Level-Set Percolation of Gaussian Random Fields on Complex Networks

We provide an explicit solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. The solution requires an analysis of the heterogeneous micro-structure of the percolation problem, i.e., a self-consistent determination of locally varying percolation probabilities. This is achieved using a cavity or message passing approach. It can be evaluated, both for single large instances of locally tree-like graphs, and in the thermodynamic limit of random graphs of finite mean degree in the configuration model class.