Community detection is one of the most important problems in network analysis. Among many algorithms proposed for this task, methods based on statistical inference are of particular interest: they are mathematically sound and were shown to provide partitions of good quality. Statistical inference methods are based on fitting some random graph model (a.k.a. null model) to the observed network by maximizing the likelihood. The choice of this model is extremely important and is the main focus of the current study. We provide an extensive theoretical and empirical analysis to compare several models: the widely used planted partition model, recently proposed degree-corrected modification of this model, and a new null model having some desirable statistical properties. We also develop and compare two likelihood optimization algorithms suitable for the models under consideration. An extensive empirical analysis on a variety of datasets shows, in particular, that the new model is the best one for describing most of the considered real-world complex networks according to the likelihood of observed graph structures.
To develop a machine sound monitoring system, a method for detecting anomalous sound is proposed. Exact likelihood estimation using Normalizing Flows is a promising technique for unsupervised anomaly detection, but it can fail at out-of-distribution detection since the likelihood is affected by the smoothness of the data. To improve the detection performance, we train the model to assign higher likelihood to target machine sounds and lower likelihood to sounds from other machines of the same machine type. We demonstrate that this enables the model to incorporate a self-supervised classificatio…
We give a statistical interpretation of entropic optimal transport by showing that performing maximum-likelihood estimation for Gaussian deconvolution corresponds to calculating a projection with respect to the entropic optimal transport distance. This structural result gives theoretical support for the wide adoption of these tools in the machine learning community.
Unlike the commonly used parametric regression models such as mixed models, that can easily violate the required statistical assumptions and result in invalid statistical inference, target maximum likelihood estimation allows more realistic data-generative models and provides double-robust, semi-parametric and efficient estimators. Target maximum likelihood estimators (TMLEs) for the causal effect of a community-level static exposure were previously proposed by Balzer et al. In this manuscript, we build on this work and present identifiability results and develop two semi-parametric efficient …
Over the past years, many applications aim to assess the causal effect of treatments assigned at the community level, while data are still collected at the individual level among individuals of the community. In many cases, one wants to evaluate the effect of a stochastic intervention on the community, where all communities in the target population receive probabilistically assigned treatments based on a known specified mechanism (e.g., implementing a community-level intervention policy that target stochastic changes in the behavior of a target population of communities). The tmleCommunity pac…
The asymptotic normality of the Maximum Likelihood Estimator (MLE) is a cornerstone of statistical theory. In the present paper, we provide sharp explicit upper bounds on Zolotarev-type distances between the exact, unknown distribution of the MLE and its limiting normal distribution. Our approach to this fundamental issue is based on a sound combination of the Delta method, Stein's method, Taylor expansions and conditional expectations, for the classical situations where the MLE can be expressed as a function of a sum of independent and identically distributed terms. This encompasses in partic…
We derive sharp thresholds for exact recovery of communities in a weighted stochastic block model, where observations are collected in the form of a weighted adjacency matrix, and the weight of each edge is generated independently from a distribution determined by the community membership of its endpoints. Our main result, characterizing the precise boundary between success and failure of maximum likelihood estimation when edge weights are drawn from discrete distributions, involves the Renyi divergence of order $\frac{1}{2}$ between the distributions of within-community and between-community …
Recently network analysis has gained more and more attentions in statistics, as well as in computer science, probability, and applied mathematics. Community detection for the stochastic block model (SBM) is probably the most studied topic in network analysis. Many methodologies have been proposed. Some beautiful and significant phase transition results are obtained in various settings. In this paper, we provide a general minimax theory for community detection. It gives minimax rates of the mis-match ratio for a wide rage of settings including homogeneous and inhomogeneous SBMs, dense and spars…
The University of Texas at Dallas（德克萨斯大学达拉斯分校）；Virginia Tech（弗吉尼亚理工大学）
This paper produces an efficient Semidefinite Programming (SDP) solution for community detection that incorporates non-graph data, which in this context is known as side information. SDP is an efficient solution for standard community detection on graphs. We formulate a semi-definite relaxation for the maximum likelihood estimation of node labels, subject to observing both graph and non-graph data. This formulation is distinct from the SDP solution of standard community detection, but maintains its desirable properties. We calculate the exact recovery threshold for three types of non-graph inf…
Sequential change-point detection for graphs is a fundamental problem for streaming network data types and has wide applications in social networks and power systems. Given fixed vertices and a sequence of random graphs, the objective is to detect the change-point where the underlying distribution of the random graph changes. In particular, we focus on the local change that only affects a subgraph. We adopt the classical Erdos-Renyi model and revisit the generalized likelihood ratio (GLR) detection procedure. The scan statistic is computed by sequentially estimating the most-likely subgraph wh…
A class of random non-stationary signals termed timbre x dynamics is introduced and studied. These signals are obtained by non-linear transformations of sta-tionary random gaussian signals, in such a way that the transformation can be approximated by translations in an appropriate representation domain. In such situations, approximate maximum likelihood estimation techniques can be de-rived, which yield simultaneous estimation of the transformation and the power spectrum of the underlying stationary signal. This paper focuses on the case of modulation and time warping of station-ary signals, a…
Computer models are used to model complex processes in various disciplines. Often, a key source of uncertainty in the behavior of complex computer models is uncertainty due to unknown model input parameters. Statistical computer model calibration is the process of inferring model parameter values, along with associated uncertainties, from observations of the physical process and from model outputs at various parameter settings. Observations and model outputs are often in the form of high-dimensional spatial fields, especially in the environmental sciences. Sound statistical inference may be co…
In this paper, we study the information theoretic bounds for exact recovery in sub-hypergraph models for community detection. We define a general model called the $m-$uniform sub-hypergraph stochastic block model ($m-$ShSBM). Under the $m-$ShSBM, we use Fano's inequality to identify the region of model parameters where any algorithm fails to exactly recover the planted communities with a large probability. We also identify the region where a Maximum Likelihood Estimation (MLE) algorithm succeeds to exactly recover the communities with high probability. Our bounds are tight and pertain to the c…
Inferring microbial community structure based on temporal metagenomics data is an important goal in microbiome studies. The deterministic generalized Lotka-Volterra differential (GLV) equations have been used to model the dynamics of microbial data. However, these approaches fail to take random environmental fluctuations into account, which may negatively impact the estimates. We propose a new stochastic GLV (SGLV) differential equation model, where the random perturbations of Brownian motion in the model can naturally account for the external environmental effects on the microbial community. …
\cite{bickel2009nonparametric} developed a general framework to establish consistency of community detection in stochastic block model (SBM). In most applications of this framework, the community label is discrete. For example, in \citep{bickel2009nonparametric,zhao2012consistency} the degree corrected SBM is assumed to have a discrete degree parameter. In this paper, we generalize the method of \cite{bickel2009nonparametric} to give consistency analysis of maximum likelihood estimator (MLE) in SBM with continuous community label. We show that there is a standard procedure to transform the $||…
Statistically sound pattern discovery harnesses the rigour of statistical hypothesis testing to overcome many of the issues that have hampered standard data mining approaches to pattern discovery. Most importantly, application of appropriate statistical tests allows precise control over the risk of false discoveries -- patterns that are found in the sample data but do not hold in the wider population from which the sample was drawn. Statistical tests can also be applied to filter out patterns that are unlikely to be useful, removing uninformative variations of the key patterns in the data. Thi…
We demonstrate an exact equivalence between two widely used methods of community detection in networks, the method of modularity maximization in its generalized form which incorporates a resolution parameter controlling the size of the communities discovered, and the method of maximum likelihood applied to the special case of the stochastic block model known as the planted partition model, in which all communities in a network are assumed to have statistically similar properties. Among other things, this equivalence provides a mathematically principled derivation of the modularity function, cl…
University of Michigan（密歇根大学）；Google, Inc.（谷歌公司）；University of California, Berkeley（加州大学伯克利分校）
Many algorithms have been proposed for fitting network models with communities, but most of them do not scale well to large networks, and often fail on sparse networks. Here we propose a new fast pseudo-likelihood method for fitting the stochastic block model for networks, as well as a variant that allows for an arbitrary degree distribution by conditioning on degrees. We show that the algorithms perform well under a range of settings, including on very sparse networks, and illustrate on the example of a network of political blogs. We also propose spectral clustering with perturbations, a meth…
Community structures detection in complex network is important for understanding not only the topological structures of the network, but also the functions of it. Stochastic block model and nonnegative matrix factorization are two widely used methods for community detection, which are proposed from different perspectives. In this paper, the relations between them are studied. The logarithm of likelihood function for stochastic block model can be reformulated under the framework of nonnegative matrix factorization. Besides the model equivalence, the algorithms employed by the two methods are di…
Pontifícia Universidade Católica do Rio de Janeiro (PUC-Rio)（里约热内卢天主教大学）
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 param…
The problem of community detection receives great attention in recent years. Many methods have been proposed to discover communities in networks. In this paper, we propose a Gaussian stochastic blockmodel that uses Gaussian distributions to fit weight of edges in networks for non-overlapping community detection. The maximum likelihood estimation of this model has the same objective function as general label propagation with node preference. The node preference of a specific vertex turns out to be a value proportional to the intra-community eigenvector centrality (the corresponding entry in pri…
Real-world networks usually have community structure, that is, nodes are grouped into densely connected communities. Community detection is one of the most popular and best-studied research topics in network science and has attracted attention in many different fields, including computer science, statistics, social sciences, among others. Numerous approaches for community detection have been proposed in literature, from ad-hoc algorithms to systematic model-based approaches. The large number of available methods leads to a fundamental question: whether a certain method can provide consistent e…
This paper presents a statistically sound method for using likelihood to assess potential models of network evolution. The method is tested on data from five real networks. Data from the internet autonomous system network, from two photo sharing sites and from a co-authorship network are tested using this framework.
We study the community detection problem on a Gaussian mixture model, in which vertices are divided into $k\geq 2$ distinct communities. The major difference in our model is that the intensities for Gaussian perturbations are different for different entries in the observation matrix, and we do not assume that every community has the same number of vertices. We explicitly find the threshold for the exact recovery of the maximum likelihood estimation. Applications include the community detection on hypergraphs.
Building on existing stochastic actor-oriented models for panel data, we employ a conditional logistic framework to explore growth mechanisms for tie creation in continuously-observed networks. This framework models the likelihood of tie formation distinguishing it from hazard models that consider time to tie formation. It enables multiple growth mechanisms for network evolution (homophily, focus constraints, reinforcement, reciprocity, triadic closure, and popularity) to be modeled simultaneously. We apply this framework to communication within a Facebook-like community. The findings exemplif…
The mean field variational Bayes method is becoming increasingly popular in statistics and machine learning. Its iterative Coordinate Ascent Variational Inference algorithm has been widely applied to large scale Bayesian inference. See Blei et al. (2017) for a recent comprehensive review. Despite the popularity of the mean field method there exist remarkably little fundamental theoretical justifications. To the best of our knowledge, the iterative algorithm has never been investigated for any high dimensional and complex model. In this paper, we study the mean field method for community detect…
We study a new connection between a technical measure called $μ$-conductance that arises in the study of Markov chains for sampling convex bodies and the network community profile that characterizes size-resolved properties of clusters and communities in social and information networks. The idea of $μ$-conductance is similar to the traditional graph conductance, but disregards sets with small volume. We derive a sequence of optimization problems including a low-rank semi-definite program from which we can derive a lower bound on the optimal $μ$-conductance value. These ideas give the first the…
We study networks that display community structure -- groups of nodes within which connections are unusually dense. Using methods from random matrix theory, we calculate the spectra of such networks in the limit of large size, and hence demonstrate the presence of a phase transition in matrix methods for community detection, such as the popular modularity maximization method. The transition separates a regime in which such methods successfully detect the community structure from one in which the structure is present but is not detected. By comparing these results with recent analyses of maximu…
Bell Laboratories, Nokia（诺基亚贝尔实验室）；Princeton University（普林斯顿大学）
We present a novel active learning algorithm for community detection on networks. Our proposed algorithm uses a Maximal Expected Model Change (MEMC) criterion for querying network nodes label assignments. MEMC detects nodes that maximally change the community assignment likelihood model following a query. Our method is inspired by detection in the benchmark Stochastic Block Model (SBM), where we provide sample complexity analysis and empirical study with SBM and real network data for binary as well as for the multi-class settings. The analysis also covers the most challenging case of sparse de…
While there has been a plethora of approaches for detecting disjoint communities from real-world complex networks, some methods for detecting overlapping community structures have also been recently proposed. In this work, we argue that, instead of developing separate approaches for detecting overlapping communities, a promising alternative is to infer the overlapping communities from multiple disjoint community structures. We propose an ensemble-based approach, called EnCoD, that leverages the solutions produced by various disjoint community detection algorithms to discover the overlapping co…
In this paper, we prove a sharp limit on the community detection problem with colored edges. We assume two equal-sized communities and there are $m$ different types of edges. If two vertices are in the same community, the distribution of edges follows $p_i=α_i\log{n}/n$ for $1\leq i \leq m$, otherwise the distribution of edges is $q_i=β_i\log{n}/n$ for $1\leq i \leq m$, where $α_i$ and $β_i$ are positive constants and $n$ is the total number of vertices. Under these assumptions, a fundamental limit on community detection is characterized using the Hellinger distance between the two distributio…
University of Toledo（托莱多大学）；The Ohio State University（俄亥俄州立大学）
The stochastic block model (SBM) is one of the most widely used generative models for network data. Many continuous-time dynamic network models are built upon the same assumption as the SBM: edges or events between all pairs of nodes are conditionally independent given the block or community memberships, which prevents them from reproducing higher-order motifs such as triangles that are commonly observed in real networks. We propose the multivariate community Hawkes (MULCH) model, an extremely flexible community-based model for continuous-time networks that introduces dependence between node p…
Community detection is a commonly used technique for identifying groups in a network based on similarities in connectivity patterns. To facilitate community detection in large networks, we recast the network to be partitioned into a smaller network of 'super nodes', each super node comprising one or more nodes in the original network. To define the seeds of our super nodes, we apply the 'CoreHD' ranking from dismantling and decycling. We test our approach through the analysis of two common methods for community detection: modularity maximization with the Louvain algorithm and maximum likelihoo…
Many empirical networks have community structure, in which nodes are densely interconnected within each community (i.e., a group of nodes) and sparsely across different communities. Like other local and meso-scale structure of networks, communities are generally heterogeneous in various aspects such as the size, density of edges, connectivity to other communities and significance. In the present study, we propose a method to statistically test the significance of individual communities in a given network. Compared to the previous methods, the present algorithm is unique in that it accepts diff…
Empirical observations suggest that in practice, community membership does not completely explain the dependency between the edges of an observation graph. The residual dependence of the graph edges are modeled in this paper, to first order, by auxiliary node latent variables that affect the statistics of the graph edges but carry no information about the communities of interest. We then study community detection in graphs obeying the stochastic block model and censored block model with auxiliary latent variables. We analyze the conditions for exact recovery when these auxiliary latent variabl…
Recovering Fairness Directly from Modularity: a New Way for Fair Community Partitioning
School of Mathematics and Statistics（数学与统计学学院）；Xidian University（西安电子科技大学）；City Faculty of Data Science（数据科学城市学院）；University of Macau（澳门大学）；College of Mathematical Medicine（数学医学学院）；Zhejiang Normal University（浙江师范大学）
Community structure analysis is a powerful tool for social networks, which can simplify their topological and functional analysis considerably. However, since community detection methods have random factors and real social networks obtained from complex systems always contain error edges, evaluating the significance of community structure partitioned is an urgent and important question. In this paper, integrating the specific characteristics of real society, we present a novel framework analyzing the significance of social community specially. The dynamics of social interactions are modeled by…
To unravel the driving patterns of networks, the most popular models rely on community detection algorithms. However, these approaches are generally unable to reproduce the structural features of the network. Therefore, attempts are always made to develop models that incorporate these network properties beside the community structure. In this work, we present a probabilistic generative model and an efficient algorithm to both perform community detection and capture reciprocity in networks. Our approach jointly models pairs of edges with exact 2-edge joint distributions. In addition, it provide…
The stochastic block model is able to generate different network partitions, ranging from traditional assortative communities to disassortative structures. Since the degree-corrected stochastic block model does not specify which mixing pattern is desired, the inference algorithms, which discover the most likely partition of the networks nodes, are likely to get trapped in the local optima of the log-likelihood. Here we introduce a new model constraining nodes' internal degrees ratios in the objective function to stabilize the inference of block models from the observed network data. Given the …
We analyse bill cosponsorship networks in the Italian Chamber of Deputies. In comparison with other parliaments, a distinguishing feature of the Chamber is the large number of political groups. Our analysis aims to infer the pattern of collaborations between these groups from data on bill cosponsorships. We propose an extension of stochastic block models for edge-valued graphs and derive measures of group productivity and of collaboration between political parties. As the model proposed encloses a large number of parameters, we pursue a penalized likelihood approach that enables us to infer a …
An important aspect of preventing fake news dissemination is to proactively detect the likelihood of its spreading. Research in the domain of fake news spreader detection has not been explored much from a network analysis perspective. In this paper, we propose a graph neural network based approach to identify nodes that are likely to become spreaders of false information. Using the community health assessment model and interpersonal trust we propose an inductive representation learning framework to predict nodes of densely-connected community structures that are most likely to spread fake news…
Dankook University（檀国大学）；Yonsei University（延世大学）；University of Konstanz（康斯坦茨大学）
The discovery of community structures in social networks has gained significant attention since it is a fundamental problem in understanding the networks' topology and functions. However, most social network data are collected from partially observable networks with both missing nodes and edges. In this paper, we address a new problem of detecting overlapping community structures in the context of such an incomplete network, where communities in the network are allowed to overlap since nodes belong to multiple communities at once. To solve this problem, we introduce KroMFac, a new framework th…
