Degree-corrected Stochastic Block Model Research Articles

Network and attribute-based clustering of tennis players and tournaments

AbstractThis paper aims at targeting some relevant issues for clustering tennis players and tournaments: (i) it considers players, tournaments and the relation between them; (ii) the relation is taken into account in the fuzzy clustering model based on the Partitioning Around Medoids (PAM) algorithm through spatial constraints; (iii) the attributes of the players and of the tournaments are of different nature, qualitative and quantitative. The proposal is novel for the methodology used, a spatial Fuzzy clustering model for players and for tournaments (based on related attributes), where the spatial penalty term in each clustering model depends on the relation between players and tournaments described in the adjacency matrix. The proposed model is compared with a bipartite players-tournament complex network model (the Degree-Corrected Stochastic Blockmodel) that considers only the relation between players and tournaments, described in the adjacency matrix, to obtain communities on each side of the bipartite network. An application on data taken from the ATP official website with regards to the draws of the tournaments, and from the sport statistics website Wheelo ratings for the performance data of players and tournaments, shows the performances of the proposed clustering model.

Distributed Pseudo-Likelihood Method for Community Detection in Large-Scale Networks

This paper proposes a distributed pseudo-likelihood method (DPL) to conveniently identify the community structure of large-scale networks. Specifically, we first propose a block-wise splitting method to divide large-scale network data into several subnetworks and distribute them among multiple workers. For simplicity, we assume the classical stochastic block model. Then, the DPL algorithm is iteratively implemented for the distributed optimization of the sum of the local pseudo-likelihood functions. At each iteration, the worker updates its local community labels and communicates with the master. The master then broadcasts the combined estimator to each worker for the new iterative steps. Based on the distributed system, DPL significantly reduces the computational complexity of the traditional pseudo-likelihood method using a single machine. Furthermore, to ensure statistical accuracy, we theoretically discuss the requirements of the worker sample size. Moreover, we extend the DPL method to estimate degree-corrected stochastic block models. The superior performance of the proposed distributed algorithm is demonstrated through extensive numerical studies and real data analysis.

Two-sample test for stochastic block models via the largest singular value

The stochastic block model is widely used for detecting community structures in network data. However, the research interest in much of the literature focuses on the study of one sample of stochastic block models. Detecting the difference between the two community structures is a less studied issue for stochastic block models. In this article, we propose a novel test statistic based on the largest singular value of a residual matrix obtained by subtracting the geometric mean of two estimated block mean effects from the sum of two observed adjacency matrices. We prove that the null distribution of the proposed test statistic converges in distribution to the Tracy-Widom distribution with index 1, and we show the difference between the two samples for stochastic block models can be tested via the proposed method. We show that the proposed test is asymptotically powerful against alternative models. Further, we extend the proposed method to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well.

Network-adjusted covariates for community detection

Abstract Community detection is a crucial task in network analysis that can be significantly improved by incorporating subject-level information, i.e., covariates. Existing methods have shown the effectiveness of using covariates on the low-degree nodes, but rarely discuss the case where communities have significantly different density levels, i.e. multiscale networks. In this paper, we introduce a novel method that addresses this challenge by constructing network-adjusted covariates, which leverage the network connections and covariates with a node-specific weight for each node. This weight can be calculated without tuning parameters. We present novel theoretical results on the strong consistency of our method under degree-corrected stochastic blockmodels with covariates, even in the presence of misspecification and multiple sparse communities. Additionally, we establish a general lower bound for the community detection problem when both network and covariates are present, and it shows our method is optimal for connection intensity up to a constant factor. Our method outperforms existing approaches in simulations and a LastFM app user network. We then compare our method with others on a statistics publication citation network where 30% of nodes are isolated, and our method produces reasonable and balanced results.

Subsampling spectral clustering for stochastic block models in large-scale networks

The rapid development of science and technology has generated large amounts of network data, leading to significant computational challenges for network community detection. A novel subsampling spectral clustering algorithm is proposed to address this issue, which aims to identify community structures in large-scale networks with limited computing resources. The algorithm constructs a subnetwork by simple random subsampling from the entire network, and then extends the existing spectral clustering to the subnetwork to estimate the community labels for entire network nodes. As a result, for large-scale datasets, the method can be realized even using a personal computer. Moreover, the proposed method can be generalized in a parallel way. Theoretically, under the stochastic block model and its extension, the degree-corrected stochastic block model, the theoretical properties of the subsampling spectral clustering method are correspondingly established. Finally, to illustrate and evaluate the proposed method, a number of simulation studies and two real data analyses are conducted.

Probabilistic program inference in network-based epidemiological simulations.

Accurate epidemiological models require parameter estimates that account for mobility patterns and social network structure. We demonstrate the effectiveness of probabilistic programming for parameter inference in these models. We consider an agent-based simulation that represents mobility networks as degree-corrected stochastic block models, whose parameters we estimate from cell phone co-location data. We then use probabilistic program inference methods to approximate the distribution over disease transmission parameters conditioned on reported cases and deaths. Our experiments demonstrate that the resulting models improve the quality of fit in multiple geographies relative to baselines that do not model network topology.

Monitoring a labeled degree‐corrected stochastic block model

AbstractThe stochastic block model (SBM) is a random graph model that focuses on partitioning the nodes into blocks or communities. A degree‐corrected stochastic block model (DCSBM) considers degree heterogeneity within nodes. Investigation of the type of edge label can be useful for studying networks. We have proposed a labeled degree‐corrected stochastic block model (LDCSBM), added the probability of the occurrence of each edge label, and monitored the behavior of this network. The LDCSBM is a dynamic network that varies over time; thus, we applied the monitoring process to both the US Senate voting network and simulated networks by defining structural changes. We used the Shewhart control chart for detecting changes and studied the effect of Phase I parameter estimation on Phase II performance. The efficiency of the model for surveillance has been evaluated using the average run length for estimated parameters.

Degree-corrected distribution-free model for community detection in weighted networks

A degree-corrected distribution-free model is proposed for weighted social networks with latent structural information. The model extends the previous distribution-free models by considering variation in node degree to fit real-world weighted networks, and it also extends the classical degree-corrected stochastic block model from un-weighted network to weighted network. We design an algorithm based on the idea of spectral clustering to fit the model. Theoretical framework on consistent estimation for the algorithm is developed under the model. Theoretical results when edge weights are generated from different distributions are analyzed. We also propose a general modularity as an extension of Newman’s modularity from un-weighted network to weighted network. Using experiments with simulated and real-world networks, we show that our method significantly outperforms the uncorrected one, and the general modularity is effective.

Randomized Spectral Clustering in Large-Scale Stochastic Block Models

Spectral clustering has been one of the widely used methods for community detection in networks. However, large-scale networks bring computational challenges to the eigenvalue decomposition therein. In this paper, we study the spectral clustering using randomized sketching algorithms from a statistical perspective, where we typically assume the network data are generated from a stochastic block model that is not necessarily of full rank. To do this, we first use the recently developed sketching algorithms to obtain two randomized spectral clustering algorithms, namely, the random projection-based and the random sampling-based spectral clustering. Then we study the theoretical bounds of the resulting algorithms in terms of the approximation error for the population adjacency matrix, the misclassification error, and the estimation error for the link probability matrix. It turns out that, under mild conditions, the randomized spectral clustering algorithms lead to the same theoretical bounds as those of the original spectral clustering algorithm. We also extend the results to degree-corrected stochastic block models. Numerical experiments support our theoretical findings and show the efficiency of randomized methods. A new R package called Rclust is developed and made available to the public. Supplementary materials for this article are available online.

Spectral Clustering on Spherical Coordinates Under the Degree-Corrected Stochastic Blockmodel

ABSTRACT Spectral clustering is a popular method for community detection in network graphs: starting from a matrix representation of the graph, the nodes are clustered on a low-dimensional projection obtained from a truncated spectral decomposition of the matrix. Estimating correctly the number of communities and the dimension of the reduced latent space is critical for good performance of spectral clustering algorithms. Furthermore, many real-world graphs, such as enterprise computer networks studied in cyber-security applications, often display heterogeneous within-community degree distributions. Such heterogeneous degree distributions are usually not well captured by standard spectral clustering algorithms. In this article, a novel spectral clustering algorithm is proposed for community detection under the degree-corrected stochastic blockmodel. The proposed method is based on a transformation of the spectral embedding to spherical coordinates, and a novel modeling assumption in the transformed space. The method allows for simultaneous and automated selection of the number of communities and the latent dimension for spectral embeddings of graphs with uneven node degrees. Results show improved performance over competing methods in representing computer networks.

Network-Based Discovery of Opioid Use Vulnerability in Rats Using the Bayesian Stochastic Block Model.

Opioid use disorder is a psychological condition that affects over 200,000 people per year in the U.S., causing the Centers for Disease Control and Prevention to label the crisis as a rapidly spreading public health epidemic. The behavioral relationship between opioid exposure and development of opioid use disorder (OUD) varies greatly between individuals, implying existence of sup-populations with varying degrees of opioid vulnerability. However, effective pre-clinical identification of these sub-populations remains challenging due to the complex multivariate measurements employed in animal models of OUD. In this study, we propose a novel non-linear network-based data analysis workflow that employs seven behavioral traits to identify opioid use sub-populations and assesses contributions of behavioral variables to opioid vulnerability and resiliency. Through this analysis workflow we determined how behavioral variables across heroin taking, refraining and seeking interact with one another to identify potentially heroin resilient and vulnerable behavioral sub-populations. Data were collected from over 400 heterogeneous stock rats in two geographically distinct locations. Rats underwent heroin self-administration training, followed by a progressive ratio and heroin-primed reinstatement test. Next, rats underwent extinction training and a cue-induced reinstatement test. To enter the analysis workflow, we integrated data from different cohorts of rats and removed possible batch effects. We then constructed a rat-rat similarity network based on their behavioral patterns and implemented community detection on this similarity network using a Bayesian degree-corrected stochastic block model to uncover sub-populations of rats with differing levels of opioid vulnerability. We identified three statistically distinct clusters corresponding to distinct behavioral sub-populations, vulnerable, resilient and intermediate for heroin use, refraining and seeking. We implement this analysis workflow as an open source R package, named mlsbm.

Assessing the use of the moving window approaches when monitoring social networks using a degree corrected stochastic block model

PurposeThis study aims to assess the effect of updating the Phase I data – to enhance the parameters' estimates – on the control charts' detection power designed to monitor social networks.Design/methodology/approachA dynamic version of the degree corrected stochastic block model (DCSBM) is used to model the network. Both the Shewhart and exponentially weighted moving average (EWMA) control charts are used to monitor the model parameters. A performance comparison is conducted for each chart when designed using both fixed and moving windows of networks.FindingsOur results show that continuously updating the parameters' estimates during the monitoring phase delays the Shewhart chart's detection of networks' anomalies; as compared to the fixed window approach. While the EWMA chart performance is either indifferent or worse, based on the updating technique, as compared to the fixed window approach. Generally, the EWMA chart performs uniformly better than the Shewhart chart for all shift sizes. We recommend the use of the EWMA chart when monitoring networks modeled with the DCSBM, with sufficiently small to moderate fixed window size to estimate the unknown model parameters.Originality/valueThis study shows that the excessive recommendations in literature regarding the continuous updating of Phase I data during the monitoring phase to enhance the control chart performance cannot generally be extended to social network monitoring; especially when using the DCSBM. That is to say, the effect of continuously updating the parameters' estimates highly depends on the nature of the process being monitored.

State Aggregations in Markov Chains and Block Models of Networks.

We consider state-aggregation schemes for Markov chains from an information-theoretic perspective. Specifically, we consider aggregating the states of a Markov chain such that the mutual information of the aggregated states separated by T time steps is maximized. We show that for T=1 this recovers the maximum-likelihood estimator of the degree-corrected stochastic block model as a particular case, which enables us to explain certain features of the likelihood landscape of this generative network model from a dynamical lens. We further highlight how we can uncover coherent, long-range dynamical modules for which considering a timescale T≫1 is essential. We demonstrate our results using synthetic flows and real-world ocean currents, where we are able to recover the fundamental features of the surface currents of the oceans.

Using Maximum Entry-Wise Deviation to Test the Goodness of Fit for Stochastic Block Models

The stochastic block model is widely used for detecting community structures in network data. How to test the goodness of fit of the model is one of the fundamental problems and has gained growing interests in recent years. In this article, we propose a novel goodness-of-fit test based on the maximum entry of the centered and rescaled adjacency matrix for the stochastic block model. One noticeable advantage of the proposed test is that the number of communities can be allowed to grow linearly with the number of nodes ignoring a logarithmic factor. We prove that the null distribution of the test statistic converges in distribution to a Gumbel distribution, and we show that both the number of communities and the membership vector can be tested via the proposed method. Furthermore, we show that the proposed test has asymptotic power guarantee against a class of alternatives. We also demonstrate that the proposed method can be extended to the degree-corrected stochastic block model. Both simulation studies and real-world data examples indicate that the proposed method works well. Supplementary materials for this article are available online.

Targeted Sampling from Massive Block Model Graphs with Personalized PageRank

SummaryThe paper provides statistical theory and intuition for personalized PageRank (called ‘PPR’): a popular technique that samples a small community from a massive network. We study a setting where the entire network is expensive to obtain thoroughly or to maintain, but we can start from a seed node of interest and ‘crawl’ the network to find other nodes through their connections. By crawling the graph in a designed way, the PPR vector can be approximated without querying the entire massive graph, making it an alternative to snowball sampling. Using the degree-corrected stochastic block model, we study whether the PPR vector can select nodes that belong to the same block as the seed node. We provide a simple and interpretable form for the PPR vector, highlighting its biases towards high degree nodes outside the target block. We examine a simple adjustment based on node degrees and establish consistency results for PPR clustering that allows for directed graphs. These results are enabled by recent technical advances showing the elementwise convergence of eigenvectors. We illustrate the method with the massive Twitter friendship graph, which we crawl by using the Twitter application programming interface. We find that the adjusted and unadjusted PPR techniques are complementary approaches, where the adjustment makes the results particularly localized around the seed node, and that the bias adjustment greatly benefits from degree regularization.

A Bayesian Inference Method Using Monte Carlo Sampling for Estimating the Number of Communities in Bipartite Networks

Community detection is an important analysis task for complex networks, including bipartite networks, which consist of nodes of two types and edges connecting only nodes of different types. Many community detection methods take the number of communities in the networks as a fixed known quantity; however, it is impossible to give such information in advance in real-world networks. In our paper, we propose a projection-free Bayesian inference method to determine the number of pure-type communities in bipartite networks. This paper makes the following contributions: (1) we present the first principle derivation of a practical method, using the degree-corrected bipartite stochastic block model that is able to deal with networks with broad degree distributions, for estimating the number of pure-type communities of bipartite networks; (2) a prior probability distribution is proposed over the partition of a bipartite network; (3) we design a Monte Carlo algorithm incorporated with our proposed method and prior probability distribution. We give a demonstration of our algorithm on synthetic bipartite networks including an easy case with a homogeneous degree distribution and a difficult case with a heterogeneous degree distribution. The results show that the algorithm gives the correct number of communities of synthetic networks in most cases and outperforms the projection method especially in the networks with heterogeneous degree distributions.

A Regularized Stochastic Block Model for the robust community detection in complex networks

The stochastic block model is able to generate random graphs with different types of network partitions, ranging from the traditional assortative structures to the disassortative structures. Since the stochastic block model does not specify which mixing pattern is desired, the inference algorithms discover the locally most likely nodes’ partition, regardless of its type. Here we introduce a new model constraining nodes’ internal degree ratios in the objective function to guide the inference algorithms to converge to the desired type of structure in the observed network data. We show experimentally that given the regularized model, the inference algorithms, such as Markov chain Monte Carlo, reliably and quickly find the assortative or disassortative structure as directed by the value of a single parameter. In contrast, when the sought-after assortative community structure is not strong in the observed network, the traditional inference algorithms using the degree-corrected stochastic block model tend to converge to undesired disassortative partitions.

Modeling and detecting change in temporal networks via the degree corrected stochastic block model

AbstractIn many applications, it is of interest to identify anomalous behavior within a dynamic interacting system. Such anomalous interactions are reflected by structural changes in the network representation of the system. We propose and investigate the use of the degree corrected stochastic block model (DCSBM) to model and monitor dynamic networks that undergo a significant structural change. We apply statistical process monitoring techniques to the estimated parameters of the DCSBM to identify significant structural changes in the network. We apply our surveillance strategy to a dynamic US Senate covoting network. We detect significant changes in the political network that reflect both times of cohesion and times of polarization among Republican and Democratic party members. Our analysis demonstrates that the DCSBM monitoring procedure effectively detects local and global structural changes in complex networks, providing useful insights into the modeled system. The DCSBM approach is an example of a general framework that combines parametric random graph models and statistical process monitoring techniques for network surveillance.

The effect of temporal aggregation level in social network monitoring.

Social networks have become ubiquitous in modern society, which makes social network monitoring a research area of significant practical importance. Social network data consist of social interactions between pairs of individuals that are temporally aggregated over a certain interval of time, and the level of such temporal aggregation can have substantial impact on social network monitoring. There have been several studies on the effect of temporal aggregation in the process monitoring literature, but no studies on the effect of temporal aggregation in social network monitoring. We use the degree corrected stochastic block model (DCSBM) to simulate social networks and network anomalies and analyze these networks in the context of both count and binary network data. In conjunction with this model, we use the Priebe scan method as the monitoring method. We demonstrate that temporal aggregation at high levels leads to a considerable decrease in the ability to detect an anomaly within a specified time period. Moreover, converting social network communication data from counts to binary indicators can result in a significant loss of information, hindering detection performance. Aggregation at an appropriate level with count data, however, can amplify the anomalous signal generated by network anomalies and improve detection performance. Our results provide both insights on the practical effects of temporal aggregation and a framework for the study of other combinations of network models, surveillance methods, and types of anomalies.

An impossibility result for reconstruction in the degree-corrected stochastic block model

We consider the Degree-Corrected Stochastic Block Model (DC-SBM): a random graph on $n$ nodes, having i.i.d. weights $(\phi_{u})_{u=1}^{n}$ (possibly heavy-tailed), partitioned into $q\geq2$ asymptotically equal-sized clusters. The model parameters are two constants $a,b>0$ and the finite second moment of the weights $\Phi^{(2)}$. Vertices $u$ and $v$ are connected by an edge with probability $\frac{\phi_{u}\phi_{v}}{n}a$ when they are in the same class and with probability $\frac{\phi_{u}\phi_{v}}{n}b$ otherwise. We prove that it is information-theoretically impossible to estimate the clusters in a way positively correlated with the true community structure when $(a-b)^{2}\Phi^{(2)}\leq q(a+b)$. As by-products of our proof we obtain $(1)$ a precise coupling result for local neighbourhoods in DC-SBMs, that we use in Gulikers, Lelarge and Massoulie (2016) to establish a law of large numbers for local-functionals and $(2)$ that long-range interactions are weak in (power-law) DC-SBMs.