Multiplex Generalized Dot Product Graph networks: theory and applications

The University Of Central Florida Board Of Trustees, Orlando FL

Investigators

Abstract

Stochastic network models appear in a variety of applications, including genetics, proteomics, medical imaging, international relationships, brain science and many more. This research project examines collections of such networks, the so called multilayer network, where each of the individual networks (layers) have the broadest possible organization and yet possess some common features that allow meaningful stochastic inference. Examples include brain networks of different individuals, protein interaction networks, and trade networks between countries in various commodities. The current project presents an integral effort of merging applications and theory and will develop techniques that will be applicable for solution of a variety of real-life problems involving graph-structured data. Results of this research will be beneficial for many domains of knowledge that rely on analysis of stochastic networks where layers belong to different groups: a) brain science research by providing tools for analysis of brain networks and their variations under various conditions; b) medical research and practice by providing model-based explanations on what makes brain networks associated with particular diseases different from normal; c) molecular biology by developing techniques for analyzing the enzymatic influences between proteins related to various functions; d) finance and international relations by analyzing world’s trade and financial networks corresponding to various modalities; e) social sciences by analyzing the similarities and the differences in communities related to different types of social connections. In addition, the project will provide ample opportunities for training through various educational activities, including mentoring Ph.D., M.S. and undergraduate students, teaching a Special Topics graduate courses, organizing interdisciplinary seminars, and promoting interdisciplinary research and diversity. In more detail, the project will study the multiplex network model where all layers have the same set of nodes, and all the edges between nodes are drawn within layers, which is true in the applications discussed above. The research will be built on the notion that the matrices of probabilities of connections between nodes in layers of the network follow the most versatile Generalized Dot Product Graph (GDPG) model. GDPG includes all popular block network models as its particular cases. Although there have been some efforts to extend GDPGs to multilayer scenarios, the multilayer GDPG formulations have been limited to the very restrictive case where all networks are generated by the same invariant subspace. The latter is a direct extension of the SBM-equipped multiplex network where communities persist in all layers. The deficiency of the above formulation is that it prevents finding partitions of layers of the network into groups according to some natural condition. Hence, it is imperative to advance the multiplex GDPG to the case where groups of layers are embedded into different subspaces. Finding those clusters of layers will allow to provide model-based assessments of the differences between networks corresponding to different conditions. In addition, GDPG will be further generalized to the multiplex Signed GDPG (SGDPG) network setting, which allows more flexible modeling of a variety of real life networks. The objective of the project is to provide various extensions to the multilayer GDPG models, to develop scalable algorithms and theoretical tools for their analysis, and to apply those findings to analysis of brain networks. Furthermore, it aims to supplement statistical procedures with precision guarantees via oracle inequalities and minimax studies. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.