Split Likelihood Method For Joint Community Detection In Multi-layer Networks

Modern society is an era of complex networks,which represent the interaction and connection between various entities in human society.In recent years,with the continuous development of complex network research,people have gradually begun to use this new tool to study large-scale complex systems in the real world.For example,complex network modeling and inference are widely used in fields such as computer science,genetics,social sciences,economics,and neuroscience,which has greatly promoted people’s interest in network modeling and statistical inference tools.A network is composed of a set of nodes(representing a set of entities)and a set of edges(representing interactions or connections between entities).Entities in modern systems often participate in multiple interactions,which enables multi-layer networks to better describe modern complex systems.Community structure is the most important feature of network structure,and like single-layer networks,detecting community structure is an important task in studying multi-layer networks.This article proposes methods and theories for joint community detection in multi-layer networks,which can be used to solve problems in real-world data.The random block model(SBM)is a traditional graph generative model.On the basis of SBM,the Random Effects Random Block Model(RESBM)is a flexible model framework for generating multi-layer networks.The idea of this model is to randomly generate the community structure of a multi-layer network through a hypothetical average community structure,so that certain layers of the generated multi-layer network have unique community structures.Accurately fitting the RESBM model in large-scale networks is an NP hard problem.Paul & Chen proposed variational expectation maximization algorithm(Var EM),two-step spectral clustering or matrix decomposition combined with maximum likelihood method to fit the RESBM model.The disadvantage of these two methods is that in complex model situations,the variable stage expectation maximization method has low computational efficiency when dealing with large-scale nodes,while the two-step method separates the characteristics of the model.In fact,the first step in the two-step method does not rely on the model being a completely non parametric step,and both methods lack theoretical guarantees.Therefore,this article proposes the RE-SL algorithm to fit the RESBM model,and the fitted model output is used to solve the community discovery problem in multi-layer networks.Based on the framework of split likelihood method,the RE-SL algorithm decouples the row and column community labels of the multi-layer network,replaces the original complete likelihood function with the split complete likelihood function,and gets the lower bound of the split likelihood function according to Jensen inequality scaling.The problem of maximizing the original likelihood function is transformed into the lower bound of the split likelihood function for optimization.The RE-SL algorithm adopts an alternating maximization method to calculate the maximum posterior expectation of the community allocation matrix given the current other community assignments and the current model parameters.In turn,the posterior expectation is used to update the model parameters,iterating repeatedly until convergence.This article demonstrates the superiority of the RE-SL algorithm from two aspects.On the one hand,the convergence theory of the RE-SL algorithm has been established based on the theoretical properties of the algorithm.At the same time,it indicates that if the initial label overlaps with the real community label,then the community label estimated by the RE-SL algorithm has strong consistency.On the other hand,in terms of algorithm performance,a large number of simulation experiments have shown that the RE-SL algorithm performs well in large-scale networks,and has advantages in community detection accuracy and computational efficiency.In addition,the application of the RE-SL algorithm to the study of resting state f MRI in schizophrenia demonstrates the characteristics of changes in the network community structure of patients with schizophrenia.