Simultaneous estimation of connectivity and dimensionality in samples of networks

TL;DR

This paper introduces a convex optimization-based method for jointly estimating connectivity matrices and their embedding dimensions in multiple networks, even with imperfect community detection, outperforming traditional methods in accuracy.

Contribution

It proposes a novel approach that simultaneously estimates network connectivity and dimensionality, handling heterogeneity and imperfect community recovery.

Findings

Method achieves accurate estimation under blockmodel assumptions.

Outperforms averaging-based methods in simulations.

Application reveals connectivity in brain networks may be low-rank.

Abstract

An overarching objective in contemporary statistical network analysis is extracting salient information from datasets consisting of multiple networks. To date, considerable attention has been devoted to node and network clustering, while comparatively less attention has been devoted to downstream connectivity estimation and parsimonious embedding dimension selection. Given a sample of potentially heterogeneous networks, this paper proposes a method to simultaneously estimate a latent matrix of connectivity probabilities and its embedding dimensionality or rank after first pre-estimating the number of communities and the node community memberships. The method is formulated as a convex optimization problem and solved using an alternating direction method of multipliers algorithm. We establish estimation error bounds under the Frobenius norm and nuclear norm for settings in which observable networks have blockmodel structure, even when node memberships are imperfectly recovered. When perfect membership recovery is possible and dimensionality is much smaller than the number of communities, the proposed method outperforms conventional averaging-based methods for estimating connectivity and dimensionality. Numerical studies empirically demonstrate the accuracy of our method across various scenarios. Additionally, analysis of a primate brain dataset demonstrates that posited connectivity is not necessarily full rank in practice, illustrating the need for flexible methodology.