3 research outputs found
Consistent polynomial-time unseeded graph matching for Lipschitz graphons
We propose a consistent polynomial-time method for the unseeded node matching
problem for networks with smooth underlying structures. Despite widely
conjectured by the research community that the structured graph matching
problem to be significantly easier than its worst case counterpart, well-known
to be NP-hard, the statistical version of the problem has stood a challenge
that resisted any solution both provable and polynomial-time. The closest
existing work requires quasi-polynomial time. Our method is based on the latest
advances in graphon estimation techniques and analysis on the concentration of
empirical Wasserstein distances. Its core is a simple yet unconventional
sampling-and-matching scheme that reduces the problem from unseeded to seeded.
Our method allows flexible efficiencies, is convenient to analyze and
potentially can be extended to more general settings. Our work enables a rich
variety of subsequent estimations and inferences.Comment: 13 page
Matchability of heterogeneous networks pairs
We consider the problem of graph matchability in non-identically distributed
networks. In a general class of edge-independent networks, we demonstrate that
graph matchability can be lost with high probability when matching the networks
directly. We further demonstrate that under mild model assumptions,
matchability is almost perfectly recovered by centering the networks using
Universal Singular Value Thresholding before matching. These theoretical
results are then demonstrated in both real and synthetic simulation settings.
We also recover analogous core-matchability results in a very general core-junk
network model, wherein some vertices do not correspond between the graph pair.Comment: 44 pages, 10 figure
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