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 .
Another recent work @cite_22 focuses on the case @math , @math for fixed constants @math and @math , and @math for @math . It is shown that no polynomial-time algorithm can attain the information-theoretic threshold of detecting the planted dense subgraph unless the planted clique problem can be solved in polynomial time (see [Hypothesis 1] HajekWuXu14 for the precise statement). For exact recovery, MLE succeeds with high probability if @math ; however, no randomized polynomial-time solver exists, conditioned on the same planted clique hardness hypothesis.
@cite_22
1613681423 1859301526 1897704527 1987681136
This paper studies the problem of detecting the presence of a small dense community planted in a large Erd o s-R 'enyi random graph @math , where the edge probability within the community exceeds @math by a constant factor. Assuming the hardness of the planted clique detection problem, we show that the computational complexity of detecting the community exhibits the following phase transition phenomenon: As the graph size @math grows and the graph becomes sparser according to @math , there exists a critical value of @math , below which there exists a computationally intensive procedure that can detect far smaller communities than any computationally efficient procedure, and above which a linear-time procedure is statistically optimal. The results also lead to the average-case hardness results for recovering the dense community and approximating the densest @math -subgraph. 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 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. We develop new techniques for deriving very strong computational lower bounds for a class of well-known NP-hard problems, including weighted satisfiability , dominating set , hitting set , set cover , clique , and independent set . For example, although a trivial enumeration can easily test in time O(nk) if a given graph of n vertices has a clique of size k, we prove that unless an unlikely collapse occurs in parameterized complexity theory, the problem is not solvable in time f(k) no(k) for any function f, even if we restrict the parameter value k to be bounded by an arbitrarily small function of n. Under the same assumption, we prove that even if we restrict the parameter values k to be Θ(μ(n)) for any reasonable function μ, no algorithm of running time no(k) can test if a graph of n vertices has a clique of size k. Similar strong lower bounds are also derived for other problems in the above class. Our techniques can be extended to derive computational lower bounds on approximation algorithms for NP-hard optimization problems. For example, we prove that the NP-hard distinguishing substring selection problem, for which a polynomial time approximation scheme has been recently developed, has no polynomial time approximation schemes of running time f(1 e)no(1 e) for any function f unless an unlikely collapse occurs in parameterized complexity theory.