The stochastic block model is a natural model for studying community detection in random networks. Its clustering properties have been extensively studied in the statistics, physics and computer science literature. Recently this area has experienced major mathematical breakthroughs, particularly for the binary (two-community) version, see [24, 25, 20]. In this paper, we introduce a variant of the binary model which we call the regular stochastic block model (RSBM). We prove rigidity of this model by showing that with high probability an exact recovery of the community structure is possible. Spectral methods exhibit a regime where this can be done efficiently. Moreover we also prove that, in this setting, any suitably good partial recovery can be bootstrapped to obtain a full recovery of the communities. Definition of the model and main resultsThe stochastic block model (SBM) is a classical clusterexhibiting random graph model that has been extensively studied, both empirically and rigorously, across numerous fields. In its simplest form, the SBM is a model of random graphs on 2n nodes with two equalsized clusters A and B such that |A| = |B| = n and A ∩ B = ∅. Edges between various pairs of vertices appear independently with probability p = p n if the two vertices belong to the same cluster and with probability q = q n otherwise. Thus, for any vertex, the expected number of same-class neighbors is a := a n := p(n − 1) ∼ pn, and the expected number of across-class neighbors is b := b n := qn.Given a realization of the graph, the broad goal is to determine whether it is possible (with high probability) to find the partition A, B; and if the answer is * Department of Mathematics, University of Washington, gerandy@math.washington.edu † Department of Mathematics, University of Washington, dumitriu@math.washington.edu ‡ Department of Mathematics, University of Washington, sganguly@math.washington.edu § Department of Mathematics, University of Washington, hoffman@math.washington.edu ¶ International University, National University Hochiminh City, tvlinh@hcmiu.edu.vn yes, whether it is possible to do so using an efficient algorithm. Otherwise, the best one can hope for is the existence of an algorithm that will output a partition which is highly (or at least positively) correlated with the underlying cluster. To this end, consider the space M of all algorithms which take as input a finite graph on 2n vertices and output a partition of the vertex set into two sets. Informally, we say that an algorithm in M allows for weak recovery if, with probability going to 1 as n goes to infinity, it outputs a partition (A ′ , B ′ ) such that |A∆A ′ | + |B∆B ′ | = o(n) (here ∆ denotes the symmetric difference). We say that an algorithm allows for strong recovery if, with probability going to 1 as n goes to infinity, it outputs the partition (A, B). Finally, an algorithm in M will be called efficient if its run time is polynomial in n.The problem of community detection described above is closely related to the min-bisection prob...

No abstract

We prove an analogue of Alon's spectral gap conjecture for random bipartite, biregular graphs. We use the Ihara-Bass formula to connect the non-backtracking spectrum to that of the adjacency matrix, employing the moment method to show there exists a spectral gap for the non-backtracking matrix. Finally, we give some applications in machine learning and coding theory.

No abstract

We study the number of random permutations needed to invariably generate the symmetric group, Sn, when the distribution of cycle counts has the strong α-logarithmic property. The canonical example is the Ewens sampling formula, for which the number of k-cycles relates to a conditioned Poisson random variable with mean α/k. The special case α = 1 corresponds to uniformly random permutations, for which it was recently shown that exactly four are needed.For strong α-logarithmic measures, and almost every α, we show that precisely (1 − α log 2) −1 permutations are needed to invariably generate Sn. A corollary is that for many other probability measures on Sn no bounded number of permutations will invariably generate Sn with positive probability. Along the way we generalize classic theorems of Erdős, Tehran, Pyber, Luczak and Bovey to permutations obtained from the Ewens sampling formula. arXiv:1610.04212v2 [math.PR]