Estimation of Subgraph Densities in Noisy Networks
While it is common practice in applied network analysis to report various standard network summary statistics, these numbers are rarely accompanied by some quantification of uncertainty. Yet any error inherent in the measurements underlying the construction of the network, or in the network construction procedure itself, necessarily must propagate to any summary statistics reported. Here we study the problem of estimating the density of an arbitrary subgraph, given a noisy version of some underlying network as data. Under a simple model of network error, we show that consistent estimation of such densities is impossible when the rates of error are unknown and only a single network is observed. Next, focusing first on the problem of estimating the density of edges from noisy networks, as a canonical prototype of the more general problem, we develop method-of-moment estimators of network edge density and error rates for the case where a minimal number of network replicates are available. These estimators are shown to be asymptotically normal as the number of vertices increases to infinity. We also provide confidence intervals for quantifying the uncertainty in these estimates based on either the asymptotic normality or a bootstrap scheme. We then present a generalization of these results to higher-order subgraph densities, and illustrate with the case of two-star and triangle densities. Bootstrap confidence intervals for those high-order densities are constructed based on a new algorithm for generating a graph with pre-determined counts for edges, two-stars, and triangles. The algorithm is based on the idea of edge-rewiring, and is of some independent interest. We illustrate the use of the proposed methods in the context of gene coexpression networks.
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