Averages of Unlabeled Networks: Geometric Characterization and Asymptotic Behavior

TL;DR

This paper develops a geometric and probabilistic framework for analyzing averages of unlabeled, undirected networks with edge weights, addressing the challenges posed by the lack of vertex labels.

Contribution

It characterizes the space of unlabeled networks, explores its geometric properties, and establishes the asymptotic behavior of an empirical mean in this space.

Findings

The space of unlabeled networks has a complex quotient geometry.

Asymptotic properties of network averages are derived.

Implications for statistical analysis of network data are discussed.

Abstract

It is becoming increasingly common to see large collections of network data objects -- that is, data sets in which a network is viewed as a fundamental unit of observation. As a result, there is a pressing need to develop network-based analogues of even many of the most basic tools already standard for scalar and vector data. In this paper, our focus is on averages of unlabeled, undirected networks with edge weights. Specifically, we (i) characterize a certain notion of the space of all such networks, (ii) describe key topological and geometric properties of this space relevant to doing probability and statistics thereupon, and (iii) use these properties to establish the asymptotic behavior of a generalized notion of an empirical mean under sampling from a distribution supported on this space. Our results rely on a combination of tools from geometry, probability theory, and statistical shape analysis. In particular, the lack of vertex labeling necessitates working with a quotient space modding out permutations of labels. This results in a nontrivial geometry for the space of unlabeled networks, which in turn is found to have important implications on the types of probabilistic and statistical results that may be obtained and the techniques needed to obtain them.