Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime @math A critical parameter is the effective signal-to-noise ratio @math , with @math corresponding to the Kesten-Stigum threshold. We show that a belief propagation algorithm achieves weak recovery if @math , beyond the Kesten-Stigum threshold by a factor of @math The belief propagation algorithm only needs to run for @math iterations, with the total time complexity @math , where @math is the iterated logarithm of @math Conversely, if @math , no local algorithm can asymptotically outperform trivial random guessing. Furthermore, a linear message-passing algorithm that corresponds to applying power iteration to the non-backtracking matrix of the graph is shown to attain weak recovery if and only if @math . In addition, the belief propagation algorithm can be combined with a linear-time voting procedure to achieve the information limit of exact recovery (correctly classify all vertices with high probability) for all @math where @math is a function of @math .
We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real-valued random matrix. Of particular interest is the setting where the number of clusters submatrices is allowed to grow unbounded with the problem size. These formulations cover several classical models such as planted clique, planted densest subgraph, planted partition, planted coloring, and stochastic block model, which are widely used for studying community detection and clustering bi-clustering. For both problems, we show that the space of the model parameters (cluster submatrix size, cluster density, and submatrix mean) can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities: (1) the regime, where all algorithms fail; (2) the regime, where the computationally expensive Maximum Likelihood Estimator (MLE) succeeds; (3) the regime, where the polynomial-time convexified MLE succeeds; (4) the regime, where a simple counting thresholding procedure succeeds. Moreover, we show that each of these algorithms provably fails in the previous harder regimes. Our theorems establish the minimax recovery limit, which are tight up to constants and hold with a growing number of clusters submatrices, and provide a stronger performance guarantee than previously known for polynomial-time algorithms. Our study demonstrates the tradeoffs between statistical and computational considerations, and suggests that the minimax recovery limit may not be achievable by polynomial-time algorithms. Preface 1. Events and probability 2. Discrete random variables and expectation 3. Moments and deviations 4. Chernoff bounds 5. Balls, bins and random graphs 6. The probabilistic method 7. Markov chains and random walks 8. Continuous distributions and the Poisson process 9. Entropy, randomness and information 10. The Monte Carlo method 11. Coupling of Markov chains 12. Martingales 13. Pairwise independence and universal hash functions 14. Balanced allocations References.  In a random graph on n vertices, the maximum clique is likely to be of size very close to 2 lg n. However, the clique produced by applying the naive “greedy” heuristic to a random graph is unlikely to have size much exceeding lg n. The factor of two separating these estimates motivates the search for more effective heuristics. This article analyzes a heuristic search strategy, the Metropolis process, which is just one step above the greedy one in its level of sophistication. It is shown that the Metropolis process takes super-polynomial time to locate a clique that is only slightly bigger than that produced by the greedy heuristic. We give a lower bound of @math for the degree-4 Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, we show that on an Erd "os-R 'enyi graph @math , with high probability there is a feasible point for the degree-4 SOS relaxation of the clique problem with an objective value of @math , so that the program cannot distinguish between a random graph and a random graph with a planted clique of size @math . This bound is tight. We build on the works of Deshpande and Montanari and , who give lower bounds of @math and @math respectively. We improve on their results by making a perturbation to the SDP solution proposed in their work, then showing that this perturbation remains PSD as the objective value approaches @math . In an independent work, Hopkins, Kothari and Potechin [HKP15] have obtained a similar lower bound for the degree- @math SOS relaxation. In the hidden clique problem, one needs to find the maximum clique in an @math -vertex graph that has a clique of size @math but is otherwise random. An algorithm of Alon, Krivelevich and Sudakov that is based on spectral techniques is known to solve this problem (with high probability over the random choice of input graph) when @math for a sufficiently large constant @math . In this manuscript we present a new algorithm for finding hidden cliques. It too provably works when @math for a sufficiently large constant @math . However, our algorithm has the advantage of being much simpler (no use of spectral techniques), running faster (linear time), and experiments show that the leading constant @math is smaller than in the spectral approach. We also present linear time algorithms that experimentally find even smaller hidden cliques, though it remains open whether any of these algorithms finds hidden cliques of size @math . We consider the following probabilistic model of a graph on n labeled vertices. First choose a random graph G(n, 1 2), and then choose randomly a subset Q of vertices of size k and force it to be a clique by joining every pair of vertices of Q by an edge. The problem is to give a polynomial time algorithm for finding this hidden clique almost surely for various values of k. This question was posed independently, in various variants, by Jerrum and by Kucera. In this paper we present an efficient algorithm for all k>cn0.5, for any fixed c>0, thus improving the trivial case k>cn0.5(log n)0.5. The algorithm is based on the spectral properties of the graph. © 1998 John Wiley & Sons, Inc. Random Struct. Alg., 13: 457–466, 1998 Given a large data matrix @math , we consider the problem of determining whether its entries are i.i.d. with some known marginal distribution @math , or instead @math contains a principal submatrix @math whose entries have marginal distribution @math . As a special case, the hidden (or planted) clique problem requires to find a planted clique in an otherwise uniformly random graph. Assuming unbounded computational resources, this hypothesis testing problem is statistically solvable provided @math for a suitable constant @math . However, despite substantial effort, no polynomial time algorithm is known that succeeds with high probability when @math . Recently Meka and Wigderson meka2013association , proposed a method to establish lower bounds within the Sum of Squares (SOS) semidefinite hierarchy. Here we consider the degree- @math SOS relaxation, and study the construction of meka2013association to prove that SOS fails unless @math . An argument presented by Barak implies that this lower bound cannot be substantially improved unless the witness construction is changed in the proof. Our proof uses the moments method to bound the spectrum of a certain random association scheme, i.e. a symmetric random matrix whose rows and columns are indexed by the edges of an Erd "os-Renyi random graph. We consider the problems of finding a maximum clique in a graph and finding a maximum-edge biclique in a bipartite graph. Both problems are NP-hard. We write both problems as matrix-rank minimization and then relax them using the nuclear norm. This technique, which may be regarded as a generalization of compressive sensing, has recently been shown to be an effective way to solve rank optimization problems. In the special case that the input graph has a planted clique or biclique (i.e., a single large clique or biclique plus diversionary edges), our algorithm successfully provides an exact solution to the original instance. For each problem, we provide two analyses of when our algorithm succeeds. In the first analysis, the diversionary edges are placed by an adversary. In the second, they are placed at random. In the case of random edges for the planted clique problem, we obtain the same bound as Alon, Krivelevich and Sudakov as well as Feige and Krauthgamer, but we use different techniques.