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Spectral inference on multiple networks is a rapidly-developing subfield ofgraph statistics. Recent work has demonstrated that joint, or simultaneous,spectral embedding of multiple independent networks can deliver more accurateestimation than individual spectral decompositions of those same networks. Suchinference procedures typically rely heavily on independence assumptions acrossthe multiple network realizations, and even in this case, little attention hasbeen paid to the induced network correlation in such joint embeddings. Here, wepresent a generalized omnibus embedding methodology and provide a detailedanalysis of this embedding across both independent and correlated networks, thelatter of which significantly extends the reach of such procedures. We describehow this omnibus embedding can itself induce correlation, leading us todistinguish between inherent correlation -- the correlation that arisesnaturally in multisample network data -- and induced correlation, which is anartifice of the joint embedding methodology. We show that the generalizedomnibus embedding procedure is flexible and robust, and prove both consistencyand a central limit theorem for the embedded points. We examine how induced andinherent correlation can impact inference for network time series data, and weprovide network analogues of classical questions such as the effective samplesize for more generally correlated data. Further, we show how an appropriatelycalibrated generalized omnibus embedding can detect changes in real biologicalnetworks that previous embedding procedures could not discern, confirming thatthe effect of inherent and induced correlation can be subtle andtransformative, with import in theory and practice.

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