Consistency of empirical distributions of sequences of graph statistics in networks with dependent edges
arxiv(2024)
摘要
One of the first steps in applications of statistical network analysis is
frequently to produce summary charts of important features of the network. Many
of these features take the form of sequences of graph statistics counting the
number of realized events in the network, examples of which include the degree
distribution, as well as the edgewise shared partner distribution, and more. We
provide conditions under which the empirical distributions of sequences of
graph statistics are consistent in the ℓ_∞-norm in settings where
edges in the network are dependent. We accomplish this by elaborating a weak
dependence condition which ensures that we can obtain exponential inequalities
which bound probabilities of deviations of graph statistics from the expected
value. We apply this concentration inequality to empirical distributions of
sequences of graph statistics and derive non-asymptotic bounds on the
ℓ_∞-error which hold with high probability. Our non-asymptotic
results are then extended to demonstrate uniform convergence almost surely in
selected examples. We illustrate theoretical results through examples,
simulation studies, and an application.
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