On Inference for Modularity Statistics in Structured Networks

Abstract

This article revisits the classical concept of network modularity and its spectral relaxations used throughout graph data analysis. We formulate and study several modularity statistic variants for which we establish asymptotic distributional results in the large-network limit for networks exhibiting nodal community structure. Our work facilitates testing for network differences and can be used in conjunction with existing theoretical guarantees for stochastic blockmodel random graphs. Our results are enabled by recent advances in the study of low-rank truncations of large network adjacency matrices. We provide confirmatory simulation studies and real data analysis pertaining to the network neuroscience study of psychosis, specifically schizophrenia. Collectively, this article contributes to the limited existing literature to date on statistical inference for modularity-based network analysis. Supplemental materials for this article are available online.

KEYWORDS:

• Blockmodel
• Latent structure
• Modularity
• Network
• Random graph

Acknowledgments

The authors thank Nicholas Theis and the entire CONCEPT lab for real data expertise. This research uses data from the UK Biobank, a major biomedical database, obtained from the U.K. Biobank Resource under application number 68923 (PI: Konasale Prasad).

Disclosure Statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported in part by the University of Pittsburgh Center for Research Computing through the resources provided. Specifically, this work used the H2P cluster which is supported by NSF award number OAC-2117681. JC gratefully acknowledges support from the University of Wisconsin–Madison, Office of the Vice Chancellor for Research and Graduate Education, with funding from the Wisconsin Alumni Research Foundation.