Networks and data supported on graphs have become ubiquitous in the sciences and engineering. This paper studies the 'blind' community detection problem, where we seek to infer the community structure of a graph model given the observation of independent graph signals on a set of nodes whose connections are unknown. We model each observation as filtered white noise, where the underlying network structure varies with every observation. These varying network structures are modeled as independent realizations of a latent planted partition model (PPM), justifying our assumption of a constant underlying community structure over all observations. Under certain conditions on the graph filter and PPM parameters, we propose algorithms for determining (i) the number of latent communities and (ii) the associated partitions of the PPM. We then prove statistical guarantees in the asymptotic and non-asymptotic sampling cases. Numerical experiments on real and synthetic data demonstrate the efficacy of our algorithms.
In community detection, the exact recovery of communities (clusters) has been mainly investigated under the general stochastic block model with edges drawn from Bernoulli distributions. This paper considers the exact recovery of communities in a complete graph in which the graph edges are drawn from either a set of Gaussian distributions with community-dependent means and variances, or a set of exponential distributions with community-dependent means. For each case, we introduce a new semi-metric that describes sufficient and necessary conditions of exact recovery. The necessary and sufficient…
We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the threshold for the exact recovery of the maximum likelihood estimation. Applications include the community detection on hypergraphs.
We consider the problem of learning latent community structure from multiple correlated networks. We study edge-correlated stochastic block models with two balanced communities, focusing on the regime where the average degree is logarithmic in the number of vertices. Our main result derives the precise information-theoretic threshold for exact community recovery using multiple correlated graphs. This threshold captures the interplay between the community recovery and graph matching tasks. In particular, we uncover and characterize a region of the parameter space where exact community recovery …
Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)（里约热内卢天主教大学）
The Degree-Corrected Stochastic Block Model (DCSBM) is a popular model to generate random graphs with community structure given an expected degree sequence. The standard approach of community detection based on the DCSBM is to search for the model parameters that are the most likely to have produced the observed network data through maximum likelihood estimation (MLE). Current techniques for the MLE problem are heuristics, and therefore do not guarantee convergence to the optimum. We present mathematical programming formulations and exact solution methods that can provably find the model param…
The Chinese University of Hong Kong（香港中文大学）；University of Michigan, Ann Arbor（密歇根大学安娜堡分校）
Signed graphs encode similarity and dissimilarity relationships among different entities with positive and negative edges. In this paper, we study the problem of community recovery over signed graphs generated by the signed stochastic block model (SSBM) with two equal-sized communities. Our approach is based on the maximum likelihood estimation (MLE) of the SSBM. Unlike many existing approaches, our formulation reveals that the positive and negative edges of a signed graph should be treated unequally. We then propose a simple two-stage iterative algorithm for solving the regularized MLE. It is…
Empirical observations suggest that in practice, community membership does not completely explain the dependency between the edges of an observation graph. The residual dependence of the graph edges are modeled in this paper, to first order, by auxiliary node latent variables that affect the statistics of the graph edges but carry no information about the communities of interest. We then study community detection in graphs obeying the stochastic block model and censored block model with auxiliary latent variables. We analyze the conditions for exact recovery when these auxiliary latent variabl…
Heuristic and exact modularity optimization with size-constrained communities
Kellogg School of Management, Northwestern University（西北大学凯洛格管理学院）；Department of Mechanical and Industrial Engineering, University of Toronto（多伦多大学机械与工业工程系）；Centre for Science and Technology Studies (CWTS), Leiden University（莱顿大学科学技术研究中心）；Center for Complex Networks and Systems Research, Luddy School of Informatics, Computing, and Engineering, Indiana University Bloomington（印第安纳大学布鲁明顿分校卢迪信息、计算与工程学院复杂网络与系统研究中心）
Markov dynamics as a zooming lens for multiscale community detection: non clique-like communities and the field-of-view limit
近年来，复杂网络社区发现算法受到了广泛的关注。已经提出了各种各样的计算几何学，其中一些具有很长的历史，用于识别社区，或者用于识别良好的图分区。在大多数情况下，算法最大限度地提高了一个特定的目标函数，从而找到了“正确的”分裂成社区。尽管仍然缺乏对算法的彻底比较，但已经努力设计基准，即，具有已知社区结构的随机图模型，可以对算法进行评估。然而，流行的社区检测方法和基准通常假设一个隐式的社区概念的基础上，类candle子图，一种形式的社区结构，并不总是真实网络的特征。具体来说，从几何约束中产生的网络可以具有自然的非类cyclo-like子结构，具有大的有效直径，这可以被解释为长距离社区。在这项工作中，我们表明，远程社区逃避检测流行的方法，这是盲目的一个有限的“视野”的限制，一个内在的上限规模上的社区，他们可以检测到。视野限制意味着远距离社区往往被过度分割。我们展示了如何通过对社区检测采用动态视角（Delvenne et al.（2010）PNAS：107：12755-12760; Lambiotte et al.（2008）arXiv：0812.1770），其中图上的马尔可夫过程的演变被用作在所有尺度上的网络结构上的变焦镜头，人们可以检测集团或非集团样社区，而无需对检测施加上限。因此，算法在固有的低直径，类candle基准上的性能可能并不总是表示在具有本地，稀疏连接的真实网络中同样好的结果…
Random-walk based network embedding algorithms like DeepWalk and node2vec are widely used to obtain Euclidean representation of the nodes in a network prior to performing downstream inference tasks. However, despite their impressive empirical performance, there is a lack of theoretical results explaining their large-sample behavior. In this paper, we study node2vec and DeepWalk through the perspective of matrix factorization. In particular, we analyze these algorithms in the setting of community detection for stochastic blockmodel graphs (and their degree-corrected variants). By exploiting the…
The University of Texas at Dallas（德克萨斯大学达拉斯分校）；Virginia Tech（弗吉尼亚理工大学）
This paper produces an efficient Semidefinite Programming (SDP) solution for community detection that incorporates non-graph data, which in this context is known as side information. SDP is an efficient solution for standard community detection on graphs. We formulate a semi-definite relaxation for the maximum likelihood estimation of node labels, subject to observing both graph and non-graph data. This formulation is distinct from the SDP solution of standard community detection, but maintains its desirable properties. We calculate the exact recovery threshold for three types of non-graph inf…
