Complex networks enable to represent and characterize the interactions between entities in various complex systems which widely exist in the real world and usually generate vast amounts of data about all the elements, their behaviors and interactions over time. The studies concentrating on new network analysis approaches and methodologies are vital because of the diversity and ubiquity of complex networks. The stochastic block model (SBM), based on Bayesian theory, is a statistical network model. SBMs are essential tools for analyzing complex networks since SBMs have the advantages of interpretability, expressiveness, flexibility and generalization. Thus, designing diverse SBMs and their learning algorithms for various networks has become an intensively researched topic in network analysis and data mining. In this article, we review, in a comprehensive and in-depth manner, SBMs for different types of networks (i.e., model extensions), existing methods (including parameter estimation and model selection) for learning optimal SBMs for given networks and SBMs combined with deep learning. Finally, we provide an outlook on the future research directions of SBMs.

Finding hidden structural patterns is a critical problem for all types of networks, including signed networks. Among all of the methods for structural analysis of complex network, stochastic blockmodel (SBM) is an important research tool because it is flexible and can generate networks with many different types of structures. However, most existing SBM learning methods for signed networks are unsupervised, leading to poor performance in terms of finding hidden structural patterns, especially when handling noisy and sparse networks. Learning SBM in a semi-supervised way is a promising avenue for overcoming the above difficulty. In this type of model, a small number of labelled nodes and a large number of unlabelled nodes, coupled with their network structures, are simultaneously used to train SBM. We propose a novel semi-supervised signed stochastic blockmodel and its learning algorithm based on variational Bayesian inference, with the goal of discovering both assortative (the nodes connect more densely in same clusters than that in different clusters) and disassortative (the nodes link more sparsely in same clusters than that in different clusters) structures from signed networks. The proposed model is validated through a number of experiments wherein it compared with the state-of-the-art methods using both synthetic and real-world data. The carefully designed tests, allowing to account for different scenarios, show our method outperforms other approaches existing in this space. It is especially relevant in the case of noisy and sparse networks as they constitute the majority of the real-world networks.

Passengers move between urban places for diverse interests and drive the metropolitan regions as the aggregation of urban places to group into network communities. This paper aims to examine the relationship between the spatial patterns (represented by the network communities) of mobility flows and places of interest (POIs). Furtherly, it intends to identify the categories of POIs that play the most significant role in shaping the spatial patterns of mobility flows. To achieve these purposes, we partition the study area into disjoint regions and construct the network with each partitioned region as a node and connection between them as links weighted by the mobility flows. The community detection algorithm is implemented on the network to discover spatial mobility patterns, and the multiclass classification based on the logistic regression method is adopted to classify spatial communities featured by POIs. Taking the taxi systems of Shanghai and Beijing as examples, we detect spatial communities based on the movement strengths among regions. Then we investigate their correlations with POIs. It finds that communities’ modularity correlates linearly with POIs; particularly governments, hotels, and the traffic facilities are of the most significance for generating the mobility patterns. This study can provide valuable insight into understanding the spatial mobility patterns from the perspective of POIs.

Signed networks with positive and negative links attract considerable interest in their studying since they contain more information than unsigned networks. Community detection and sign (or attitude) prediction are still primary challenges, as the fundamental problems of signed network analysis. For this, a generative Bayesian approach is presented wherein 1) a signed stochastic blockmodel is proposed to characterize the community structure in the context of signed networks, by explicit formulating the distributions of the density and frustration of signed links from a stochastic perspective, and 2) a model learning algorithm is advanced by theoretical deriving a variational Bayes EM for the parameter estimation and variation-based approximate evidence for the model selection. The comparison of the above approach with the state-of-the-art methods on synthetic and real-world networks, shows its advantage in the community detection and sign prediction for the exploratory networks.

Stochastic blockmodel (SBM) enables us to decompose and analyze an exploratory network without a priori knowledge about its intrinsic structure. However, the task of effectively and efficiently learning a SBM from a large-scale network is still challenging due to the high computational cost of its model selection and parameter estimation. To address this issue, we present a novel SBM learning algorithm referred to as BLOS (BLOckwise Sbm learning). Distinct from the literature, the model selection and parameter estimation of SBM are concurrently, rather than alternately, executed in BLOS by embedding the minimum message length criterion into a block-wise EM algorithm, which greatly reduces the time complexity of SBM learning without losing learning accuracy and modeling flexibility. Its effectiveness and efficiency have been tested through rigorous comparisons with the state-of-the-art methods on both synthetic and real-world networks.

There has been an increasing interest in exploring signed networks with positive and negative links in that they contain more information than unsigned networks. As fundamental problems of signed network analysis, community detection and sign (or attitude) prediction are still primary challenges. To address them, we propose a generative Bayesian approach, in which 1) a signed stochastic blockmodel is proposed to characterize the community structure in context of signed networks, by means of explicitly formulating the distributions of both density and frustration of signed links from a stochastic perspective, and 2) a model learning algorithm is proposed by theoretically deriving a variational Bayes EM for parameter estimation and a variation based approximate evidence for model selection. Through the comparisons with state-of-the-art methods on synthetic and real-world networks, the proposed approach shows its superiority in both community detection and sign prediction for exploratory networks.