Multilayer and multiplex networks are becoming common network data sets in recent times. We consider the problem of identifying the common community structure for a special type of multilayer networks called multi-relational networks. We consider extensions of the spectral clustering methods for multi-relational networks and give theoretical guarantees that the spectral clustering methods recover community structure consistently for multi-relational networks generated from multilayer versions of both stochastic and degree-corrected block models even with dependence between network layers. The methods are shown to work under optimal conditions on the degree parameter of the networks to detect both assortative and disassortative community structures with vanishing error proportions even if individual layers of the multi-relational network has the network structures below community detectability threshold. We reinforce the validity of the theoretical results via simulations too.
Complex networks contain various interactions among similar or different entities. These kinds of networks are called multi-relational networks, in which each layer corresponds to a special type of interaction. Multi-relational networks are a particular type of multilayer networks in which nodes are similar entities; however, edges or communications demonstrate different types of interactions among similar entities. In this survey, we study community detection methods for multi-relational networks. The considered models are divided into two main groups, namely, direct methods and indirect meth…
There has been a surge of interest in community detection in homogeneous single-relational networks which contain only one type of nodes and edges. However, many real-world systems are naturally described as heterogeneous multi-relational networks which contain multiple types of nodes and edges. In this paper, we propose a new method for detecting communities in such networks. Our method is based on optimizing the composite modularity, which is a new modularity proposed for evaluating partitions of a heterogeneous multi-relational network into communities. Our method is parameter-free, scalabl…
University of Toledo（托莱多大学）；The Ohio State University（俄亥俄州立大学）
The stochastic block model (SBM) is one of the most widely used generative models for network data. Many continuous-time dynamic network models are built upon the same assumption as the SBM: edges or events between all pairs of nodes are conditionally independent given the block or community memberships, which prevents them from reproducing higher-order motifs such as triangles that are commonly observed in real networks. We propose the multivariate community Hawkes (MULCH) model, an extremely flexible community-based model for continuous-time networks that introduces dependence between node p…
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 …
Motivated by applications in domains such as social networks and computational biology, we study the problem of community recovery in graphs with locality. In this problem, pairwise noisy measurements of whether two nodes are in the same community or different communities come mainly or exclusively from nearby nodes rather than uniformly sampled between all nodes pairs, as in most existing models. We present an algorithm that runs nearly linearly in the number of measurements and which achieves the information theoretic limit for exact recovery.
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.
University of Illinois, Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）
This paper addresses the question of identifying the best candidate answer to a question on Community Question Answer (CQA) forums. The problem is important because Individuals often visit CQA forums to seek answers to nuanced questions. We develop a novel induced relational graph convolutional network (IR-GCN) framework to address the question. We make three contributions. First, we introduce a modular framework that separates the construction of the graph with the label selection mechanism. We use equivalence relations to induce a graph comprising cliques and identify two label assignment me…
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…
Community identification of network components enables us to understand the mesoscale clustering structure of networks. A number of algorithms have been developed to determine the most likely community structures in networks. Such a probabilistic or stochastic nature of this problem can naturally involve the ambiguity in resultant community structures. More specifically, stochastic algorithms can result in different community structures for each realization in principle. In this study, instead of trying to "solve" this community degeneracy problem, we turn the tables by taking the degeneracy a…
Twitter bot detection is an important and challenging task. Existing bot detection measures fail to address the challenge of community and disguise, falling short of detecting bots that disguise as genuine users and attack collectively. To address these two challenges of Twitter bot detection, we propose BotRGCN, which is short for Bot detection with Relational Graph Convolutional Networks. BotRGCN addresses the challenge of community by constructing a heterogeneous graph from follow relationships and applies relational graph convolutional networks. Apart from that, BotRGCN makes use of multi-…
Recent years have witnessed a growing number of researches on community characterization. In contrast to the large body of researches on the categorical measures (rise or decline) for evaluating the community development, we propose to estimate the community development strength (to which degree the rise or decline is). More specifically, given already known categorical information of community development, we are attempting to quantify the community development strength, which is of great interest. Motivated by the increasing availability of large-scale data on the network between entities am…
Targeting the right group of workers for crowdsourcing often achieves better quality results. One unique example of targeted crowdsourcing is seeking community-situated workers whose familiarity with the background and the norms of a particular group can help produce better outcome or accuracy. These community-situated crowd workers can be recruited in different ways from generic online crowdsourcing platforms or from online recovery communities. We evaluate three different approaches to recruit generic and community-situated crowd in terms of the time and the cost of recruitment, and the accu…
In this paper, we study the information-theoretic limits of community detection in the symmetric two-community stochastic block model, with intra-community and inter-community edge probabilities $\frac{a}{n}$ and $\frac{b}{n}$ respectively. We consider the sparse setting, in which $a$ and $b$ do not scale with $n$, and provide upper and lower bounds on the proportion of community labels recovered on average. We provide a numerical example for which the bounds are near-matching for moderate values of $a - b$, and matching in the limit as $a-b$ grows large.
We study a semidefinite programming (SDP) relaxation of the maximum likelihood estimation for exactly recovering a hidden community of cardinality $K$ from an $n \times n$ symmetric data matrix $A$, where for distinct indices $i,j$, $A_{ij} \sim P$ if $i, j$ are both in the community and $A_{ij} \sim Q$ otherwise, for two known probability distributions $P$ and $Q$. We identify a sufficient condition and a necessary condition for the success of SDP for the general model. For both the Bernoulli case ($P={\rm Bern}(p)$ and $Q={\rm Bern}(q)$ with $p>q$) and the Gaussian case ($P=\mathcal{N}(μ,1)$…
The ability to measure and track the speed and trajectory of a community's post-disaster recovery is essential to inform resource allocation and prioritization. The current survey-based approaches to examining community recovery, however, have significant lags and put the burden of data collection on affected people. Also, the existing literature lacks quantitative measures for important milestones to inform the assessment of recovery trajectory. Recognizing these gaps, this study uses location-based data related to visitation patterns and credit card transactions to specify critical recovery …
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…
Recently network analysis has gained more and more attentions in statistics, as well as in computer science, probability, and applied mathematics. Community detection for the stochastic block model (SBM) is probably the most studied topic in network analysis. Many methodologies have been proposed. Some beautiful and significant phase transition results are obtained in various settings. In this paper, we provide a general minimax theory for community detection. It gives minimax rates of the mis-match ratio for a wide rage of settings including homogeneous and inhomogeneous SBMs, dense and spars…
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…
We propose a dynamic network sampling scheme to optimize block recovery for stochastic blockmodel (SBM) in the case where it is prohibitively expensive to observe the entire graph. Theoretically, we provide justification of our proposed Chernoff-optimal dynamic sampling scheme via the Chernoff information. Practically, we evaluate the performance, in terms of block recovery, of our method on several real datasets from different domains. Both theoretically and practically results suggest that our method can identify vertices that have the most impact on block structure so that one can only chec…
We derive sharp thresholds for exact recovery of communities in a weighted stochastic block model, where observations are collected in the form of a weighted adjacency matrix, and the weight of each edge is generated independently from a distribution determined by the community membership of its endpoints. Our main result, characterizing the precise boundary between success and failure of maximum likelihood estimation when edge weights are drawn from discrete distributions, involves the Renyi divergence of order $\frac{1}{2}$ between the distributions of within-community and between-community …
We introduce the nonparametric metadata dependent relational (NMDR) model, a Bayesian nonparametric stochastic block model for network data. The NMDR allows the entities associated with each node to have mixed membership in an unbounded collection of latent communities. Learned regression models allow these memberships to depend on, and be predicted from, arbitrary node metadata. We develop efficient MCMC algorithms for learning NMDR models from partially observed node relationships. Retrospective MCMC methods allow our sampler to work directly with the infinite stick-breaking representation o…
Boston University（波士顿大学）；Carnegie Mellon University（卡内基梅隆大学）
We propose and analyze the problems of \textit{community goodness-of-fit and two-sample testing} for stochastic block models (SBM), where changes arise due to modification in community memberships of nodes. Motivated by practical applications, we consider the challenging sparse regime, where expected node degrees are constant, and the inter-community mean degree ($b$) scales proportionally to intra-community mean degree ($a$). Prior work has sharply characterized partial or full community recovery in terms of a "signal-to-noise ratio" ($\mathrm{SNR}$) based on $a$ and $b$. For both problems, w…
We study the problem of community detection (CD) on Euclidean random geometric graphs where each vertex has two latent variables: a binary community label and a $\mathbb{R}^d$ valued location label which forms the support of a Poisson point process of intensity $λ$. A random graph is then drawn with edge probabilities dependent on both the community and location labels. In contrast to the stochastic block model (SBM) that has no location labels, the resulting random graph contains many more short loops due to the geometric embedding. We consider the recovery of the community labels, partial an…
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…
Neural node embeddings have recently emerged as a powerful representation for supervised learning tasks involving graph-structured data. We leverage this recent advance to develop a novel algorithm for unsupervised community discovery in graphs. Through extensive experimental studies on simulated and real-world data, we demonstrate that the proposed approach consistently improves over the current state-of-the-art. Specifically, our approach empirically attains the information-theoretic limits for community recovery under the benchmark Stochastic Block Models for graph generation and exhibits b…
This paper is motivated by the reconstruction problem on the sparse stochastic block model. The paper "Belief Propagation, robust reconstruction and optimal recovery of block models" by Mossel, Neeman, and Sly provided and proved a reconstruction algorithm that recovers an optimal fraction of the communities in the 2 community case. The main step in their proof was to show that when the signal to noise ratio is sufficiently large, in particular $θ^2d > C$, the reconstruction accuracy on a regular tree with or without noise on the leaves is the same. This paper will generalize their results, in…
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 $\text{SBM}(n,p,Q)$, $n$ vertices are split into $k$ communities of relative size $\{p_i\}_{i \in [k]}$, and vertices in community $i$ and $j$ connect independently with probability $\{Q_{i,j}\}_{i…
Despite the rising importance of enhancing community resilience to disasters, our understanding on how communities recover from catastrophic events is limited. Here we study the population recovery dynamics of disaster affected regions by observing the movements of over 2.5 million mobile phone users across three countries before, during and after five major disasters. We find that, although the regions affected by the five disasters have significant differences in socio-economic characteristics, we observe a universal recovery pattern where displaced populations return in an exponential manne…
Community detection is considered for a stochastic block model graph of n vertices, with K vertices in the planted community, edge probability p for pairs of vertices both in the community, and edge probability q for other pairs of vertices. The main focus of the paper is on weak recovery of the community based on the graph G, with o(K) misclassified vertices on average, in the sublinear regime $n^{1-o(1)} \leq K \leq o(n).$ A critical parameter is the effective signal-to-noise ratio $λ=K^2(p-q)^2/((n-K)q)$, with $λ=1$ corresponding to the Kesten-Stigum threshold. We show that a belief propaga…
A message passing algorithm is derived for recovering communities within a graph generated by a variation of the Barabási-Albert preferential attachment model. The estimator is assumed to know the arrival times, or order of attachment, of the vertices. The derivation of the algorithm is based on belief propagation under an independence assumption. Two precursors to the message passing algorithm are analyzed: the first is a degree thresholding (DT) algorithm and the second is an algorithm based on the arrival times of the children (C) of a given vertex, where the children of a given vertex are …
Multilayer networks are a useful data structure for simultaneously capturing multiple types of relationships between a set of nodes. In such networks, each relational definition gives rise to a layer. While each layer provides its own set of information, community structure across layers can be collectively utilized to discover and quantify underlying relational patterns between nodes. To concisely extract information from a multilayer network, we propose to identify and combine sets of layers with meaningful similarities in community structure. In this paper, we describe the "strata multilaye…
Mining patterns from multi-relational data is a problem attracting increasing interest within the data mining community. Traditional data mining approaches are typically developed for highly simplified types of data, such as an attribute-value table or a binary database, such that those methods are not directly applicable to multi-relational data. Nevertheless, multi-relational data is a more truthful and therefore often also a more powerful representation of reality. Mining patterns of a suitably expressive syntax directly from this representation, is thus a research problem of great importan…
University of Illinois at Urbana-Champaign（伊利诺伊大学厄巴纳-香槟分校）；University of Chicago（芝加哥大学）
Community detection and orthogonal group synchronization are both fundamental problems with a variety of important applications in science and engineering. In this work, we consider the joint problem of community detection and orthogonal group synchronization which aims to recover the communities and perform synchronization simultaneously. To this end, we propose a simple algorithm that consists of a spectral decomposition step followed by a blockwise column pivoted QR factorization (CPQR). The proposed algorithm is efficient and scales linearly with the number of edges in the graph. We also l…
