Gaussian level-set percolation on complex networks

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 treelike 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 nonbacktracking matrix satisfies λ_{max}(B)|_{h_{c}}=1. We present level-dependent distributions of local percolation probabilities for Erdős-Rényi 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↗0 for the system with power-law degree distribution, and generally also for negative values of h, if the multivariate Gaussian acquires a nonzero 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)≡1 the critical percolation threshold h_{c} decreases to 0, as the degree c of the random regular graph diverges. For K=(c)=1/c, however, the critical percolation threshold h_{c} is shown to diverge as c→∞.