Unifying Sparsest Cut, Cluster Deletion, and Modularity Clustering Objectives with Correlation Clustering

Graph clustering, or community detection, is the task of identifying groups of closely related objects in a large network. In this paper we introduce a new community-detection framework called LambdaCC that is based on a specially weighted version of correlation clustering. A key component in our methodology is a clustering resolution parameter, $\lambda$, which implicitly controls the size and structure of clusters formed by our framework. We show that, by increasing this parameter, our objective effectively interpolates between two different strategies in graph clustering: finding a sparse cut and forming dense subgraphs. Our methodology unifies and generalizes a number of other important clustering quality functions including modularity, sparsest cut, and cluster deletion, and places them all within the context of an optimization problem that has been well studied from the perspective of approximation algorithms. Our approach is particularly relevant in the regime of finding dense clusters, as it leads to a 2-approximation for the cluster deletion problem. We use our approach to cluster several graphs, including large collaboration networks and social networks

Stochastic Block Models are a Discrete Surface Tension

Networks, which represent agents and interactions between them, arise in myriad applications throughout the sciences, engineering, and even the humanities. To understand large-scale structure in a network, a common task is to cluster a network's nodes into sets called "communities", such that there are dense connections within communities but sparse connections between them. A popular and statistically principled method to perform such clustering is to use a family of generative models known as stochastic block models (SBMs). In this paper, we show that maximum likelihood estimation in an SBM is a network analog of a well-known continuum surface-tension problem that arises from an application in metallurgy. To illustrate the utility of this relationship, we implement network analogs of three surface-tension algorithms, with which we successfully recover planted community structure in synthetic networks and which yield fascinating insights on empirical networks that we construct from hyperspectral videos.Comment: to appear in Journal of Nonlinear Scienc

A Triad Percolation Method for Detecting Communities in Social Networks

For the purpose of detecting communities in social networks, a triad percolation method is proposed, which first locates all close-triads and open-triads from a social network, then a specified close-triad or open-triad is selected as the seed to expand by utilizing the triad percolation method, such that a community is found when this expanding process meet a particular threshold. This approach can efficiently detect communities not only from a densely social network, but also from the sparsely one. Experimental results performing on real-world social benchmark networks and artificially simulated networks give a satisfactory correspondence

Statistical Methods for Networks with Node Covariates

Network data, which represent relations or interactions between individual entities, together with nodal covariates information, arise in many scientific and engineering fields such as biology and social science. This dissertation focuses on developing statistical models and theory that utilize information from both the network structure and node covariates to improve statistical learning tasks, such as community detection and missing value imputation. The first project studies the problem of community detection for degree-heterogeneous networks with covariates, where we aim to cluster the nodes into groups that share similar patterns in link connectivity and/or covariates distribution. We consider incorporating node covariates via a flexible degree-corrected block model by allowing the community memberships to depend on node covariates, while the link probabilities are determined by both node community memberships and degree parameters. We develop two algorithms, one using the variational inference and the other based on the pseudo-likelihood for estimating the proposed model. Simulation studies indicate that the proposed model can obtain better community detection results compared to methods that only utilize the network information. Further, we show that under mild conditions, the community memberships and the covariate parameters can be estimated consistently. The second project considers the problem of missing value imputation when individuals are linked through a network. We assume the edges in the network are related with the distances in the covariates of the individuals through a latent space network model. We propose an iterative imputation algorithm that is flexible and utilizes both the correlation among node variables and the connectivity between observations given by the network. We relate the proposed method to a Bayesian model and discuss the convergence of the imputation distribution when the specified conditional models for imputation are compatible with the true underlying model of the covariates. We also use simulation studies and a data example to illustrate empirically that the imputation accuracy can be improved by incorporating network information. The final contribution of this dissertation is on incorporating covariates under the edge exchangeable framework. Edge exchangeable models have attractive theoretical and practical properties which make them appropriate for modeling many sparse real-world interaction networks constructed through edge sampling mechanisms. However, as far as we know, there is no edge exchangeable network model that allows for node covariates. In the third project, we propose a model that incorporates node covariates under the edge exchangeable model framework and show that it enjoys properties such as sparsity, and partial exchangeability. We further develop a maximum likelihood estimation method to estimate the model parameters and demonstrate its performance through both simulation studies and a data example.PHDStatisticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/163165/1/liuyumu_1.pd