Community detection and percolation of information in a geometric setting
This work addresses foundational challenges in network science by extending community detection to geometric frameworks, offering incremental theoretical advances for researchers in machine learning and statistical physics.
The paper tackles the problem of generalizing stochastic block models to geometric settings by proposing a geometric random graph model where connection probabilities depend on distance, and it provides conditions for recovering vertex locations in sparse regimes and for information percolation in a branching random walk model.
We make the first steps towards generalizing the theory of stochastic block models, in the sparse regime, towards a model where the discrete community structure is replaced by an underlying geometry. We consider a geometric random graph over a homogeneous metric space where the probability of two vertices to be connected is an arbitrary function of the distance. We give sufficient conditions under which the locations can be recovered (up to an isomorphism of the space) in the sparse regime. Moreover, we define a geometric counterpart of the model of flow of information on trees, due to Mossel and Peres, in which one considers a branching random walk on a sphere and the goal is to recover the location of the root based on the locations of leaves. We give some sufficient conditions for percolation and for non-percolation of information in this model.