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

arXiv: https://arxiv.org/abs/2408.03461 · HTML: https://mlsanity.com/paper/2408.03461/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 13 of its 44 references that are in the map.
- First posted on arXiv 6 Aug 2024.

## Builds on (most cited first, 13 of 13)

- [Community Detection and Stochastic Block Models](https://mlsanity.com/paper/1703.10146/) (2017)
- [Graph-based Anomaly Detection and Description: A Survey](https://mlsanity.com/paper/1404.4679/) (2014)
- [Fr'echet Regression for Random Objects with Euclidean Predictors](https://mlsanity.com/paper/1608.03012/) (2016)
- [Stochastic blockmodel approximation of a graphon: Theory and consistent estimation](https://mlsanity.com/paper/1311.1731/) (2013)
- [Metrics for Graph Comparison: A Practitioner's Guide](https://mlsanity.com/paper/1904.07414/) (2019)
- [Gromov-Wasserstein Factorization Models for Graph Clustering](https://mlsanity.com/paper/1911.08530/) (2019)
- [Strong Consistency, Graph Laplacians, and the Stochastic Block Model](https://mlsanity.com/paper/2004.09780/) (2020)
- [Network histograms and universality of blockmodel approximation](https://mlsanity.com/paper/1312.5306/) (2013)
- [Hypothesis Testing For Network Data in Functional Neuroimaging](https://mlsanity.com/paper/1407.5525/) (2014)
- [Two-sample Hypothesis Testing for Inhomogeneous Random Graphs](https://mlsanity.com/paper/1707.00833/) (2017)
- [Modeling Network Populations via Graph Distances](https://mlsanity.com/paper/1904.07367/) (2019)
- [The Resistance Perturbation Distance: A Metric for the Analysis of Dynamic Networks](https://mlsanity.com/paper/1605.01091/) (2016)
- [Change Point Methods on a Sequence of Graphs](https://mlsanity.com/paper/1805.07113/) (2018)

## Built on by (most cited first, 1 of 1)

- [The Spectral Barycentre of a Set of Graphs with Community Structure](https://mlsanity.com/paper/2502.00038/) (2025)

## Abstract (arXiv)

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