Title: What is a degree distribution?

Abstract: The most studied aspect of statistical network models is their degree structure, reflecting the propensity of nodes within a network to form connections with other nodes. Yet many simple random graph models are understood only asymptotically; for finitely many nodes, they do not yield either closed-form statistical likelihoods or precise forward generating mechanisms. In contrast, we provide exact statistical results, limit theorems, and large-sample approximations that govern the behavior of networks based on random weights whose pairwise products parameterize independent Bernoulli trials. For power-law degree sequences we make the important observation that the frequently observed exponential cutoff behavior can be explained as an effect of this sampling. This enables us for the first time to understand, from a statistical perspective, the heterogeneity of network degrees observed in practice, and to explore how properties of degree sequences scale with network size. Our results thus provide new tools to quantify degree variation within and across network populations.