Torus Graph Inference for Detection of Localized Activity

Our goal is to detect localized regions of excessive activity in a network, distinguishing networks that contain such regions from networks whose activity is more homogeneous. We consider inference on random graphs from a latent position model whose latent position space is a torus, using edge density and maximum degree on as test statistics. 1 Torus Latent Position Models By now, it goes without saying that random graph models are a popular and fruitful area of study, with applications to physics, biology, sociology, and other fields. We consider a random graph model whose edge probabilities depend on the latent positions of the graph’s vertices; in our model, these latent positions lie on a torus. 1.1 Latent Position Models Latent position models, or latent space models, were introduced by Hoff, Raftery and Handcock [3]. In general, these random graph models propose a latent position (in some space of interest) for each vertex; the probability of the edge ij is a function of the distance between the latent positions of the vertices i and j. To generate a random graph from a latent position model, we use a two-stage process: First, latent positions, `1, . . . , `n, are drawn (i.i.d.) from a specified distribution on the latent position space. Given the latent positions, we generate a random graph G(`1, . . . , `n) by drawing ( n 2 ) Bernoulli random variables Y12, . . . , Yn−1,n for the edges. The edge variables are conditionally independent (conditioning on the positions `1, . . . , `n), with Yij having success probability pij , some function of the distance between the latent positions `i and `j . Formally, a latent (fixed) position model for graphs with n vertices is a sample space on Gn;. The collection is defined by a metric space (X , d) (the space in which the vertices take their latent positions) and a function f : R→ [0, 1] (used to convert distances between points into probabilities of vertex adjacency). Thus, we may call the model LPM(X , d, f). This model consists of 1Note that Hoff, Raftery, and Handcock do not require (X , d) to be a metric space; indeed, one of their examples makes use of non-symmetric projections. the samples spaces (Gn, P`), where ` : [n] → X assigns a latent position to each vertex. In a particular sample space (Gn, p`) ∈ LPM(X , d, f), the probability of a specific graph G is