How to compute the barycenter of a weighted graph

TL;DR

This paper introduces a stochastic algorithm based on noisy simulated annealing to compute the Frechet mean of weighted undirected graphs, addressing a key gap in graph analysis for data science.

Contribution

It presents a novel stochastic method for finding the barycenter of weighted graphs, leveraging homogenization and simulated annealing techniques.

Findings

Successfully computes barycenters for large weighted graphs

Demonstrates effectiveness on social network and citation network subgraphs

Provides a new tool for statistical analysis of complex graph data

Abstract

Discrete structures like graphs make it possible to naturally and flexibly model complex phenomena. Since graphs that represent various types of information are increasingly available today, their analysis has become a popular subject of research. The graphs studied in the field of data science at this time generally have a large number of nodes that are not fairly weighted and connected to each other, translating a structural specification of the data. Yet, even an algorithm for locating the average position in graphs is lacking although this knowledge would be of primary interest for statistical or representation problems. In this work, we develop a stochastic algorithm for finding the Frechet mean of weighted undirected metric graphs. This method relies on a noisy simulated annealing algorithm dealt with using homogenization. We then illustrate our algorithm with two examples (subgraphs of a social network and of a collaboration and citation network).