We give upper and lower bounds on the information-theoretic threshold for community detection in the stochastic block model. Specifically, consider the symmetric stochastic block model with $q$ groups, average degree $d$, and connection probabilities $c_\text{in}/n$ and $c_\text{out}/n$ for within-group and between-group edges respectively; let $λ= (c_\text{in}-c_\text{out})/(qd)$. We show that, when $q$ is large, and $λ= O(1/q)$, the critical value of $d$ at which community detection becomes possible---in physical terms, the condensation threshold---is \[ d_\text{c} = Θ\!\left( \frac{\log q}{q λ^2} \right) \, , \] with tighter results in certain regimes. Above this threshold, we show that any partition of the nodes into $q$ groups which is as `good' as the planted one, in terms of the number of within- and between-group edges, is correlated with it. This gives an exponential-time algorithm that performs better than chance; specifically, community detection becomes possible below the Kesten-Stigum bound for $q \ge 5$ in the disassortative case $λ< 0$, and for $q \ge 11$ in the assortative case $λ>0$ (similar upper bounds were obtained independently by Abbe and Sandon). Conversely, below this threshold, we show that no algorithm can label the vertices better than chance, or even distinguish the block model from an \ER\ random graph with high probability.
Our lower bound on $d_\text{c}$ uses Robinson and Wormald's small subgraph conditioning method, and we also give (less explicit) results for non-symmetric stochastic block models. In the symmetric case, we obtain explicit results by using bounds on certain functions of doubly stochastic matrices due to Achlioptas and Naor; indeed, our lower bound on $d_\text{c}$ is their second moment lower bound on the $q$-colorability threshold for random graphs with a certain effective degree.
Community detection in the sparse hypergraph stochastic block model
University of Washington（华盛顿大学）；University of California, San Diego（加州大学圣地亚哥分校）
本文研究了稀疏随机超图中的社团检测问题。Angelini et al.（2015）证明了在由超图随机块模型生成的稀疏超图中，社区检测的模型参数存在一个尖锐的阈值。我们解决了两个块的情况下的猜想的积极的一部分：在阈值以上，有一个谱算法，渐近几乎肯定构建一个分区的超图与真正的分区。我们的方法是Massoulié（2014）为稀疏随机图开发的方法的随机超图的推广。
Graph energy as a measure of community detectability in networks
Frankfurt School of Finance and Management（法兰克福金融管理学院）；University of Florida（佛罗里达大学）；University of California, Los Angeles（加州大学洛杉矶分校）；Santa Fe Institute（圣塔菲研究所）；Indiana University（印第安纳大学）
