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On Modularity Asymptotics and Inference in Large Structured Networks and Random Matrix Analysis

Mitra, Anirban (2023) On Modularity Asymptotics and Inference in Large Structured Networks and Random Matrix Analysis. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Networks are used in multiple disciplines to study connectivity among its constituent members and can be modeled with the help of numerous random graph models, one of many being the Stochastic Blockmodel (SBM). Networks generated from such models often posses low rank latent structures and spectral methods are used to estimate and study said properties. On the other hand modularity functions are used as optimization criteria to detect clusters of network nodes. This work proposes variant formulations of modularity functions and builds on existing works to provide asymptotic normality under SBM networks. The limit theorems further facilitate study of network differences. Confirmatory simulations are shown under different SBM frameworks to support the theory. Extensive studies of different formulations for higher order approximations of eigen structures are shown which can be used to study special cases of modularity functions’ asymptotics. Strict bounds for functions of higher order products involving residual matrices of SBM networks are provided. The analyses of such higher order matrix products are also of separate interest and can be used independently for future research. In the process of analyzing third and fourth order products of the residual matrices, challenges in such random matrix analysis are identified. Simulation studies on random graphs generated from other, more general, low rank probabilistic models suggest that spectral clustering provide more accurate node-cluster estimates as opposed to using existing modularity functions. This further provides validation to study modularity functions formulated using the spectral analysis of networks. Real data on human brain networks are analyzed to study subject level modularity values. Projection of singular subspace is utilized to study network differences and analyze patient versus control network-clusters. These projections are used to compute distances among subjects’ brain networks and Multidimensional scaling along with Varimax rotation are utilized to map the samples of brain networks onto twodimensional scatter plots. It is found, in multiple cases, that brain networks of patients suffering from Schizophrenia are clustered together and towards the edge of the data cloud in some examples, thereby suggesting that network-based analysis of the human brain are effective to study Schizophrenia.


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Item Type: University of Pittsburgh ETD
Status: Unpublished
CreatorsEmailPitt UsernameORCID
Mitra, Anirbananm303@pitt.eduANM3030000-0001-5485-1108
ETD Committee:
TitleMemberEmail AddressPitt UsernameORCID
Committee ChairCape,
Committee MemberIyengar,
Committee MemberChen,
Committee MemberPrasad,
Date: 10 May 2023
Date Type: Publication
Defense Date: 2023
Approval Date: 10 May 2023
Submission Date: 22 March 2023
Access Restriction: 1 year -- Restrict access to University of Pittsburgh for a period of 1 year.
Number of Pages: 118
Institution: University of Pittsburgh
Schools and Programs: Dietrich School of Arts and Sciences > Statistics
Degree: PhD - Doctor of Philosophy
Thesis Type: Doctoral Dissertation
Refereed: Yes
Uncontrolled Keywords: Network, Spectral embedding, Modularity, Asymptotics, Latent Space Analysis.
Date Deposited: 10 May 2023 17:59
Last Modified: 10 May 2023 17:59


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