Several probabilistic models from high-dimensional statistics and machine learning reveal an intriguing --and yet poorly understood-- dichotomy. Either simple local algorithms succeed in estimating the object of interest, or even sophisticated semi-definite programming (SDP) relaxations fail. In order to explore this phenomenon, we study a classical SDP relaxation of the minimum graph bisection problem, when applied to Erd o s-Renyi random graphs with bounded average degree @math , and obtain several types of results. First, we use a dual witness construction (using the so-called non-backtracking matrix of the graph) to upper bound the SDP value. Second, we prove that a simple local algorithm approximately solves the SDP to within a factor @math of the upper bound. In particular, the local algorithm is at most @math suboptimal, and @math suboptimal for large degree. We then analyze a more sophisticated local algorithm, which aggregates information according to the harmonic measure on the limiting Galton-Watson (GW) tree. The resulting lower bound is expressed in terms of the conductance of the GW tree and matches surprisingly well the empirically determined SDP values on large-scale Erd o s-Renyi graphs. We finally consider the planted partition model. In this case, purely local algorithms are known to fail, but they do succeed if a small amount of side information is available. Our results imply quantitative bounds on the threshold for partial recovery using SDP in this model.
Several papers applied sophisticated spectral methods to the stochastic block model near the detection threshold @cite_37 @cite_29 @cite_39 . Our upper bound in Theorem is based on a duality argument, where we establish feasibility of a certain dual witness construction using an argument similar to @cite_39 .
@cite_37 @cite_29 @cite_39
2023348178 1914749871
[1] conjectured the existence of a sharp threshold on model parameters for community detection in sparse random graphs drawn from the stochastic block model. Mossel, Neeman and Sly [2] established the negative part of the conjecture, proving impossibility of non-trivial reconstruction below the threshold. In this work we solve the positive part of the conjecture. To that end we introduce a modified adjacency matrix B which counts self-avoiding paths of a given length e between pairs of nodes. We then prove that for logarithmic length e, the leading eigenvectors of this modified matrix provide a non-trivial reconstruction of the underlying structure, thereby settling the conjecture. A key step in the proof consists in establishing a weak Ramanujan property of the constructed matrix B. Namely, the spectrum of B consists in two leading eigenvalues ρ(B), λ2 and n -- 2 eigenvalues of a lower order O(ne √ρ(B) for all e 0, ρ(B) denoting B's spectral radius. We study a random graph model named the "block model" in statistics and the "planted partition model" in theoretical computer science. In its simplest form, this is a random graph with two equal-sized clusters, with a between-class edge probability of @math and a within-class edge probability of @math . A striking conjecture of Decelle, Krzkala, Moore and Zdeborov 'a based on deep, non-rigorous ideas from statistical physics, gave a precise prediction for the algorithmic threshold of clustering in the sparse planted partition model. In particular, if @math and @math , @math and @math then conjectured that it is possible to efficiently cluster in a way correlated with the true partition if @math and impossible if @math for some sufficiently large @math . In a previous work, we proved that indeed it is information theoretically impossible to to cluster if @math . A different independent proof of the same result was recently obtained by Laurent Massoulie.