergence from the configuration model multigraph to the uniform random simple graph with the given vertex degrees. As examples, we give a few applications to asymptotic normality. We show also a weaker result yielding contiguity when the maximum degree is too large for the main theorem to hold.     

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.     

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     

Further results on random cubic planar graphs

We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.     

Gromov hyperbolicity of periodic planar graphs

The study of hyperbolic graphs is an interesting topic since the hyperbolicity of a geodesic metric space is equivalent to the hyperbolicity of a graph related to it.The main result in this paper is a very simple characterization of the hyperbolicity of a large class of periodic planar graphs.     

Random dense bipartite graphs and directed graphs with specified degrees

Let s and t be vectors of positive integers with the same sum. We study the uniform distribution on the space of simple bipartite graphs with degree sequence s in one part and t in the other; equivalently, binary matrices with row sums s and column sums t . In particular, we find precise formulae for the probabilities that a given bipartite graph is edge‐disjoint from, a subgraph of, or an induced subgraph of a random graph in the class. We also give similar formulae for the uniform distribution on the set of simple directed graphs with out‐degrees s and in‐degrees t . In each case, the graphs or digraphs are required to be sufficiently dense, with the degrees varying within certain limits, and the subgraphs are required to be sufficiently sparse. Previous results were restricted to spaces of sparse graphs. Our theorems are based on an enumeration of bipartite graphs avoiding a given set of edges, proved by multidimensional complex integration. As a sample application, we determine the expected permanent of a random binary matrix with row sums s and column sums t . © 2009 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     

Cover time and mixing time of random walks on dynamic graphs

The application of simple random walks on graphs is a powerful tool that is useful in many algorithmic settings such as network exploration, sampling, information spreading, and distributed computing. This is due to the reliance of a simple random walk on only local data, its negligible memory requirements, and its distributed nature. It is well known that for static graphs the cover time, that is, the expected time to visit every node of the graph, and the mixing time, that is, the time to sample a node according to the stationary distribution, are at most polynomial relative to the size of the graph. Motivated by real world networks, such as peer‐to‐peer and wireless networks, the conference version of this paper was the first to study random walks on arbitrary dynamic networks. We study the most general model in which an oblivious adversary is permitted to change the graph after every step of the random walk. In contrast to static graphs, and somewhat counter‐intuitively, we show that there are adversary strategies that force the expected cover time and the mixing time of the simple random walk on dynamic graphs to be exponentially long, even when at each time step the network is well connected and rapidly mixing. To resolve this, we propose a simple strategy, the lazy random walk, which guarantees, under minor conditions, polynomial cover time and polynomial mixing time regardless of the changes made by the adversary.     

Random walks on random simple graphs

This paper looks at random regular simple graphs and considers nearest neighbor random walks on such graphs. This paper considers walks where the degree d of each vertex is around (log n)^a where a is a constant which is at least 2 and where n is the number of vertices. By extending techniques of Dou, this paper shows that for most such graphs, the position of the random walk becomes close to uniformly distributed after slightly more than log n/log d steps. This paper also gets similar results for the random graph G(n, p), where p = d/(n − 1). © 1996 John Wiley & Sons, Inc.     

Small maximal matchings of random cubic graphs

We consider the expected size of a smallest maximal matching of cubic graphs. Firstly, we present a randomized greedy algorithm for finding a small maximal matching of cubic graphs. We analyze the average‐case performance of this heuristic on random n‐vertex cubic graphs using differential equations. In this way, we prove that the expected size of the maximal matching returned by the algorithm is asymptotically almost surely (a.a.s.) less than 0.34623n. We also give an existence proof which shows that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. less than 0.3214n. It is known that the size of a smallest maximal matching of a random n‐vertex cubic graph is a.a.s. larger than 0.3158n. © 2009 Wiley Periodicals, Inc. J Graph Theory 62: 293–323, 2009     

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     

Competing first passage percolation on random graphs with finite variance degrees

We study the growth of two competing infection types on graphs generated by the configuration model with a given degree sequence. Starting from two vertices chosen uniformly at random, the infection types spread via the edges in the graph in that an uninfected vertex becomes type 1 (2) infected at rate λ₁ (λ₂) times the number of nearest neighbors of type 1 (2). Assuming (essentially) that the degree of a randomly chosen vertex has finite second moment, we show that if λ₁ = λ₂, then the fraction of vertices that are ultimately infected by type 1 converges to a continuous random variable V ∈ (0,1), as the number of vertices tends to infinity. Both infection types hence occupy a positive (random) fraction of the vertices. If λ₁ ≠ λ₂, on the other hand, then the type with the larger intensity occupies all but a vanishing fraction of the vertices. Our results apply also to a uniformly chosen simple graph with the given degree sequence.     

Asymptotic enumeration of sparse 2‐connected graphs

We determine an asymptotic formula for the number of labelled 2‐connected (simple) graphs on n vertices and m edges, provided that m ‐ n →∞ and m = O(nlog n) as n →∞. This is the entire range of m not covered by previous results. The proof involves determining properties of the core and kernel of random graphs with minimum degree at least 2. The case of 2‐edge‐connectedness is treated similarly. We also obtain formulae for the number of 2‐connected graphs with given degree sequence for most (“typical”) sequences. Our main result solves a problem of Wright from 1983. © 2012 Wiley Periodicals, Inc. Random Struct. Alg., 2013     

Merging percolation on Zd and classical random graphs: Phase transition

We study a random graph model which is a superposition of bond percolation on Z^d with parameter p, 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 case” of inhomogeneous random graphs. This allows us to use the newly developed theory of inhomogeneous random graphs to describe the phase diagram on the set of parameters c ≥ 0 and 0 ≤ p < p_c, where p_c = p_c(d) is the critical probability for the bond percolation on Z^d. The phase transition is of second order as in the classical random graph. We find the scaled size of the largest connected component in the supercritical regime. We also provide a sharp upper bound for the largest connected component in the subcritical regime. The latter is a new result for inhomogeneous random graphs with unbounded kernels. © 2009 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Meyniel's conjecture holds for random d‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.     

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     

Modularity in random regular graphs and lattices

Given a graph G, the modularity of a partition of the vertex set measures the extent to which edge density is higher within parts than between parts; and the modularity of G is the maximum modularity of a partition.We give an upper bound on the modularity of r-regular graphs as a function of the edge expansion (or isoperimetric number) under the restriction that each part in our partition has a sub-linear numbers of vertices. This leads to results for random r-regular graphs. In particular we show the modularity of a random cubic graph partitioned into sub-linear parts is almost surely in the interval (0.66, 0.88).The modularity of a complete rectangular section of the integer lattice in a fixed dimension was estimated in Guimer et. al. [R. Guimerà, M. Sales-Pardo and L.A. Amaral, Modularity from fluctuations in random graphs and complex networks, Phys. Rev. E 70 (2) (2004) 025101]. We extend this result to any subgraph of such a lattice, and indeed to more general graphs.     

Connectivity of inhomogeneous random graphs

We find conditions for the connectivity of inhomogeneous random graphs with intermediate density. Our results generalize the classical result for G(n, p), when . We draw n independent points X_i from a general distribution on a separable metric space, and let their indices form the vertex set of a graph. An edge (i, j) is added with probability , where is a fixed kernel. We show that, under reasonably weak assumptions, the connectivity threshold of the model can be determined. © 2013 Wiley Periodicals, Inc. Random Struct. Alg., 45, 408‐420, 2014