Document Type

Dissertation

Degree

Doctor of Philosophy (PhD)

Major/Program

Applied Mathematical Sciences

First Advisor's Name

Edgar J. Fuller

First Advisor's Committee Title

Committee chair

Second Advisor's Name

Ian Dryden

Second Advisor's Committee Title

committee member

Third Advisor's Name

Mirroslav Yotov

Third Advisor's Committee Title

committee member

Fourth Advisor's Name

Misak Sargsian

Fourth Advisor's Committee Title

committee member

Keywords

complex networks, diffusion, diffusion similarity, clustering, algorithm, diffusion on graphs, classification, molecular structure, connectomes, C.elegans

Date of Defense

6-6-2023

Abstract

Complex networks arise in a wealth of areas and their structure can yield important insights into the relationships between the objects that are connected within them. Applications to social networks, such as Facebook and Twitter, have been studied extensively as well as citation networks and customer behaviors. Other, sometimes more subtle, interactions exist, such as metabolic processes, evolutionary networks, and cellular communication networks. Configurations of objects in space, which includes chemical compounds, proteins, and polymers, can be modeled as knots (curves $K \in \mathbb{R}^3$) and in turn can be realized as graphs. These representations offer an opportunity to determine the mathematical properties of idealized versions of these objects that are related to real world properties.

The properties of such graphs can then, in some cases, be shown to correspond to the ability of the chemicals to function in a given way or the ways in which communities form within social networks. Another important application is to the study of networks of neurons formed in the central nervous systems of animals. These neural connectomes develop structures, including communities and layered networks that provide memory storage and response to stimuli. Herein the spectral and diffusion approaches are used to identify structures in these large-scale networks and present some results for the nematode \emph{C.elegans}.

The application of spectral tools to the normalized Laplacian of a number of synthetic graphs facilitates the consideration of the relationships between these matrices and their graph structure. Here, a tool that uses the automorphisms of graphs to help identify the eigenvalues of these automorphism compatible matrices is expanded on. Next, a different approach is implemented on graphs to identify community structure that focuses on the newly introduced diffusion and diffusion similarity matrices. Their use will provide results that support the diffusion matrix's applicability to a range of toy graph examples and large networks from biology and chemistry. Additional results are used to determine the time parameter of this methodology where the matrix begins to saturate a graph during the diffusion process. These tools are first applied to several synthetic graphs and then graphs derived from molecules and connectomes.

Identifier

FIDC011051

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