Date of Thesis

Spring 2025

Description

Analyzing dynamics that take place on graphs (networks) is fundamental to modern network theory, with applications spanning biology, social networks, and engineering systems. A significant body of research exists that considers the similarity and scaling of network topologies. However, understanding how dynamic behavior varies across different graph topologies - particularly how signals propagate and dynamic behaviors scale between small and large networks - remains a significant challenge.

To address this challenge, this thesis presents a comprehensive framework to analyze how dynamic signals propagate and scale on network topologies using temporal distance theory. Three research goals were accomplished. First, a statistical analysis of dynamic signal propagation patterns on the complete set of non-isomorphic graphs with 9 or less vertices (nodes) was performed. Using temporal distance theory to generate the data eliminated the need for over a hundred million simulations across 273,191 network topologies and three distinct dynamic models. Second, a novel algorithm was developed to quantitatively measure the similarity between dynamics taking place on different networks called TD-Match. TD-Match is robust to graph isomorphs and can be used as a look-up function for dynamically similar networks in complete sets of network topologies. By comparing the similarity of the dynamics on graphs, this methodology can reveal graph sets that contain "same dynamics, different graphs." Finally, this work developed a novel methodology for dimensionally scaling, or renormalizing, a network that focuses on the characteristics of the dynamics propagating on that network, independent of topology. As far as the author is aware, this is the first renormalization method that prioritizes dynamic signals on the network instead of network topology. A new tool called TD-MORPH was developed to explore TD-Match and dynamic renormalization.

The statistical analyses, TD-Match algorithm, and dynamic renormalization method offer a new framework for network analysis. This work provides a computationally efficient foundation for future studies seeking insight into the interplay between graph structures and network dynamics.

Keywords

graphs, complex networks, network similarity, network dynamics, temporal distance, graph renormalization

Access Type

Masters Thesis

Degree Type

Master of Science in Mechanical Engineering

Major

Mechanical Engineering

First Advisor

Andrew Sloboda

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