On the Number of Edges of the Frechet Mean and Median Graphs

TL;DR

This paper investigates whether the structural property of edge density is preserved in the Frechet mean and median graphs derived from a sample, establishing that it is an hereditary property.

Contribution

It proves that edge density is inherited by the Frechet mean and median graphs, regardless of the estimation method used, providing a foundational theoretical insight.

Findings

Edge density is an hereditary property in graphs.

Frechet mean and median graphs inherit edge density.

The inheritance holds regardless of the estimation method.

Abstract

The availability of large datasets composed of graphs creates an unprecedented need to invent novel tools in statistical learning for graph-valued random variables. To characterize the average of a sample of graphs, one can compute the sample Frechet mean and median graphs. In this paper, we address the following foundational question: does a mean or median graph inherit the structural properties of the graphs in the sample? An important graph property is the edge density; we establish that edge density is an hereditary property, which can be transmitted from a graph sample to its sample Frechet mean or median graphs, irrespective of the method used to estimate the mean or the median. Because of the prominence of the Frechet mean in graph-valued machine learning, this novel theoretical result has some significant practical consequences.