We consider a random sparse graph with bounded average degree, in which a subset of vertices has higher connectivity than the background. In particular, the average degree inside this subset of vertices is larger than outside (but still bounded). Given a realization of such graph, we aim at identifying the hidden subset of vertices. This can be regarded as a model for the problem of finding a tightly knitted community in a social network, or a cluster in a relational dataset.
In this paper we present two sets of contributions: $(i)$ We use the cavity method from spin glass theory to derive an exact phase diagram for the reconstruction problem. In particular, as the difference in edge probability increases, the problem undergoes two phase transitions, a static phase transition and a dynamic one. $(ii)$ We establish rigorous bounds on the dynamic phase transition and prove that, above a certain threshold, a local algorithm (belief propagation) correctly identify most of the hidden set. Below the same threshold \emph{no local algorithm} can achieve this goal. However, in this regime the subset can be identified by exhaustive search.
For small hidden sets and large average degree, the phase transition for local algorithms takes an intriguingly simple form. Local algorithms succeed with high probability for ${\rm deg}_{\rm in} - {\rm deg}_{\rm out} > \sqrt{{\rm deg}_{\rm out}/e}$ and fail for ${\rm deg}_{\rm in} - {\rm deg}_{\rm out} < \sqrt{{\rm deg}_{\rm out}/e}$ (with ${\rm deg}_{\rm in}$, ${\rm deg}_{\rm out}$ the average degrees inside and outside the community). We argue that spectral algorithms are also ineffective in the latter regime.
It is an open problem whether any polynomial time algorithms might succeed for ${\rm deg}_{\rm in} - {\rm deg}_{\rm out} < \sqrt{{\rm deg}_{\rm out}/e}$.
Community detection in sparse time-evolving graphs with a dynamical Bethe-Hessian
GIPSA-lab（吉卜赛实验室）；UGA（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立理工学院）；L2S（信号与系统实验室）；CentraleSupélec（中央理工-高等电力学院）；University of Paris Saclay（巴黎萨克雷大学）
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。
Fudan University（复旦大学）；Shanghai Center for Mathematical Sciences（上海数学中心）；University of Pennsylvania（宾夕法尼亚大学）；Shanghai University of Finance and Economics（上海财经大学）
Revisiting the Bethe-Hessian: Improved Community Detection in Sparse Heterogeneous Graphs
GIPSA-lab（GIPSA实验室）；UGA（格勒诺布尔阿尔卑斯大学）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔国立高等物理、数学与信息学院）；L2S（信号与系统实验室）；CentraleSupélec（中央理工-高等电力学院）；University of Paris Saclay（巴黎萨克雷大学）
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）
Finding the Sparsest Vectors in a Subspace: Theory, Algorithms, and Applications
New York University（纽约大学）；University of Denver（丹佛大学）；Chinese University of Hong Kong（香港中文大学）；ShanghaiTech University（上海科技大学）；Columbia University（哥伦比亚大学）；Johns Hopkins University（约翰斯·霍普金斯大学）
Finding sparse solutions of systems of polynomial equations via group-sparsity optimization
LORIA（洛林信息学及其应用研究实验室）；Université de Lorraine（洛林大学）；CNRS（法国国家科学研究中心）；Inria（法国国家信息与自动化研究所）；University of California, Berkeley（加州大学伯克利分校）；Linköping University（林雪平大学）
