Community Detection For Mixed Membership Stochastic Block Model With Covariates

As one of the most important features of social network,community structure can not only help us understand the topological characteristics and essence of the network,but also play an important role in exploring the behavior characteristics of users and the logical relationship between users in a micro level.Therefore,Community detection as an important social network research attracts many researchers in the field researching from different parts,but there are still many problems cannot be solved in community detection research,such as the overlapping community detection,the problem of heterogeneous network,the application of community detection and so on.In the field of community detection,the early community detection methods are mainly about the social networks of excluding covariate or other information,and don't make detection about overlap network community.However,Mixed Membership Stochastic Block Model(MMSB)can be used to divide overlap network community,at the same time,each node can be obtained in the mixed membership degree of each community.However,the existing MMSB model only considers the connection information of the social network itself without considering the covariate information of nodes,which limits the accuracy and application of the model's community detection.Roy et al.(2018)considered using Stochastic Model(SBM)with node covariables to study Facebook data,but SBM Model cannot be used for overlapping communities.Based on this model,this paper proposes an overlapping community partition method with covariate information based on MMSB model based on existing social networking community detection methods.This paper mainly completes the following research contents:1)Based on the MMSB model,an overlapping community detection model is proposed,which includes the nodes' covariables.Firstly,the model simulates the generation process of the observation network by establishing a generation model.Then,the maximum likelihood estimation criterion is used to estimate the parameters in the model.According to the parameters,the mixed membership degree of nodes,the social network is divided into communities.Due to the high complexity of the likelihood function of this model and the fact that the parameters are not mutually independent,it is not convenient to use the conventional maximum likelihood estimation method for parameter estimation.Therefore,this paper uses the variational maximum expectation algorithm for parameter estimation.We will explain the rationality of the model construction from the mathematical point of view.Then the iterative formula of VBEM algorithm is derived.2)As we all know,there are many nodes covariate information exist in social networks,such as Facebook users' age,gender,school,occupation,preference,hometown and so on.Some covariates do not play an important role in community detection.Especially when there are enough nodes in the social network,filtering covariates can greatly reduce the running time of the model on the premise of guaranteeing the effect of community detection.In this paper,The Least Absolute Shrinkage and Selection Operator(Lasso)method is adopted to filter covariables,which can filter variables effectively and reduce the complexity of the model.3)The performance of this model is evaluated by numerical simulation.This paper mainly evaluates the performance of the model under the conditions of uniform and non-uniform communities,low,medium and high OIR and average degree.The main indicator used in the evaluation model is standardized mutual information.The results show that the network community generated by MMSB model with covariates is better under high average degree.4)Using the MMSB model with covariates made community detection on Facebook data,empirical analysis was carried out,and the results of community detection were displayed by visual method.The empirical results show that the likelihood function of the model is converged,and the partition effect of the model can be demonstrated by drawing the heat map of the adjacency matrix and the mixed membership degree of the display node.Finally,using the modularity method,we quantified the community detection effects of the MMSB model with covariates and the basic MMSB model and SBM model.As a result,we found that the community detection effect of MMSB model with covariates was better than other two models.