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
Recommended Citation
Hughes, Mark Anthony Hugo, "Automorphisms and Spectral Properties of Graphs with Applications to Classification and Community Detection for Complex Networks" (2023). FIU Electronic Theses and Dissertations. 5270.
https://digitalcommons.fiu.edu/etd/5270
Included in
Computational Chemistry Commons, Computational Neuroscience Commons, Other Applied Mathematics Commons, Theory and Algorithms Commons
Rights Statement
In Copyright. URI: http://rightsstatements.org/vocab/InC/1.0/
This Item is protected by copyright and/or related rights. You are free to use this Item in any way that is permitted by the copyright and related rights legislation that applies to your use. For other uses you need to obtain permission from the rights-holder(s).