Community detection on euclidean random graphs

Abstract

We study the problem of community detection on Euclidean random geometric graphs where each vertex has two latent variables: a binary community label and a R d valued location label which forms the support of a Poisson point process of intensity λ. A random graph is then drawn with edge probabilities dependent on both the community and location labels. In contrast to the stochastic block model (SBM) that has no location labels, the resulting random graph contains many more short loops due to the geometric embedding. We consider the recovery of the community labels, partial and exact, using the random graph and the location labels. We establish phase transitions for both sparse and logarithmic degree regimes, and provide bounds on the location of the thresholds, conjectured to be tight in the case of exact recovery. We also show that the threshold of the distinguishability problem, i.e., the testing between our model and the null model without community labels exhibits no phase-transition and in particular, does not match the weak recovery threshold (in contrast to the SBM).

Learning random points from geometric graphs or orderings

“…The related problem trying to detect latent information on communities in a geometric framework was studied by [18]. In this case, points of a Poisson process in the unit square are equipped with an additional label indicating to which of two hidden communities they belong.…”

Community Detection on Euclidean Random Graphs

“…The proof of this follows from absolute continuity of infinite measures [41]. Refer to [38] for the details.…”

“…Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php process of T-Good cells to that of dependent site percolation on Z d to conclude that if a single cell is T-Good with sufficiently high probability, then there exists a giant T-Good component. Owing to space constraints, we only give the sketch here and refer to the full version [38] for details.…”

“…We skip the proof as it follows from straightforward arguments of independent thinning properties of the PPP. Refer to [38] for details. In what follows, we will need the following classical result on the ergodic property of marks of a stationary point process.…”

“…But since θ(H λ,f in (·)−f out (·),d ) = 0, there exists an almost-surely finite r such that V I (0) is contained within a ball of radius r, and hence the Information Flow from Infinity cannot be solved. The details can be found in [38].…”