Dynamic Stochastic Block Model Research Articles

HB-DSBM: Modeling the Dynamic Complex Networks From Community Level to Node Level.

A variety of methods have been proposed for modeling and mining dynamic complex networks, in which the topological structure varies with time. As the most popular and successful network model, the stochastic block model (SBM) has been extended and applied to community detection, link prediction, anomaly detection, and evolution analysis of dynamic networks. However, all current models based on the SBM for modeling dynamic networks are designed at the community level, assuming that nodes in each community have the same dynamic behavior, which usually results in poor performance on temporal community detection and loses the modeling of node abnormal behavior. To solve the above-mentioned problem, this article proposes a hierarchical Bayesian dynamic SBM (HB-DSBM) for modeling the node-level and community-level dynamic behavior in a dynamic network synchronously. Based on the SBM, we introduce a hierarchical Dirichlet generative mechanism to associate the global community evolution with the microscopic transition behavior of nodes near-perfectly and generate the observed links across the dynamic networks. Meanwhile, an effective variational inference algorithm is developed and we can easy to infer the communities and dynamic behaviors of the nodes. Furthermore, with the two-level evolution behaviors, it can identify nodes or communities with abnormal behavior. Experiments on simulated and real-world networks demonstrate that HB-DSBM has achieved state-of-the-art performance on community detection and evolution. In addition, abnormal evolutionary behavior and events on dynamic networks can be effectively identified by our model.

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.

Approaches to blockmodeling dynamic networks: A Monte Carlo simulation study

Blockmodeling refers to a variety of statistical methods for reducing and simplifying large and complex networks. While methods for blockmodeling networks observed at one time point are well established, it is only recently that researchers have proposed several methods for analysing dynamic networks (i.e., networks observed at multiple time points). The considered approaches are based on k-means or stochastic blockmodeling, with different ways being used to model time dependency among time points. Their novelty means they have yet to be extensively compared and evaluated and the paper therefore aims to compare and evaluate them using Monte Carlo simulations. Different network characteristics are considered, including whether tie formation is random or governed by local network mechanisms. The results show the Dynamic Stochastic Blockmodel (Matias and Miele 2017) performs best if the blockmodel does not change; otherwise, the Stochastic Blockmodel for Multipartite Networks (Bar-Hen et al. 2020) does.

Sharded Blockchain for Collaborative Computing in the Internet of Things: Combined of Dynamic Clustering and Deep Reinforcement Learning Approach

Immutability, decentralization, and linear promoted scalability make the sharded blockchain a promising solution, which can effectively address the trust issue in the large-scale Internet of Things (IoT). However, currently, the throughput of sharded blockchains is still limited when it comes to high proportion of cross-shard transactions (CSTs). On the other hand, the assemblage characteristic of the collaborative computing in IoT has not been received attention. Therefore, in this article, we present a clustering-based sharded blockchain strategy for collaborative computing in the IoT, where the sharding of the blockchain system is implemented in two steps: <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K-means</i> -clustering-based user grouping and the assignment of consensus nodes. In this framework, how to reasonably group the IoT users while simultaneously guaranteeing the system performance is the key point. Specifically, we describe the data transactions among IoT devices by data transaction flow graph (DTFG) based on a dynamic stochastic block model. Then, formed as a Markov decision process (MDP), the optimization of the cluster number (shard number) and the adjustment of consensus parameters are jointly trained by deep reinforcement learning (DRL). Simulation results show that the proposed scheme improves the scalability of the sharded blockchain in the IoT application.

Poisson degree corrected dynamic stochastic block model

Stochastic Block Model (SBM) provides a statistical tool for modeling and clustering network data. In this paper, we propose an extension of this model for discrete-time dynamic networks that takes into account the variability in node degrees, allowing us to model a broader class of networks. We develop a probabilistic model that generates temporal graphs with a dynamic cluster structure and time-dependent degree corrections for each node. Thanks to these degree corrections, the nodes can have variable in- and out-degrees, allowing us to model complex cluster structures as well as interactions that decrease or increase over time. We compare the proposed model to a model without degree correction and highlight its advantages in the case of inhomogenous degree distributions in the clusters and in the recovery of unstable cluster dynamics. We propose an inference procedure based on Variational Expectation-Maximization (VEM) that also provides the means to estimate the time-dependent degree corrections. Extensive experiments on simulated and real datasets confirm the benefits of our approach and show the effectiveness of the proposed algorithm.

Sparse and smooth: Improved guarantees for spectral clustering in the dynamic stochastic block model

In this paper, we analyze classical variants of the Spectral Clustering (SC) algorithm in the Dynamic Stochastic Block Model (DSBM). Existing results show that, in the relatively sparse case where the expected degree grows logarithmically with the number of nodes, guarantees in the static case can be extended to the dynamic case and yield improved error bounds when the DSBM is sufficiently smooth in time, that is, the communities do not change too much between two time steps. We improve over these results by drawing a new link between the sparsity and the smoothness of the DSBM: the smoother the DSBM is, the more sparse it can be, while still guaranteeing consistent recovery. In particular, a mild condition on the smoothness allows to treat the sparse case with bounded degree. These guarantees are valid for the SC applied to the adjacency matrix or the normalized Laplacian. As a by-product of our analysis, we obtain to our knowledge the best spectral concentration bound available for the normalized Laplacian of matrices with independent Bernoulli entries.

Interpretable Clustering on Dynamic Graphs with Recurrent Graph Neural Networks

We study the problem of clustering nodes in a dynamic graph, where the connections between nodes and nodes' cluster memberships may change over time, e.g., due to community migration. We first propose a dynamic stochastic block model that captures these changes, and a simple decay-based clustering algorithm that clusters nodes based on weighted connections between them, where the weight decreases at a fixed rate over time. This decay rate can then be interpreted as signifying the importance of including historical connection information in the clustering. However, the optimal decay rate may differ for clusters with different rates of turnover. We characterize the optimal decay rate for each cluster and propose a clustering method that achieves almost exact recovery of the true clusters. We then demonstrate the efficacy of our clustering algorithm with optimized decay rates on simulated graph data. Recurrent neural networks (RNNs), a popular algorithm for sequence learning, use a similar decay-based method, and we use this insight to propose two new RNN-GCN (graph convolutional network) architectures for semi-supervised graph clustering. We finally demonstrate that the proposed architectures perform well on real data compared to state-of-the-art graph clustering algorithms.

Community Detection in Dynamic Networks: Equivalence Between Stochastic Blockmodels and Evolutionary Spectral Clustering

Community detection aims to identify densely connected groups of nodes in complex networks. Although a variety of methods have been proposed for community detection, the relationship between them is not well understood. Recently, researchers have shown the equivalence between modularity optimization and likelihood maximization in stochastic block models (SBMs) for static networks. Showing this equivalence is important for both understanding the different community detection methods and selecting the hyperparameters in the different algorithms in a more principled way. In this paper, we extend this equivalence for dynamic community detection algorithms. In particular, we show the equivalence of evolutionary spectral clustering to a variant of dynamic stochastic blockmodel. For this purpose, we first introduce a novel dynamic SBM where the evolution of communities over time is modeled with pairwise Markov random fields. We then show that the log-posterior of the proposed model is equivalent to the quality function of evolutionary spectral clustering. This equivalence is used to determine the forgetting factor in evolutionary spectral clustering and to develop two new algorithms for dynamic community detection. Compared to original evolutionary spectral clustering, the forgetting factor is time-dependent and derived directly from the parameters of the proposed dynamic SBM. The proposed algorithms are shown to be superior to state-of-the-art dynamic community detection methods for both simulated and real-world dynamic networks.

An exact algorithm for time-dependent variational inference for the dynamic stochastic block model

Abstract An exact algorithm for estimating the dynamic stochastic block model is proposed. This model assumes a hidden Markov chain for the evolution of the social behavior of a group of individuals at repeated time occasions and may be used to assign these individuals to the latent blocks in a dynamic fashion. For the estimation of this model, the proposed exact algorithm maximizes the target function introduced by Matias and Miele [7]. This function is derived from a variational approximation of the model log-likelihood, based on the assumption that the latent variables identifying the blocks are a posteriori independent across individuals, but not across time occasions. A simulation study is performed to compare the exact algorithm with the approximate maximization algorithm proposed by Matias and Miele [7]. Results show that there is a certain advantage of the first in terms of dynamic assignment of individuals to the latent blocks in comparison to the true blocking structure, as measured by the adjusted Rand index.

Core-periphery dynamics in a plant-pollinator network.

Mutualistic networks are highly dynamic, characterized by high temporal turnover of species and interactions. Yet, we have a limited understanding of how the internal structure of these networks and the roles species play in them vary through time. We used 6years of observation data and a novel statistical method (dynamic stochastic block models) to assess how network structure and species' structural position within the network change throughout subseasons of the flowering season and across years in a quantitative plant-pollinator network from a dryland ecosystem in Argentina. Our analyses revealed a core-periphery structure persistent through subseasons and years. Yet, species structural position as core or peripheral was highly dynamic: virtually all species that were at the core in some subseasons were also peripheral in other subseasons, while many other species always remained peripheral. Our results illuminate our understanding of the dynamics of mutualistic networks and have important implications for ecosystem management and conservation.

Dynamic network models and graphon estimation

In the present paper, we consider a dynamic stochastic network model. The objective is estimation of the tensor of connection probabilities $\mathbf{{\Lambda}}$ when it is generated by a Dynamic Stochastic Block Model (DSBM) or a dynamic graphon. In particular, in the context of the DSBM, we derive a penalized least squares estimator $\widehat{\boldsymbol{\Lambda}}$ of $\mathbf{{\Lambda}}$ and show that $\widehat{\boldsymbol{\Lambda}}$ satisfies an oracle inequality and also attains minimax lower bounds for the risk. We extend those results to estimation of $\mathbf{{\Lambda}}$ when it is generated by a dynamic graphon function. The estimators constructed in the paper are adaptive to the unknown number of blocks in the context of the DSBM or to the smoothness of the graphon function. The technique relies on the vectorization of the model and leads to much simpler mathematical arguments than the ones used previously in the stationary set up. In addition, all results in the paper are nonasymptotic and allow a variety of extensions.

Nonparametric Identification in the Dynamic Stochastic Block Model

We show nonparametric identification of the parameters in the dynamic stochastic block model as recently introduced by Matias and Miele (2017) in the case of binary, finitely weighted, and general edge states. We formulate conditions on the true parameters, which guarantee actual point identification instead of mere generic identification, and which also lead to novel conclusions in the static case. In particular, our results justify in terms of the identification of the applications by Matias and Miele to finitely weighted edges with three edge states. We also give numerical illustrations via the variational EM algorithm in simulation settings covered by our identification analysis.

Consistency of the maximum likelihood and variational estimators in a dynamic stochastic block model

We consider a dynamic version of the stochastic block model, in which the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution is modeled through a hidden Markov chain on the nodes memberships. We prove the consistency (as the number of nodes and time steps increase) of the maximum likelihood and variational estimators of the model parameters, and obtain upper bounds on the rates of convergence of these estimators. We also explore the particular case where the number of time steps is fixed and connectivity parameters are allowed to vary.

Disentangling group and link persistence in dynamic stochastic block models

We study the inference of a model of dynamic networks in which both communities and links keep memory of previous network states. By considering maximum likelihood inference from single snapshot observations of the network, we show that link persistence makes the inference of communities harder, decreasing the detectability threshold, while community persistence tends to make it easier. We analytically show that communities inferred from single network snapshot can share a maximum overlap with the underlying communities of a specific previous instant in time. This leads to time-lagged inference: the identification of past communities rather than present ones. Finally we compute the time lag and propose a corrected algorithm, the lagged snapshot dynamic algorithm, for community detection in dynamic networks. We analytically and numerically characterize the detectability transitions of such algorithm as a function of the memory parameters of the model and we make a comparison with a full dynamic inference.

Differentially private nonlinear observer design using contraction analysis

SummaryReal‐time information processing applications such as those enabling a more intelligent infrastructure are increasingly focused on analyzing privacy‐sensitive data obtained from individuals. To produce accurate statistics about the habits of a population of users of a system, this data might need to be processed through model‐based estimators. Moreover, models of population dynamics, originating for example from epidemiology or the social sciences, are often necessarily nonlinear. Motivated by these trends, this paper presents an approach to design nonlinear privacy‐preserving model‐based observers, relying on additive input or output noise to give differential privacy guarantees to the individuals providing the input data. For the case of output perturbation, contraction analysis allows us to design convergent observers as well as set the level of privacy‐preserving noise appropriately. Two examples illustrate the proposed approach: estimating the edge formation probabilities in a social network using a dynamic stochastic block model, and syndromic surveillance relying on an epidemiological model.

The dynamic stochastic topic block model for dynamic networks with textual edges

The present paper develops a probabilistic model to cluster the nodes of a dynamic graph, accounting for the content of textual edges as well as their frequency. Ver-tices are clustered in groups which are homogeneous both in terms of interaction frequency and discussed topics. The dynamic graph is considered stationary on a latent time interval if the proportions of topics discussed between each pair of node groups do not change in time during that interval. A classification variational expectation-maximization (C-VEM) algorithm is adopted to perform inference. A model selection criterion is also derived to select the number of node groups, time clusters and topics. Experiments on simulated data are carried out to assess the proposed methodology. We finally illustrate an application to the Enron dataset.

Dealing with reciprocity in dynamic stochastic block models

A stochastic block model for dynamic network data is introduced, where directed relations among a set of nodes are observed at different time occasions and the blocks are represented by a sequence of latent variables following a Markov chain. Dyads are explicitly modeled conditional on the states occupied by both nodes involved in the relation. With respect to the approaches already available in the literature, the main focus is on reciprocity. In this regard, three different parameterizations are proposed in which: (i) reciprocity is allowed to depend on the blocks of the nodes in the dyad; (ii) reciprocity is assumed to be constant across blocks; and (iii) reciprocity is ruled out. The assumption of conditional independence between dyads (referred to different pairs of nodes and time occasions) given the latent blocks is always retained. Given the complexity of the model, inference on its parameters is based on a variational approach, where a lower bound of the log-likelihood function is maximized instead of the intractable full model log-likelihood. An approximate likelihood ratio test statistic is proposed which compares the value at convergence of this lower bound under different model specifications. This allows us to formally test for both the hypothesis of no reciprocity and that of constant reciprocity with respect to the latent blocks. The proposed approach is illustrated via a simulation study based on different scenarios. The application to two benchmark datasets in the social network literature is also proposed to illustrate the effectiveness of the proposal in studying reciprocity and identifying groups of nodes having a similar social behavior.

Covariate-assisted Spectral Clustering in Dynamic Networks

In this paper, we study the community detection problem in the dynamic stochastic blockmodel and dynamic stochastic co-blockmodel with node covariates. Covariate-assisted spectral clustering methods for estimating group membership in dynamic directed and undirected graphs are developed respectively. Weak consistency property is proved and degree correction is studied as well. Simulation shows that our new method outperforms existing spectral clustering methods for studying dynamic networks.

Revealing the hidden structure of dynamic ecological networks.

In ecology, recent technological advances and long-term data studies now provide longitudinal interaction data (e.g. between individuals or species). Most often, time is the parameter along which interactions evolve but any other one-dimensional gradient (temperature, altitude, depth, humidity, etc.) can be considered. These data can be modelled through a sequence of different snapshots of an evolving ecological network, i.e. a dynamic network. Here, we present how the dynamic stochastic block model approach developed by Matias & Miele (Matias & Miele In press J. R. Stat. Soc. B (doi:10.1111/rssb.12200)) can capture the complexity and dynamics of these networks. First, we analyse a dynamic contact network of ants and we observe a clear high-level assembly with some variations in time at the individual level. Second, we explore the structure of a food web evolving during a year and we detect a stable predator–prey organization but also seasonal differences in the prey assemblage. Our approach, based on a rigorous statistical method implemented in the R package dynsbm, can pave the way for exploration of evolving ecological networks.

Statistical Clustering of Temporal Networks Through a Dynamic Stochastic Block Model

SummaryStatistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges. Here, we explore statistical properties and frequentist inference in a model that combines a stochastic block model for its static part with independent Markov chains for the evolution of the nodes groups through time. We model binary data as well as weighted dynamic random graphs (with discrete or continuous edges values). Our approach, motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within-group connectivity behaviour. We study identifiability of the model parameters and propose an inference procedure based on a variational expectation–maximization algorithm as well as a model selection criterion to select the number of groups. We carefully discuss our initialization strategy which plays an important role in the method and we compare our procedure with existing procedures on synthetic data sets. We also illustrate our approach on dynamic contact networks: one of encounters between high school students and two others on animal interactions. An implementation of the method is available as an R package called dynsbm.