We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $G \sim G(n,1/2,k)$ be a random graph on $n$ vertices with a planted clique of size $k$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $G$ is likely to find the planted clique. On the other hand, when $k \geq (2+\epsilon) \log_2 n$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $q = O( (n^2 / k^2) \log^2 n + n \log n)$ adaptive queries. For detection, the additive $n$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors).
Recovering a hidden community beyond the Kesten–Stigum threshold in O(Elog*V) time
Abstract 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 n 1 o (1) ≤ K ≤ o ( n ). A critical parameter is the effective signaltonoise ratio λ = K 2 ( p  q ) 2 / (( n  K ) q ), with λ = 1 corresponding to the Kesten–Stigum threshold. We show that a belief propagation (BP) algorithm achieves weak recovery if λ > 1 / e, beyond the Kesten–Stigum threshold by a factor of 1 / e. The BP algorithm only needs to run for log * n + O (1) iterations, with the total time complexity O ( E log * n ), where log * n is the iterated logarithm of n . Conversely, if λ ≤ 1 / e, no local algorithm can asymptotically outperform trivial random more »
 Award ID(s):
 1651588
 Publication Date:
 NSFPAR ID:
 10089330
 Journal Name:
 Journal of Applied Probability
 Volume:
 55
 Issue:
 02
 Page Range or eLocationID:
 325 to 352
 ISSN:
 00219002
 Sponsoring Org:
 National Science Foundation
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