Gaussian Level-Set Percolation on Complex Networks

Abstract

We present a solution of the problem of level-set percolation for multivariate Gaussians defined in terms of weighted graph Laplacians on complex networks. It is achieved using a cavity or message passing approach, which allows one to obtain the essential ingredient required for the solution, viz. a self-consistent determination of locally varying percolation probabilities. The cavity solution 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. The critical level $h_c$ of the percolation transition is obtained through the condition that the largest eigenvalue of a weighted version $B$ of a non-backtracking matrix satisfies $\lambda_{\rm max}(B)|_{h_c} =1$. We present level-dependent distributions of local percolation probabilities for Erd\H{o}s-R\'enyi networks and and for networks with degree distributions described by power laws. We find that there is a strong correlation between marginal single-node variances of a massless multivariate Gaussian and local percolation probabilities at a given level $h$, which is nearly perfect at negative values $h$, but weakens as $h\nearrow 0$ for the system with power law degree distribution, and generally also for negative values of $h$, if the multivariate Gaussian acquires a non-zero mass. The theoretical analysis simplifies in the case of random regular graphs with uniform edge-weights of the weighted graph Laplacian of the system and uniform mass parameter of the Gaussian field. An asymptotic analysis reveals that for edge-weights $K=K(c)\equiv 1$ the critical percolation threshold $h_c$ decreases to 0, as the degree $c$ of the random regular graph diverges. For $K=K(c)=1/c$, on the other hand, the critical percolation threshold $h_c$ is shown to diverge as $c\to\infty$.