Applications of Hodge Theory on Networks (23w5116)


(University of Chicago)

David Rosenbluth (Lockheed Martin)

Anthea Monod (Imperial College London)


The Banff International Research Station will host the "Applications of Hodge Theory on Networks" workshop in Banff from January 29 to February 3, 2023.

The discrete Helmholtz Hodge decomposition (HHD) separates an edge flow on a graph into meaningful components. The interpretation of the graph and edge flow depend on the application area, which makes the technique flexible. That flexibility allows for cross-pollination between apparently disparate areas of study, an increasingly important strategy for solving the most pressing and challenging data science problems.

For example, the graph and edge flow may represent competitive interactions between agents, comparisons of objects, currency exchanges, odds of a directed transition in a random walk, or the energy used when a transition occurs in a microscopic physical system. When the edge flow represents the characteristics of a system, say the electorate’s aggregate preferences between pairs of candidates, then the HHD can be used to analyze the structure of that system.

In a social choice setting, this structure may explain which candidate is most preferred, and whether the election is prone to paradoxes of choice. When the edge flow represents the log-odds of one agent beating another agent in a competitive event, the HHD can be used to identify strong competitors, competitors who successfully exploit weaknesses of opponents, and rock-paper-scissor cycles.These properties can be used to predict selection dynamics amongst the agents, and to guide training protocols. Similarly, if the edge flow is related to transition probabilities, or energy expended during transitions, then the HHD may be used to analyze the dynamics of a random walk or molecular process.

We aim to unite the study of structure and dynamics by fostering conversations between researchers who use the HHD in areas including economics, game theory, statistical physics, and machine learning. We also aim to promote collaboration between industrial and academic teams using the HHD.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the U.S. National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnología (CONACYT).