Achieving Exact Cluster Recovery Threshold via Semidefinite Programming

• Published in: IEEE transactions on information theory Vol. 62; no. 5; pp. 2788 - 2797
• Main Authors: Hajek, Bruce, Yihong Wu, Jiaming Xu
• Format: Journal Article
• Language: English
• Published: New York IEEE 01.05.2016; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Asymptotic methods; Clustering algorithms; Community detection; Erdos-Renyi random graph; Linear matrix inequalities; Maximum likelihood detection; Maximum likelihood estimation; Probability distribution; Programming; Semidefinite programming; Stochastic block model; Stochastic models; Stochastic processes; Symmetric matrices; stochastic block model; semidefinite programming; Community detection; Erdős-Rényi random graph
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2016.2546280

The binary symmetric stochastic block model deals with a random graph of n vertices partitioned into two equal-sized clusters, such that each pair of vertices is independently connected with probability p within clusters and q across clusters. In the asymptotic regime of p = a log n/n and q = b log n/n for fixed a, b, and n → ∞, we show that the semidefinite programming relaxation of the maximum likelihood estimator achieves the optimal threshold for exactly recovering the partition from the graph with probability tending to one, resolving a conjecture of Abbe et al. Furthermore, we show that the semidefinite programming relaxation also achieves the optimal recovery threshold in the planted dense subgraph model containing a single cluster of size proportional to n.