2024/09/20 by Reimer Kühn, Kuehn, Reimer
Physics and Astronomy · Social Sciences · #Complex Network Analysis Techniques #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Human Mobility and Location-Based Analysis #Mathematical Physics (math-ph) #Opinion Dynamics and Social Influence #Statistical Mechanics (cond-mat.stat-mech)
paper · pdf · doi:10.48550/arxiv.2410.03681
openalex publication_date 2024/09/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 hc of the percolation transition is obtained through the condition that the largest eigenvalue of a weighted version B of a non-backtracking matrix satisfies λ\rm max(B)|hc =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\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)≡ 1 the critical percolation threshold hc 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 hc is shown to diverge as c→∞.