When does the mean network capture the topology of a sample of networks?

TL;DR

This paper analyzes how different metrics affect the ability of the Fréchet mean to accurately represent the topology of networks, providing analytical insights for the stochastic blockmodel.

Contribution

It offers the first analytical estimates of the sample Fréchet mean for the stochastic blockmodel and compares the effects of Hamming and resistance distances on topology recovery.

Findings

Hamming distance fails to capture network topology in the mean.

Effective resistance distance accurately recovers network partitions.

Guides metric choice for topology-preserving network averaging.

Abstract

The notion of Fr\'echet mean (also known as "barycenter") network is the workhorse of most machine learning algorithms that require the estimation of a "location" parameter to analyse network-valued data. In this context, it is critical that the network barycenter inherits the topological structure of the networks in the training dataset. The metric - which measures the proximity between networks - controls the structural properties of the barycenter. This work is significant because it provides for the first time analytical estimates of the sample Fr\'echet mean for the stochastic blockmodel, which is at the cutting edge of rigorous probabilistic analysis of random networks. We show that the mean network computed with the Hamming distance is unable to capture the topology of the networks in the training sample, whereas the mean network computed using the effective resistance distance recovers the correct partitions and associated edge density. From a practical standpoint, our work informs the choice of metrics in the context where the sample Fr\'echet mean network is used to characterise the topology of networks for network-valued machine learning