Accurate Community Detection in the Stochastic Block Model via Spectral Algorithms

Abstract

We consider the problem of community detection in the Stochastic Block Model with a finite number $K$ of communities of sizes linearly growing with the network size $n$. This model consists in a random graph such that each pair of vertices is connected independently with probability $p$ within communities and $q$ across communities. One observes a realization of this random graph, and the objective is to reconstruct the communities from this observation. We show that under spectral algorithms, the number of misclassified vertices does not exceed $s$ with high probability as $n$ grows large, whenever $pn=\omega (1)$, $s=o(n)$ and \begin{equation*} \lim\inf_{n\to\infty} {n(α_1 p+α_2 q-(α_1 + α_2)p^{\frac{α_1}{α_1 + α_2}}q^{\frac{α_2}{α_1 + α_2}})\over \log (\frac{n}{s})} >1,\quad\quad(1) \end{equation*} where ${\alpha }_1$ and ${\alpha }_2$ denote the (fixed) proportions of vertices in the two smallest communities. In view of recent work by Abbe et al. and Mossel et al., this establishes that the proposed spectral algorithms are able to exactly recover communities whenever this is at all possible in the case of networks with two communities with equal sizes. We conjecture that condition (1) is actually necessary to obtain less than $s$ misclassified vertices asymptotically, which would establish the optimality of spectral method in more general scenarios.

References in corpus (1)

• Phase transition in the detection of modules in sparse networks

Cited by in corpus (4)

• Recovering communities in the general stochastic block model without knowing the parameters
• Non-Asymptotic Chernoff Lower Bound and Its Application to Community Detection in Stochastic Block Model
• Exponential error rates of SDP for block models: Beyond Grothendieck's inequality
• Achieving the Bayes Error Rate in Synchronization and Block Models by SDP, Robustly