# Higher-Order Graphon Theory: Fluctuations, Degeneracies, and Inference

arxiv（2024）

Abstract

Exchangeable random graphs, which include some of the most widely studied
network models, have emerged as the mainstay of statistical network analysis in
recent years. Graphons, which are the central objects in graph limit theory,
provide a natural way to sample exchangeable random graphs. It is well known
that network moments (motif/subgraph counts) identify a graphon (up to an
isomorphism), hence, understanding the sampling distribution of subgraph counts
in random graphs sampled from a graphon is pivotal for nonparametric network
inference. In this paper, we derive the joint asymptotic distribution of any
finite collection of network moments in random graphs sampled from a graphon,
that includes both the non-degenerate case (where the distribution is Gaussian)
as well as the degenerate case (where the distribution has both Gaussian or
non-Gaussian components). This provides the higher-order fluctuation theory for
subgraph counts in the graphon model. We also develop a novel multiplier
bootstrap for graphons that consistently approximates the limiting distribution
of the network moments (both in the Gaussian and non-Gaussian regimes). Using
this and a procedure for testing degeneracy, we construct joint confidence sets
for any finite collection of motif densities. This provides a general framework
for statistical inference based on network moments in the graphon model. To
illustrate the broad scope of our results we also consider the problem of
detecting global structure (that is, testing whether the graphon is a constant
function) based on small subgraphs. We propose a consistent test for this
problem, invoking celebrated results on quasi-random graphs, and derive its
limiting distribution both under the null and the alternative.

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