We investigate the asymptotic and statistical properties of latent position random graph models. The motivation for these models comes from network theory, particularly the study of social networks. In particular, we present latent position random graph models as a more realistic model for social networks than the well-studied Erdo&huml;s-Renyi random graphs, yet still a simple enough model to admit some rigorous asymptotic analysis.;In a latent position random graph, each vertex i is assigned a position ℓi in some (latent) space xi, and the probability of an edge between two vertices i and j depends on the distance between the latent positions ℓ i and ℓj. The positions ℓ 1,...,ℓn, may be fixed, or they may be drawn from a specific probability distribution. If we condition on the vertex positions ℓ1,...,ℓn, the edges are (conditionally) independent.;We investigate two latent position models in which the latent space xi is the k-torus Sk. (Nearly always, we set xi = S1, and embed the torus as the (glued) unit interval R/Z .) In the scaled uniform torus model, the vertex latent positions are drawn i.i.d. according to a uniform distribution on R/Z , and the probability of the edge ij is 1-2dℓiℓj sn for some specified function s. In the (unscaled) mixed-uniform torus model, the vertex latent positions are drawn i.i.d. according to a mixture of two uniform distributions whose supports partition R/Z , and the probability of the edge ij is 1 - 2d(ℓi, ℓ j).;We investigate several asymptotic properties of these random graphs as the number of vertices, n, increases to infinity. We prove central limit theorems for certain small subgraph counts, and determine thresholds for almost-sure connectivity. We then use these limiting theorems to construct power analyses of simple graph statistics (maximum degree and small subgraph counts) for a statistical inference problem of interest: distinguishing graphs which contain a localized subregion of excessive edge activity from those whose edges are more homogeneously distributed. Finally, we present examples of our analytic tools as applied to two kinds of social networks: one constructed from Science News articles (where an edge indicates that two articles discuss similar topics), and one constructed from email data gathered from Enron employees (where an edge indicates that two people communicated via email).
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