Topological data analysis for investigation of dynamics and networks

Start: 11/09/2015 - 4:15pm
End  : 11/09/2015 - 5:15pm

Applied Math Seminar

Heather Harrington (Oxford University)


Persistent homology (PH) is a technique in topological data analysis that allows one to examine features in data across multiple scales in a robust and mathematically principled manner, and it is being applied to an increasingly diverse set of applications. We investigate applications of PH to dynamics and networks, focusing on two settings: dynamics {\em on} a network and dynamics {\em of} a network.

Dynamics on a network: a contagion spreading on a network is influenced by the spatial embeddedness of the network. In modern day, contagions can spread as a wave and through the appearance of new clusters via long-range edges, such as international flights. We study contagions by constructing Ňícontagion maps¹ that use multiple contagions on a network to map the nodes as a point cloud. By analyzing the topology, geometry, and dimensionality of manifold structure in such point clouds, we reveal insights to aid in the modelling, forecast, and control of spreading processes.

Dynamics of a network: one can construct static graph snapshots to represent a network that changes in time (e.g. edges are added/removed). We show that partitionings of a network of random-graph ensembles into snapshots using existing methods often lack meaningful temporal structure that corresponds to features of the underlying system. We apply persistent homology to track the topology of a network over time and distinguish important temporal features from trivial ones. We define two types of topological spaces derived from temporal networks and use persistent homology to generate a temporal profile for a network. We show that the methods we apply from computational topology can distinguish temporal distributions and  provide a high-level summary of temporal structure.

Together, these two investigations illustrate that persistent homology can be very illuminating in the study of networks and their applications.



Emmy Noether Room, Millikan 1021, Pomona College