Estimation of Global Network Statistics from Incomplete Data [arxiv]

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Logline

Empirically studied network data is often incomplete due to hidden vertices or interactions. We present techniques for inferring global network statistics, including the degree distribution, from knowledge of the proportion of observed nodes or interactions. These techniques do not require us to make any assumptions on candidate network growth models.

Abstract

Complex networks underlie an enormous variety of social, biological, physical, and virtual systems. A profound complication for the science of complex networks is that in most cases, observing all nodes and all network interactions is impossible. Yet previous work addressing the impacts of partial network data is surprisingly limited, focuses primarily on missing nodes, and suggests that network statistics derived from subsampled data are not suitable estimators for the same network statistics describing the overall network topology. Our aim here is to generate scaling methods to predict true network statistics from only partial knowledge of nodes, links, or weights. We validate analytical results on four simulated network classes and empirical data sets of various sizes. We perform subsampling experiments by varying proportions of sampled data and demonstrate that our scaling methods can provide very good estimates of true network statistics while acknowledging limits. Lastly, we apply our techniques to a set of rich and evolving large-scale social networks, Twitter reply networks. Based on 100 million tweets, we use our scaling techniques to propose a statistical characterization of the Twitter Interactome from September 2008 to February 2009. Our treatment allows us to find support for Dunbar's hypothesis in detecting an upper threshold for the number of active social contacts that individuals maintain over the course of one week.

Highlights

• We present four sampling strategies: subgraphs induced on randomly selected nodes, subgraphs generated by failing links, subgraphs generated by randomly sampled links and (weighted) subgraphs generated by randomly sampled interactions.
• We demonstrate how global network statistics, such as the number of nodes, number of edges, maximum degree, average degree and degree distribution scale under these sampling regimes.
• We present scaling techniques for inferring global network statistics from sub-network data and validate our predicted network statistics with the true network statistics for our simulated and empirical datasets.
• Other researchers have noted that predictors for network statistics (sampled by links) have been elusive because of the difficulty in estimating the true degree distribution. We overcome this obstacle by developing a technique that predicts the degree distribution, free of any assumptions on the network model.
• We apply the techniques developed in this paper to Twitter reply networks. These networks are based on approximately 25% to 55% of all tweets authored between 9/9/08 and 11/17/08. As such, these networks a "incomplete" in that many interactions are unobserved. We present a characterization of the Twitter interactome for time period of this study and find evidence further supporting Dunbar's hypothesis.