The second largest eigenvalue of a transition matrix @math has connections with many properties of the underlying Markov chain, and especially its convergence rate towards the stationary distribution. In this paper, we give an asymptotic upper bound for the second eigenvalue when @math is the transition matrix of the simple random walk over a random directed graph with given degree sequence. This is the first result concerning the asymptotic behavior of the spectral gap for sparse non-reversible Markov chains with an unknown stationary distribution. An immediate consequence of our result is a generalization of the well-known Friedman theorem, for undirected regular graphs. Our result is based on a variation of the trace method introduced by Bordenave (2015).
In this paper, we consider random directed (multi)graphs with a specified sequence of in-degrees and out-degrees; when all the degrees are equal to @math , this model reduces to the directed @math -regular case. Our construction with half-edges is a directed variant of the classical configuration model (see @cite_33 ). When the degrees are bounded independently of the size of the graph, such multigraphs are sparse, meaning they have few edges. Even if digraphs are much more difficult to handle than undirected graphs, they are also one step closer to reality when modelling real-life situtations: see @cite_24 @cite_21 and references for (many) examples of graph-modelling that go beyond the Internet graph.
@cite_24 @cite_21 @cite_33
2169015768 1714396858 2020270524
Recent work on the structure of social networks and the internet has focused attention on graphs with distributions of vertex degree that are significantly different from the Poisson degree distributions that have been widely studied in the past. In this paper we develop in detail the theory of random graphs with arbitrary degree distributions. In addition to simple undirected, unipartite graphs, we examine the properties of directed and bipartite graphs. Among other results, we derive exact expressions for the position of the phase transition at which a giant component first forms, the mean component size, the size of the giant component if there is one, the mean number of vertices a certain distance away from a randomly chosen vertex, and the average vertex-vertex distance within a graph. We apply our theory to some real-world graphs, including the worldwide web and collaboration graphs of scientists and Fortune 1000 company directors. We demonstrate that in some cases random graphs with appropriate distributions of vertex degree predict with surprising accuracy the behavior of the real world, while in others there is a measurable discrepancy between theory and reality, perhaps indicating the presence of additional social structure in the network that is not captured by the random graph. The aim of this article is to discuss some applications of random processes in searching and reaching consensus on finite graphs. The topics covered are: Why random walks?, Speeding up random walks, Random and deterministic walks, Interacting particles and voting, Searching changing graphs. Given a graph G = (V, E) and a set of κ pairs of vertices in V, we are interested in finding, for each pair (ai, bi), a path connecting ai to bi such that the set of κ paths so found is edge-disjoint. (For arbitrary graphs the problem is NP-complete, although it is in P if κ is fixed.)We present a polynomial time randomized algorithm for finding edge-disjoint paths in the random regular graph Gn,r, for sufficiently large r. (The graph is chosen first, then an adversary chooses the pairs of end-points.) We show that almost every Gn,r is such that all sets of κ = Ω(n log n) pairs of vertices can be joined. This is within a constant factor of the optimum.