NDLI: Community Detection in General Stochastic Block models: Fundamental Limits and Efficient Algorithms for Recovery

Content Provider	IEEE Xplore Digital Library
Author	Abbe, E. Sandon, C.
Copyright Year	2015
Description	Author affiliation: Dept. Princeton, Univ. Princeton, Princeton, NJ, USA (Abbe, E.) \|\| Math. Dept., Princeton Univ., Princeton, NJ, USA (Sandon, C.)
Abstract	New phase transition phenomena have recently been discovered for the stochastic block model, for the special case of two non-overlapping symmetric communities. This gives raise in particular to new algorithmic challenges driven by the thresholds. This paper investigates whether a general phenomenon takes place for multiple communities, without imposing symmetry. In the general stochastic block model SBM(n,p,W), n vertices are split into k communities of relative size $\{p_{i}\}_{i∈[k]},$ and vertices in community i and j connect independently with probability $\{W_{ij}\}_{i,j∈[k]}.$ This paper investigates the partial and exact recovery of communities in the general SBM (in the constant and logarithmic degree regimes), and uses the generality of the results to tackle overlapping communities. The contributions of the paper are: (i) an explicit characterization of the recovery threshold in the general SBM in terms of a new f-divergence function $D_{+},$ which generalizes the Hellinger and Chernoff divergences, and which provides an operational meaning to a divergence function analog to the KL-divergence in the channel coding theorem, (ii) the development of an algorithm that recovers the communities all the way down to the optimal threshold and runs in quasi-linear time, showing that exact recovery has no information-theoretic to computational gap for multiple communities, (iii) the development of an efficient algorithm that detects communities in the constant degree regime with an explicit accuracy bound that can be made arbitrarily close to 1 when a prescribed signal-to-noise ratio [defined in terms of the spectrum of diag(p)W] tends to infinity.
Starting Page	670
Ending Page	688
File Size	506466
Page Count	19
File Format	PDF
ISBN	9781467381918
ISSN	02725428
DOI	10.1109/FOCS.2015.47
Language	English
Publisher	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Publisher Date	2015-10-17
Publisher Place	USA
Access Restriction	Subscribed
Rights Holder	Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Subject Keyword	Stochastic processes Clustering algorithms Mathematical model Computational modeling Complexity theory Computer science Signal to noise ratio graph-based codes Community detection stochastic block models phase transitions clustering algorithms information measures
Content Type	Text
Resource Type	Article

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