The problem of efficiently sampling from a set of (undirected, or directed) graphs with a given degree sequence has many applications. One approach to this problem uses a simple Markov chain, which we call the switch chain, to perform the sampling. The switch chain is known to be rapidly mixing for regular degree sequences, both in the undirected and directed setting.We prove that the switch chain for undirected graphs is rapidly mixing for any degree sequence with minimum degree at least 1 and with maximum degree d max which satisfies 3 ≤ d max ≤ 1 3 √ M , where M is the sum of the degrees. The mixing time bound obtained is only a factor n larger than that established in the regular case, where n is the number of vertices. Our result covers a wide range of degree sequences, including power-law density-bounded graphs with parameter γ > 5/2 and sufficiently many edges.For directed degree sequences such that the switch chain is irreducible, we prove that the switch chain is rapidly mixing when all in-degrees and out-degrees are positive and bounded above by 1 4 √ m, where m is the number of arcs, and not all in-degrees and out-degrees equal 1. The mixing time bound obtained in the directed case is an order of m 2 larger than that established in the regular case.Here the running time of the sampling algorithm must be (deterministically) polynomially bounded but the output need not be exactly uniform: however, the user can specify how far from the uniform distribution the samples may be. Other approaches to the problem of sampling graphs (or directed graphs) are discussed in Section 1.1.The switch chain is a natural and well-studied Markov chain for sampling from a set of graphs with a given degree sequence. Each move of the switch chain selects two distinct edges uniformly at random and attempts to replace these edges by a perfect matching of the four endvertices, chosen uniformly at random. The proposed move is rejected if the four endvertices are not distinct or if a multiple edge would be formed. We call each such move a switch. The precise definitions of the transitions for the switch chain for undirected and directed graphs are given at the start of Sections 2 and 3, respectively.Ryser [34] used switches to study the structure of 0-1 matrices. Markov chains based on switches have been introduced by Besag and Clifford [5] for 0-1 matrices (bipartite graphs), Diaconis and Sturmfels [9] for contingency tables and Rao, Jana and Bandyopadhyay [33] for directed graphs.The switch chain is aperiodic and its transition matrix is symmetric. It is well-known that the switch chain is irreducible for any (undirected) degree sequence: see [32,37]. Irreducibility for the directed chain is not guaranteed, see Rao et al. [33]. However, Berger and Müller-Hanneman [4] and LaMar [24,26] gave characterisations of directed degree sequences for which the switch chain is irreducible. In particular, the switch chain is irreducible for regular directed graphs (see for example Greenhill [15, Lemma 2.2]).In order for the switch chain to b...

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Inhomogeneous Erdős-Rényi random graphs GN on N vertices in the non-dense regime are considered in this paper. The edge between the pair of vertices {i, j} is retained with probability εN f ( i N , j N ),We study the empirical distribution of both the adjacency matrix AN and the Laplacian matrix ∆N associated with GN in the limit as N → ∞ when limN→∞ εN = 0 and limN→∞ N εN = ∞. In particular, it is shown that the empirical spectral distributions of AN and ∆N , after appropriate scaling and centering, converge to deterministic limits weakly in probability. For the special case where f (x, y) = r(x)r(y) with r : [0, 1] → [0, ∞) a continuous function, we give an explicit characterization of the limiting distributions. Furthermore, applications of the results to constrained random graphs, Chung-Lu random graphs and social networks are shown.MSC 2010 subject classifications: Primary 60B20, 05C80 ; secondary 46L54

We consider random-access networks where each node represents a server with a queue. Each node can be either active or inactive. A node deactivates at unit rate, while activates a rate that depends on its queue, provided none of its neighbors is active.We consider arbitrary bipartite graphs in the limit as the queues become large. We identify the transition time between the two states where one half of the network is active and the other half is inactive. We decompose the transition into a succession of transitions on complete bipartite subgraphs. We formulate a greedy algorithm that takes the graph as input and gives as output the set of transition paths the system is most likely to follow. Along each path we determine the mean transition time and its law on the scale of its mean. Depending on the activation rate functions, we identify three regimes of behavior.

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