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Topological Data Analysis of Temporal Networks
Repository for my Master's thesis project. See the dissertation.
Abstract
Temporal networks are a mathematical model to represent interactions evolving over time. As such, they have a multitude of applications, from biology to physics to social networks. The study of dynamics on networks is an emerging field, with many challenges in modelling and data analysis.
An important issue is to uncover meaningful temporal structure in a network. We focus on the problem of periodicity detection in temporal networks, by partitioning the time range of the network and clustering the resulting subnetworks.
For this, we leverage methods from the field of topological data analysis and persistent homology. These methods have begun to be employed with static graphs in order to provide a summary of topological features, but applications to temporal networks have never been studied in detail.
We cluster temporal networks by computing the evolution of topological features over time. Applying persistent homology to temporal networks and comparing various approaches has never been done before, and we examine their performance side-by-side with a simple clustering algorithm. Using a generative model, we show that persistent homology is able to detect periodicity in the topological structure of a network.
We define two types of topological features, with and without aggregating the temporal networks, and multiple ways of embedding them in a feature space suitable for machine-learning applications. In particular, we examine the theoretical guarantees and empirical performance of kernels defined on topological features.
Topological insights prove to be useful in statistical learning applications. Combined with the recent advances in network science, they lead to a deeper understanding of the structure of temporal networks.