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We study random digraphs on sequences of expanders with bounded average degree and weak local limit. The threshold for the existence of a giant strongly connected component, as well as the asymptotic fraction of nodes with giant fan-in or giant fan-out are local, in the sense that they are the same for two sequences with the same weak local limit. The digraph has a bow-tie structure, with all but a vanishing fraction of nodes lying either in the unique strongly connected giant and its fan-in and fan-out, or in sets with small fan-in and small fan-out. All local quantities are expressed in terms of percolation on the limiting rooted graph, without any structural assumptions on the limit, allowing, in particular, for non tree-like limits. In the course of proving these results, we prove that for unoriented percolation, there is a unique giant above criticality, whose size and critical threshold are again local. An application of our methods shows that the critical threshold for bond percolation and random digraphs on preferential attachment graphs is $p_c=0$, with an infinite order phase transition at $p_c$.
We study the typical behavior of a generalized version of Googles PageRank algorithm on a large family of inhomogeneous random digraphs. This family includes as special cases direct
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 direct
We discuss transpose (sometimes called universal exchange or all-to-all) on vertex symmetric networks. We provide a method to compare the efficiency of transpose schemes on two different networks with a cost function based on the number processors an
For $Delta ge 5$ and $q$ large as a function of $Delta$, we give a detailed picture of the phase transition of the random cluster model on random $Delta$-regular graphs. In particular, we determine the limiting distribution of the weights of the orde
The cutoff phenomenon was recently confirmed for random walks on Ramanujan graphs by the first author and Peres. In this work, we obtain analogs in higher dimensions, for random walk operators on any Ramanujan complex associated with a simple group $