The geometry of the space of generalised networks

Speaker: Stephen Y Zhang
Affiliation: University of Melbourne

Abstract

Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world.  The “space of networks”, however, has a complex structure that cannot be adequately described using conventional statistical tools.

We introduce a measure-theoretic framework for generalised networks and propose a metric that extends the Gromov-Wasserstein distance between graphs. We characterise the geometry of this space, thereby providing a unified theoretical treatment of generalised networks that encompasses the cases of pairwise, as well as higher-order, relations. In particular, we show that our generalised network metric is equivalent to the Gromov-Wasserstein and co-optimal transport metrics when restricted to the subspaces of measure networks (pairwise relations) and measure hypernetworks (higher-order relations), respectively. We extend our analysis to the setting where vertices and hyperedges have additional label information, and derive efficient computational schemes to use in practice. Equipped with these theoretical and computational tools, we demonstrate the utility of our framework in a suite of applications, including hypergraph alignment, clustering and dictionary learning from ensemble data, multi-omics alignment, as well as multiscale network alignment.

Work together with Agnese Barbensi, Tom Needham, Bei Wang, Youjia Zhou, and Fangfei Lan