When does the mean network capture the topology of a sample of networks?
This work addresses a critical issue for researchers and practitioners in network-valued machine learning by informing metric choice for topology characterization, though it is incremental as it builds on existing Fréchet mean concepts.
The paper tackled the problem of ensuring that the Fréchet mean network accurately captures the topological structure of a training dataset of networks, showing that using the Hamming distance fails to recover correct partitions and edge density, while the effective resistance distance successfully does so.
The notion of Fréchet 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 Fréchet 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 Fréchet mean network is used to characterise the topology of networks for network-valued machine learning