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Studies Of Several Problems In Network Models

Posted on:2020-09-03Degree:DoctorType:Dissertation
Country:ChinaCandidate:LAALA ZEYNEBFull Text:PDF
GTID:1367330578452133Subject:Statistics
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Nowadays,network data are common in a wide variety of fields such as social sciences,biology,economics and computer science.It is of interest to study the generative mechanism of networks and to explore various properties of the network structure.A number of network models are proposed to understand the network features and fit network data.The non-standard structure of network data makes statistical inference difficult,especially in the asymptotic theory.This thesis mainly studies three issues in network models,which are stated below.We first study the asymptotic distribution of all linear combination of maximum likelihood estimators(MLEs)in the p0 model.The p0 model puts an exponential family distribution on directed graphs with a bi-degree sequence as the exclusively sufficient statistics.The uniform consistency of the MLE and the asymptotic normality for a fixed number of the MLEs have been derived in the p0 model.Built on the previous work.we further derive a central limit theorem for a linear combination of all the MLEs with an increasing dimension when the network edges take binary,continuous and discrete values.We illustrated our theoretical results by simulation studies.Second,we study the equivalent problem between the logistic-linear model and the implicit log-linear model.The logistic-linear model is actually the p0 model.We use the notation "logistic-linear" since it has a logistic-linear representation.The implicit log-linear model can be considered as a directed version of the expected degree model,in which the edge forming probability pij between vertices i and j is given as di+b+j/g++,where di=?j?i ai,j is the out-degree of vertex i and bj=?i?j ai,j is the in-degree of vertex j,and ?in=1 di=?jn=1 bj=g++.This model is implicit since its specified edge probabilities depend on the observed data.In the undirected case,Perry and Wolfe[102]demonstrated that the two models give rise to essentially the same likelihood-based estimates of link probabilities in the sparse finite-sample regimes.Here,we show that the asymptotic equivalent between the MLE in the logistic-linear model and an explicit estimator in the implicit log-linear model under some sparse conditions.Simulation studies and real data examples are conducted to demonstrate the theoretical results.Third,we study how to choose the number of the communities in the community detection.It is an important problem in network data analysis,which splits the graphs into communities.We use the eigenvalues of the Bethe Hessian matrix and the non-backtracking matrix of a graph to decide the number of communities.We propose a refined method that can estimate the number of communities better than other methods.Simulation studies and a real data application are provided to illustrate the comparison between the refined method and other methods.
Keywords/Search Tags:Random graph models, Exponential random graph model, Community detection, Central limit theorem, Directed networks, Maximum likelihood estimator
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