Stochastic Block Model Research Articles

Hierarchical community structure in networks.

Modular and hierarchical community structures are pervasive in real-world complex systems. A great deal of effort has gone into trying to detect and study these structures. Important theoretical advances in the detection of modular have included identifying fundamental limits of detectability by formally defining community structure using probabilistic generative models. Detecting hierarchical community structure introduces additional challenges alongside those inherited from community detection. Here we present a theoretical study on hierarchical community structure in networks, which has thus far not received the same rigorous attention. We address the following questions. (1) How should we define a hierarchy of communities? (2) How do we determine if there is sufficient evidence of a hierarchical structure in a network? (3) How can we detect hierarchical structure efficiently? We approach these questions by introducing a definition of hierarchy based on the concept of stochastic externally equitable partitions and their relation to probabilistic models, such as the popular stochastic block model. We enumerate the challenges involved in detecting hierarchies and, by studying the spectral properties of hierarchical structure, present an efficient and principled method for detecting them.

Consistently estimating network statistics using aggregated relational data

Collecting complete network data is expensive, time-consuming, and often infeasible. Aggregated Relational Data (ARD), which ask respondents questions of the form "How many people with trait X do you know?" provide a low-cost option when collecting complete network data is not possible. Rather than asking about connections between each pair of individuals directly, ARD collect the number of contacts the respondent knows with a given trait. Despite widespread use and a growing literature on ARD methodology, there is still no systematic understanding of when and why ARD should accurately recover features of the unobserved network. This paper provides such a characterization by deriving conditions under which statistics about the unobserved network (or functions of these statistics like regression coefficients) can be consistently estimated using ARD. We first provide consistent estimates of network model parameters for three commonly used probabilistic models: the beta-model with node-specific unobserved effects, the stochastic block model with unobserved community structure, and latent geometric space models with unobserved latent locations. A key observation is that cross-group link probabilities for a collection of (possibly unobserved) groups identify the model parameters, meaning ARD are sufficient for parameter estimation. With these estimated parameters, it is possible to simulate graphs from the fitted distribution and analyze the distribution of network statistics. We can then characterize conditions under which the simulated networks based on ARD will allow for consistent estimation of the unobserved network statistics, such as eigenvector centrality, or response functions by or of the unobserved network, such as regression coefficients.

Log-linear stochastic block modeling and monitoring of directed sparse weighted network systems

Networks have been widely employed to reflect the relationships of entities in complex systems. In a weighted network, each node corresponds to one entity while the edge weight between two nodes can represent the number of interactions between two associated entities. More and more schemes have been established to monitor the networks, which help identify the possible changes or anomalies in corresponding systems. However, limited works can comprehensively reflect the community structure, node heterogeneity, interaction sparsity and direction of weighted networks in the literature. This article proposes a log-linear stochastic block model with latent features of nodes based on the mixture of Bernoulli distribution and Poisson distribution to characterize the sparse directional interaction counts within network systems. Explicit matrices and vectors are designed to incorporate community structure and enable straightforward maximum likelihood estimation of parameters. We further construct a monitoring statistic based on the generalized likelihood ratio test for change detection of sparse weighted networks. Comparative studies based on simulations and real data are conducted to validate the high efficiency of proposed model and monitoring scheme.

Mesoscale Structure in Urban–Rural Mobility Networks in the Pearl River Delta Area: A Weighted Stochastic Block Modeling Analysis

Understanding the spatial structure of a megaregion with urban and rural areas is crucial for promoting sustainable urbanization and urban–rural integration. Compared to the city network (or the network of urban areas), however, fewer studies focus on the network connecting rural areas or on the comparison of regional structures between urban and rural networks. Using weighted daily mobility flows from the massive mobile-phone signaling data, this study constructs an urban–urban mobility (UUM) network and an urban–rural mobility (URM) network in the Pearl River Delta (PRD) region. A weighted stochastic block model (WSBM) was adopted to identify and compare the latent mesoscale structures in the two networks. Results investigated a gradient community mesoscale structure nested with typical core–periphery (CP) structures in the UUM network and an asymmetric bipartite mesoscale structure mixed with CP hierarchies in the URM network. In a comparison of the different spatial configuration of urban/rural nodes and groupings of their roles, positions, and linkages, the study yielded empirical insights for renewed urban–rural interaction and potential planning pathways towards urban–rural integration.

Graph model selection by edge probability prequential inference

Abstract Graphs are widely used for describing systems made of many interacting components and for understanding the structure of their interactions. Various statistical models exist, which describe this structure as the result of a combination of constraints and randomness. In this article, we introduce edge probability prequential inference, a new approach to perform model selection, which relies on probability distributions on edge ensembles. From a theoretical point of view, we show that this methodology provides a more consistent ground for statistical inference with respect to existing techniques, due to the fact that it relies on multiple realizations of the random variable. It also provides better guarantees against overfitting, by making it possible to lower the number of parameters of the model below the number of observations. Experimentally, we illustrate the benefits of this methodology in two situations: to infer the partition of a stochastic blockmodel and to identify the most relevant model for a given graph between the stochastic blockmodel and the configuration model.

Sequential importance sampling for estimating expectations over the space of perfect matchings

This paper makes three contributions to estimating the number of perfect matching in bipartite graphs. First, we prove that the popular sequential importance sampling algorithm works in polynomial time for dense bipartite graphs. More carefully, our algorithm gives a (1±ϵ)-approximation for the number of perfect matchings of a λ-dense bipartite graph, using O(n1−2λλϵ−2) samples. With size n on each side and for 1 2>λ>0, a λ-dense bipartite graph has all degrees greater than (λ+1 2)n. Second, practical applications of the algorithm requires many calls to matching algorithms. A novel preprocessing step is provided which makes significant improvements. Third, three applications are provided. The first is for counting Latin squares, the second is a practical way of computing the greedy algorithm for a card guessing game with feedback and the third is for stochastic block models. In all three examples, sequential importance sampling allows treating practical problems of reasonably large sizes.

Community Detection in Multilayer Networks Based on Matrix Factorization and Spectral Embedding Method

Community detection remains a challenging research hotspot in network analysis. With the complexity of the network data structures increasing, multilayer networks, in which entities interact through multiple types of connections, prove to be effective in describing complex networks. The layers in a multilayer network may not share a common community structure. In this paper, we propose a joint method based on matrix factorization and spectral embedding to recover the groups not only for the layers but also for nodes. Specifically, the layers are grouped via the matrix factorization method with layer similarity-based regularization in the perspective of a mixture multilayer stochastic block model, and then the node communities within a layer group are revealed by clustering a combination of the spectral embedding derived from the adjacency matrices and the shared approximation matrix. Numerical studies show that the proposed method achieves competitive clustering results as the number of nodes and/or number of layers vary, together with different topologies of network layers. Additionally, we apply the proposed method on two real-world multilayer networks and obtain interesting findings which again highlight the effectiveness of our method.

Variational kinetic clustering of complex networks.

Efficiently identifying the most important communities and key transition nodes in weighted and unweighted networks is a prevalent problem in a wide range of disciplines. Here, we focus on the optimal clustering using variational kinetic parameters, linked to Markov processes defined on the underlying networks, namely, the slowest relaxation time and the Kemeny constant. We derive novel relations in terms of mean first passage times for optimizing clustering via the Kemeny constant and show that the optimal clustering boundaries have equal round-trip times to the clusters they separate. We also propose an efficient method that first projects the network nodes onto a 1D reaction coordinate and subsequently performs a variational boundary search using a parallel tempering algorithm, where the variational kinetic parameters act as an energy function to be extremized. We find that maximization of the Kemeny constant is effective in detecting communities, while the slowest relaxation time allows for detection of transition nodes. We demonstrate the validity of our method on several test systems, including synthetic networks generated from the stochastic block model and real world networks (Santa Fe Institute collaboration network, a network of co-purchased political books, and a street network of multiple cities in Luxembourg). Our approach is compared with existing clustering algorithms based on modularity and the robust Perron cluster analysis, and the identified transition nodes are compared with different notions of node centrality.

Constrained expectation-maximisation for inference of social graphs explaining online user–user interactions

Current network inference algorithms fail to generate graphs with edges that can explain whole sequences of node interactions in a given dataset or trace. To quantify how well an inferred graph can explain a trace, we introduce feasibility, a novel quality criterion, and suggest that it is linked to the result’s accuracy. In addition, we propose CEM-*, a network inference method that guarantees 100% feasibility given online social media traces, which are a non-trivial extension of the Expectation-Maximization algorithm developed by Newman (Nature Phys 14:67–75, 2018). We propose a set of linear optimization updates that incorporate a set of auxiliary variables and a set of feasibility constraints; the latter takes into consideration all the hidden paths that are possible between users based on their timestamps of interaction and guides the inference toward feasibility. We provide two CEM-* variations, that assume either an Erdős–Rényi (ER) or a Stochastic Block Model (SBM) prior for the underlying graph’s unknown distribution. Extensive experiments on one synthetic and one real-world Twitter dataset show that for both priors CEM-* can generate a posterior distribution of graphs that explains the whole trace while being closer to the ground truth. As an additional benefit, the use of the SBM prior infers and clusters users simultaneously during optimization. CEM-* outperforms baseline and state-of-the-art methods in terms of feasibility, run-time, and precision of the inferred graph and communities. Finally, we propose a heuristic to adapt the inference to lower feasibility requirements and show how it can affect the precision of the result.

PPISB: A Novel Network-Based Algorithm of Predicting Protein-Protein Interactions With Mixed Membership Stochastic Blockmodel.

Protein-protein interactions (PPIs) play an essential role for most of biological processes in cells. Many computational algorithms have thus been proposed to predict PPIs. However, most of them heavily rest on the biological information of proteins while ignoring the latent structural features of proteins presented in a PPI network. In this paper, we propose an efficient network-based prediction algorithm, namely PPISB, based on a mixed membership stochastic blockmodel. By simulating the generative process of a PPI network, PPISB is able to capture the latent community structures. The inference procedure adopted by PPISB further optimizes the membership distributions of proteins over different complexes. After that, a distance measure is designed to compute the similarity between two proteins in terms of their likelihoods of being in the same complex, thus verifying whether they interact with each other or not. To evaluate the performance of PPISB, a series of extensive experiments have been conducted with five PPI networks collected from different species and the results demonstrate that PPISB has a promising performance when applied to predict PPIs in terms of several evaluation metrics. Hence, we reason that PPISB is preferred over state-of-the-art network-based prediction algorithms especially for predicting potential PPIs.

Distribution-free model for community detection

AbstractCommunity detection for unweighted networks has been widely studied in network analysis, but the case of weighted networks remains a challenge. This paper proposes a general distribution-free model (DFM) for weighted networks in which nodes are partitioned into different communities. DFM can be seen as a generalization of the famous stochastic block models from unweighted networks to weighted networks. DFM does not require prior knowledge of a specific distribution for elements of the adjacency matrix but only the expected value. In particular, signed networks with latent community structures can be modeled by DFM. We build a theoretical guarantee to show that a simple spectral clustering algorithm stably yields consistent community detection under DFM. We also propose a four-step data generation process to generate adjacency matrices with missing edges by combining DFM, noise matrix, and a model for unweighted networks. Using experiments with simulated and real datasets, we show that some benchmark algorithms can successfully recover community membership for weighted networks generated by the proposed data generation process.

Assessing competitive balance in the English Premier League for over forty seasons using a stochastic block model

Abstract Competitive balance is of much interest in the sports analytics literature and beyond. We develop a statistical network model based on an extension of the stochastic block model to assess the balance between teams in a league. We represent the outcome of all matches in a football season as a dense network with nodes identified by teams and categorical edges representing the outcome of each game as a win, draw, or a loss. The main focus and motivation for this paper is to assess the issue of competitive balance in the context of the English First Division / Premier League over more than 40 seasons. Our analysis provides evidence suggesting a structural change around the early 2000s from a reasonably balanced league to a two-tier league.

An Improved Link Prediction Approach for Directed Complex Networks Using Stochastic Block Modeling

Link prediction finds the future or the missing links in a social–biological complex network such as a friendship network, citation network, or protein network. Current methods to link prediction follow the network properties, such as the node’s centrality, the number of edges, or the weights of the edges, among many others. As the properties of the networks vary, the link prediction methods also vary. These methods are inaccurate since they exploit limited information. This work presents a link prediction method based on the stochastic block model. The novelty of our approach is the three-step process to find the most-influential nodes using the m-PageRank metric, forming blocks using the global clustering coefficient and, finally, predicting the most-optimized links using maximum likelihood estimation. Through the experimental analysis of social, ecological, and biological datasets, we proved that the proposed model outperforms the existing state-of-the-art approaches to link prediction.

Mixed membership estimation for social networks

In economics and social science, network data are regularly observed, and a thorough understanding of the network community structure facilitates the comprehension of economic patterns and activities. Consider an undirected network with n nodes and K communities. We model the network using the Degree-Corrected Mixed-Membership (DCMM) model, where for each node i=1,2,…,n, there exists a membership vector πi=(πi(1),πi(2),…,πi(K))′, where πi(k) is the weight that node i puts on community k, 1≤k≤K. In comparison to the well-known stochastic block model (SBM), the DCMM permits both severe degree heterogeneity and mixed memberships, making it more realistic and general. We present an efficient approach, Mixed-SCORE, for estimating the mixed membership vectors of all nodes and the other DCMM parameters. This approach is inspired by the discovery of a delicate simplex structure in the spectral domain. We derive explicit error rates for the Mixed-SCORE algorithm and demonstrate that it is rate-optimal over a broad parameter space. Our findings provide a novel statistical tool for network community analysis, which can be used to understand network formations, extract nodal features, identify unobserved covariates in dyadic regressions, and estimate peer effects. We applied Mixed-SCORE to a political blog network, two trade networks, a co-authorship network, and a citee network, and obtained interpretable results.

Bootstrap percolation on the stochastic block model

We analyze the bootstrap percolation process on the stochastic block model (SBM), a natural extension of the Erdős–Rényi random graph that incorporates the community structure observed in many real systems. In the SBM, nodes are partitioned into two subsets, which represent different communities, and pairs of nodes are independently connected with a probability that depends on the communities they belong to. Under mild assumptions on the system parameters, we prove the existence of a sharp phase transition for the final number of active nodes and characterize the sub-critical and the super-critical regimes in terms of the number of initially active nodes, which are selected uniformly at random in each community.

Hawkes Processes With Stochastic Exogenous Effects for Continuous-Time Interaction Modelling.

Continuous-time interaction data is usually generated under time-evolving environment. Hawkes processes (HP) are commonly used mechanisms for the analysis of such data. However, typical model implementations (such as e.g., stochastic block models) assume that the exogenous (background) interaction rate is constant, and so they are limited in their ability to adequately describe any complex time-evolution in the background rate of a process. In this paper, we introduce a stochastic exogenous rate Hawkes process (SE-HP) which is able to learn time variations in the exogenous rate. The model affiliates each node with a piecewise-constant membership distribution with an unknown number of changepoint locations, and allows these distributions to be related to the membership distributions of interacting nodes. The time-varying background rate function is derived through combinations of these membership functions. We introduce a stochastic gradient MCMC algorithm for efficient, scalable inference. The performance of the SE-HP is explored on real world, continuous-time interaction datasets, where we demonstrate that the SE-HP strongly outperforms comparable state-of-the-art methods.

A Compressed Sensing Based Least Squares Approach to Semi-supervised Local Cluster Extraction

A least squares semi-supervised local clustering algorithm based on the idea of compressed sensing is proposed to extract clusters from a graph with known adjacency matrix. The algorithm is based on a two-stage approach similar to the one proposed by Lai and Mckenzie (SIAM J Math Data Sci 2:368–395, 2020). However, under a weaker assumption and with less computational complexity, our algorithms are shown to be able to find a desired cluster with high probability. The “one cluster at a time" feature of our method distinguishes it from other global clustering methods. Numerical experiments are conducted on the synthetic data such as stochastic block model and real data such as MNIST, political blogs network, AT &T and YaleB human faces data sets to demonstrate the effectiveness and efficiency of our algorithms.

Dynamic network sampling for community detection

We propose a dynamic network sampling scheme to optimize block recovery for stochastic blockmodel 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 check whether there are edges between them to save significant resources but still recover the block structure.

Efficiency evaluation for urban industrial metabolism through the methodologies of emergy analysis and dynamic network stochastic block model

Frequent economic activities in cities have exacerbated irreversible environmental degradation. It is urgent to balance the relationship between economic growth, resource investment and environmental protection. With the concept of urban metabolism, this study attempts to clarify the internal structure of urban system metabolism and the metabolic relationship between specific economic sectors. And then, it determines the metabolic level of current urban industrial development by means of metabolic efficiency evaluation, which can help more accurately reveal the sustainability of urban metabolic activities. Based on them, this study proposes a metabolic efficiency evaluation model combined with emergy analysis and slack-based measure dynamic network Data Envelopment Analysis (DEA). Through the multi-period and multi-stage network DEA model, the internal structure of urban industrial metabolism is opened. And considerring industrial waste as an undesirable carryover will be regarded as additional inputs in future period to reflect the impact on industrial metabolic efficiency. Using the proposed model, the industrial metabolic efficiency of 18 first-tier cities in China from 2016 to 2020 was analyzed. It is found that all cities are inefficient, which indicates that all evaluated cities need to make appropriate adjustments to improve their weak stages. In response to the specific problems of inefficiency in each city, an improvement framework is proposed to help each city better realize the sustainable development of the urban economy.

Profile‐pseudo likelihood methods for community detection of multilayer stochastic block models

The multilayer stochastic block model is one of the fundamental models in multilayer networks and is often used to represent multiple types of relations between different individuals. In this paper, we extend the profile‐pseudo likelihood method for the single‐layer stochastic block model to the case of the multilayer stochastic block model. Specifically, by assuming all network layers have identical community membership labels, we investigate the multilayer stochastic block model with a common community structure. In this paper, we develop a profile‐pseudo likelihood algorithm to fit a multilayer stochastic block model and estimate the community label. Meantime, we prove that the algorithm has convergence guarantee and that the estimated community label is strongly consistent. Further, for estimating the number of communities , we extend the corrected Bayesian information criterion to multilayer stochastic block models. We also extend this algorithm to fit the multilayer degree‐corrected stochastic block model. Both simulation studies and real‐world data examples indicate that the proposed method works well.