The Importance of Being Correlated: Implications of Dependence in Joint Spectral Inference across Multiple Networks

Spectral inference on multiple networks is a rapidly-developing subfield of graph statistics. Recent work has demonstrated that joint, or simultaneous, spectral embedding of multiple independent network realizations can deliver more accurate estimation than individual spectral decompositions of those same networks. Little attention has been paid, however, to the network correlation that such joint embedding procedures necessarily induce. In this paper, we present a detailed analysis of induced correlation in a {\em generalized omnibus} embedding for multiple networks. We show that our embedding procedure is flexible and robust, and, moreover, we prove a central limit theorem for this embedding and explicitly compute the limiting covariance. We examine how this covariance can impact inference in a network time series, and we construct an appropriately calibrated omnibus embedding that can detect changes in real biological networks that previous embedding procedures could not discern. Our analysis confirms that the effect of induced correlation can be both subtle and transformative, with import in theory and practice.