The Degree-Corrected Stochastic Block Model (DCSBM) is a popular model to generate random graphs with community structure given an expected degree sequence. The standard approach of community detection based on the DCSBM is to search for the model parameters that are the most likely to have produced the observed network data through maximum likelihood estimation (MLE). Current techniques for the MLE problem are heuristics, and therefore do not guarantee convergence to the optimum. We present mathematical programming formulations and exact solution methods that can provably find the model parameters and community assignments of maximum likelihood given an observed graph. We compare these exact methods with classical heuristic algorithms based on expectation-maximization (EM). The solutions given by exact methods give us a principled way of measuring the experimental performance of classical heuristics and comparing different variations thereof.
Network homophily, the tendency of similar nodes to be connected, and transitivity, the tendency of two nodes being connected if they share a common neighbor, are conflated properties in network analysis, since one mechanism can drive the other. Here we present a generative model and corresponding inference procedure that are capable of distinguishing between both mechanisms. Our approach is based on a variation of the stochastic block model (SBM) with the addition of triadic closure edges, and its inference can identify the most plausible mechanism responsible for the existence of every edge in the network, in addition to the underlying community structure itself. We show how the method can evade the detection of spurious communities caused solely by the formation of triangles in the network, and how it can improve the performance of edge prediction when compared to the pure version of the SBM without triadic closure.
Latent space models are frequently used for modeling single-layer networks and include many popular special cases, such as the stochastic block model and the random dot product graph. However, they are not well-developed for more complex network structures, which are becoming increasingly common in practice. Here we propose a new latent space model for multiplex networks: multiple, heterogeneous networks observed on a shared node set. Multiplex networks can represent a network sample with shared node labels, a network evolving over time, or a network with multiple types of edges. The key feature of our model is that it learns from data how much of the network structure is shared between layers and pools information across layers as appropriate. We establish identifiability, develop a fitting procedure using convex optimization in combination with a nuclear norm penalty, and prove a guarantee of recovery for the latent positions as long as there is sufficient separation between the shared and the individual latent subspaces. We compare the model to competing methods in the literature on simulated networks and on a multiplex network describing the worldwide trade of agricultural products.
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.
Chromatin communities stabilized by protein machinery play essential role in gene regulation and refine global polymeric folding of the chromatin fiber. However, treatment of these communities in the framework of the classical network theory (stochastic block model, SBM) does not take into account intrinsic linear connectivity of the chromatin loci. Here we propose the "polymer" block model, paving the way for community detection in polymer networks. On the basis of this new model we modify the non-backtracking flow operator and suggest the first protocol for annotation of compartmental domains in sparse single cell Hi-C matrices. In particular, we prove that our approach corresponds to the maximum entropy principle. The benchmark analyses demonstrates that the spectrum of the polymer non-backtracking operator resolves the true compartmental structure up to the theoretical detectability threshold, while all commonly used operators fail above it. We test various operators on real data and conclude that the sizes of the non-backtracking single cell domains are most close to the sizes of compartments from the population data. Moreover, the found domains clearly segregate in the gene density and correlate with the population compartmental mask, corroborating biological significance of our annotation of the chromatin compartmental domains in single cells Hi-C matrices.
A growing number of systems are represented as networks whose architecture conveys significant information and determines many of their properties. Examples of network architecture include modular, bipartite, and core-periphery structures. However inferring the network structure is a non trivial task and can depend sometimes on the chosen null model. Here we propose a method for classifying network structures and ranking its nodes in a statistically well-grounded fashion. The method is based on the use of Belief Propagation for learning through Entropy Maximization on both the Stochastic Block Model (SBM) and the degree-corrected Stochastic Block Model (dcSBM). As a specific application we show how the combined use of the two ensembles -SBM and dcSBM- allows to disentangle the bipartite and the core-periphery structure in the case of the e-MID interbank network. Specifically we find that, taking into account the degree, this interbank network is better described by a bipartite structure, while using the SBM the core-periphery structure emerges only when data are aggregated for more than a week.
We study community detection in the contextual stochastic block model arXiv:1807.09596 [cs.SI], arXiv:1607.02675 [stat.ME]. In arXiv:1807.09596 [cs.SI], the second author studied this problem in the setting of sparse graphs with high-dimensional node-covariates. Using the non-rigorous cavity method from statistical physics, they conjectured the sharp limits for community detection in this setting. Further, the information theoretic threshold was verified, assuming that the average degree of the observed graph is large. It is expected that the conjecture holds as soon as the average degree exceeds one, so that the graph has a giant component. We establish this conjecture, and characterize the sharp threshold for detection and weak recovery.
For community detection problem, spectral clustering is a widely used method for detecting clusters in networks. In this paper, we propose an improved spectral clustering (ISC) approach under the degree corrected stochastic block model (DCSBM). ISC is designed based on the k-means clustering algorithm on the weighted leading K + 1 eigenvectors of a regularized Laplacian matrix where the weights are their corresponding eigenvalues. Theoretical analysis of ISC shows that under mild conditions the ISC yields stable consistent community detection. Numerical results show that ISC outperforms classical spectral clustering methods for community detection on both simulated and eight empirical networks. Especially, ISC provides a significant improvement on two weak signal networks Simmons and Caltech, with error rates of 121/1137 and 96/590, respectively.
We consider the stochastic block model where connection between vertices is perturbed by some latent (and unobserved) random geometric graph. The objective is to prove that spectral methods are robust to this type of noise, even if they are agnostic to the presence (or not) of the random graph. We provide explicit regimes where the second eigenvector of the adjacency matrix is highly correlated to the true community vector (and therefore when weak/exact recovery is possible). This is possible thanks to a detailed analysis of the spectrum of the latent random graph, of its own interest.
Spectral clustering methods are widely used for detecting clusters in networks for community detection, while a small change on the graph Laplacian matrix could bring a dramatic improvement. In this paper, we propose a dual regularized graph Laplacian matrix and then employ it to three classical spectral clustering approaches under the degree-corrected stochastic block model. If the number of communities is known as $K$, we consider more than $K$ leading eigenvectors and weight them by their corresponding eigenvalues in the spectral clustering procedure to improve the performance. Three improved spectral clustering methods are dual regularized spectral clustering (DRSC) method, dual regularized spectral clustering on Ratios-of-eigenvectors (DRSCORE) method, and dual regularized symmetrized Laplacian inverse matrix (DRSLIM) method. Theoretical analysis of DRSC and DRSLIM show that under mild conditions DRSC and DRSLIM yield stable consistent community detection, moreover, DRSCORE returns perfect clustering under the ideal case. We compare the performances of DRSC, DRSCORE and DRSLIM with several spectral methods by substantial simulated networks and eight real-world networks.
We study the problem of $k$-way clustering in signed graphs. Considerable attention in recent years has been devoted to analyzing and modeling signed graphs, where the affinity measure between nodes takes either positive or negative values. Recently, Cucuringu et al. [CDGT 2019] proposed a spectral method, namely SPONGE (Signed Positive over Negative Generalized Eigenproblem), which casts the clustering task as a generalized eigenvalue problem optimizing a suitably defined objective function. This approach is motivated by social balance theory, where the clustering task aims to decompose a given network into disjoint groups, such that individuals within the same group are connected by as many positive edges as possible, while individuals from different groups are mainly connected by negative edges. Through extensive numerical simulations, SPONGE was shown to achieve state-of-the-art empirical performance. On the theoretical front, [CDGT 2019] analyzed SPONGE and the popular Signed Laplacian method under the setting of a Signed Stochastic Block Model (SSBM), for $k=2$ equal-sized clusters, in the regime where the graph is moderately dense. In this work, we build on the results in [CDGT 2019] on two fronts for the normalized versions of SPONGE and the Signed Laplacian. Firstly, for both algorithms, we extend the theoretical analysis in [CDGT 2019] to the general setting of $k \geq 2$ unequal-sized clusters in the moderately dense regime. Secondly, we introduce regularized versions of both methods to handle sparse graphs -- a regime where standard spectral methods underperform -- and provide theoretical guarantees under the same SSBM model. To the best of our knowledge, regularized spectral methods have so far not been considered in the setting of clustering signed graphs. We complement our theoretical results with an extensive set of numerical experiments on synthetic data.
The stochastic block model is one of the most studied network models for community detection. It is well-known that most algorithms proposed for fitting the stochastic block model likelihood function cannot scale to large-scale networks. One prominent work that overcomes this computational challenge is Amini et al.(2013), which proposed a fast pseudo-likelihood approach for fitting stochastic block models to large sparse networks. However, this approach does not have convergence guarantee, and is not well suited for small- or medium- scale networks. In this article, we propose a novel likelihood based approach that decouples row and column labels in the likelihood function, which enables a fast alternating maximization; the new method is computationally efficient, performs well for both small and large scale networks, and has provable convergence guarantee. We show that our method provides strongly consistent estimates of the communities in a stochastic block model. As demonstrated in simulation studies, the proposed method outperforms the pseudo-likelihood approach in terms of both estimation accuracy and computation efficiency, especially for large sparse networks. We further consider extensions of our proposed method to handle networks with degree heterogeneity and bipartite properties.
We consider an approach for community detection in time-varying networks. At its core, this approach maintains a small sketch graph to capture the essential community structure found in each snapshot of the full network. We demonstrate how the sketch can be used to explicitly identify six key community events which typically occur during network evolution: growth, shrinkage, merging, splitting, birth and death. Based on these detection techniques, we formulate a community detection algorithm which can process a network concurrently exhibiting all processes. One advantage afforded by the sketch-based algorithm is the efficient handling of large networks. Whereas detecting events in the full graph may be computationally expensive, the small size of the sketch allows changes to be quickly assessed. A second advantage occurs in networks containing clusters of disproportionate size. The sketch is constructed such that there is equal representation of each cluster, thus reducing the possibility that the small clusters are lost in the estimate. We present a new standardized benchmark based on the stochastic block model which models the addition and deletion of nodes, as well as the birth and death of communities. When coupled with existing benchmarks, this new benchmark provides a comprehensive suite of tests encompassing all six community events. We provide analysis and a set of numerical results demonstrating the advantages of our approach both in run time and in the handling of small clusters.
We consider the problem of estimating overlapping community memberships in a network, where each node can belong to multiple communities. More than a few communities per node are difficult to both estimate and interpret, so we focus on sparse node membership vectors. Our algorithm is based on sparse principal subspace estimation with iterative thresholding. The method is computationally efficient, with a computational cost equivalent to estimating the leading eigenvectors of the adjacency matrix, and does not require an additional clustering step, unlike spectral clustering methods. We show that a fixed point of the algorithm corresponds to correct node memberships under a version of the stochastic block model. The methods are evaluated empirically on simulated and real-world networks, showing good statistical performance and computational efficiency.
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? and 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.
Block modeling is widely used in studies on complex networks. The cornerstone model is the stochastic block model (SBM), widely used over the past decades. However, the SBM is limited in analyzing complex networks as the model is, in essence, a random graph model that cannot reproduce the basic properties of many complex networks, such as sparsity and heavy-tailed degree distribution. In this paper, we provide an edge exchangeable block model that incorporates such basic features and simultaneously infers the latent block structure of a given complex network. Our model is a Bayesian nonparametric model that flexibly estimates the number of blocks and takes into account the possibility of unseen nodes. Using one synthetic dataset and one real-world stock ownership dataset, we show that our model outperforms state-of-the-art SBMs for held-out link prediction tasks.
The $p$-tensor Ising model is a one-parameter discrete exponential family for modeling dependent binary data, where the sufficient statistic is a multi-linear form of degree $p \geq 2$. This is a natural generalization of the matrix Ising model, that provides a convenient mathematical framework for capturing higher-order dependencies in complex relational data. In this paper, we consider the problem of estimating the natural parameter of the $p$-tensor Ising model given a single sample from the distribution on $N$ nodes. Our estimate is based on the maximum pseudo-likelihood (MPL) method, which provides a computationally efficient algorithm for estimating the parameter that avoids computing the intractable partition function. We derive general conditions under which the MPL estimate is $\sqrt N$-consistent, that is, it converges to the true parameter at rate $1/\sqrt N$. In particular, we show the $\sqrt N$-consistency of the MPL estimate in the $p$-spin Sherrington-Kirkpatrick (SK) model, spin systems on general $p$-uniform hypergraphs, and Ising models on the hypergraph stochastic block model (HSBM). In fact, for the HSBM we pin down the exact location of the phase transition threshold, which is determined by the positivity of a certain mean-field variational problem, such that above this threshold the MPL estimate is $\sqrt N$-consistent, while below the threshold no estimator is consistent. Finally, we derive the precise fluctuations of the MPL estimate in the special case of the $p$-tensor Curie-Weiss model. An interesting consequence of our results is that the MPL estimate in the Curie-Weiss model saturates the Cramer-Rao lower bound at all points above the estimation threshold, that is, the MPL estimate incurs no loss in asymptotic efficiency, even though it is obtained by minimizing only an approximation of the true likelihood function for computational tractability.
This article studies the estimation of latent community memberships from pairwise interactions in a network of $N$ nodes, where the observed interactions can be of arbitrary type, including binary, categorical, and vector-valued, and not excluding even more general objects such as time series or spatial point patterns. As a generative model for such data, we introduce a stochastic block model with a general measurable interaction space $\mathcal S$, for which we derive information-theoretic bounds for the minimum achievable error rate. These bounds yield sharp criteria for the existence of consistent and strongly consistent estimators in terms of data sparsity, statistical similarity between intra- and inter-block interaction distributions, and the shape and size of the interaction space. The general framework makes it possible to study temporal and multiplex networks with $\mathcal S = \{0,1\}^T$, in settings where both $N \to \infty$ and $T \to \infty$, and the temporal interaction patterns are correlated over time. For temporal Markov interactions, we derive sharp consistency thresholds. We also present fast online estimation algorithms which fully utilise the non-binary nature of the observed data. Numerical experiments on synthetic and real data show that these algorithms rapidly produce accurate estimates even for very sparse data arrays.
Communities are a common and widely studied structure in networks, typically under the assumption that the network is fully and correctly observed. In practice, network data are often collected by querying nodes about their connections. In some settings, all edges of a sampled node will be recorded, and in others, a node may be asked to name its connections. These sampling mechanisms introduce noise and bias which can obscure the community structure and invalidate assumptions underlying standard community detection methods. We propose a general model for a class of network sampling mechanisms based on recording edges via querying nodes, designed to improve community detection for network data collected in this fashion. We model edge sampling probabilities as a function of both individual preferences and community parameters, and show community detection can be performed by spectral clustering under this general class of models. We also propose, as a special case of the general framework, a parametric model for directed networks we call the nomination stochastic block model, which allows for meaningful parameter interpretations and can be fitted by the method of moments. Both spectral clustering and the method of moments in this case are computationally efficient and come with theoretical guarantees of consistency. We evaluate the proposed model in simulation studies on both unweighted and weighted networks and apply it to a faculty hiring dataset, discovering a meaningful hierarchy of communities among US business schools.
We propose a novel stochastic network model, called Fractal Gaussian Network (FGN), that embodies well-defined and analytically tractable fractal structures. Such fractal structures have been empirically observed in diverse applications. FGNs interpolate continuously between the popular purely random geometric graphs (a.k.a. the Poisson Boolean network), and random graphs with increasingly fractal behavior. In fact, they form a parametric family of sparse random geometric graphs that are parametrized by a fractality parameter which governs the strength of the fractal structure. FGNs are driven by the latent spatial geometry of Gaussian Multiplicative Chaos (GMC), a canonical model of fractality in its own right. We asymptotically characterize the expected number of edges, triangles, cliques and hub-and-spoke motifs in FGNs, unveiling a distinct pattern in their scaling with the size parameter of the network. We then examine the natural question of detecting the presence of fractality and the problem of parameter estimation based on observed network data, in addition to fundamental properties of the FGN as a random graph model. We also explore fractality in community structures by unveiling a natural stochastic block model in the setting of FGNs. Finally, we substantiate our results with phenomenological analysis of the FGN in the context of available scientific literature for fractality in networks, including applications to real-world massive network data.