Percolation phase transition in weight-dependent random connection models

• Published in: Advances in applied probability Vol. 53; no. 4; pp. 1090 - 1114
• Main Authors: Gracar, Peter, Lüchtrath, Lukas, Mörters, Peter
• Format: Journal Article
• Language: English
• Published: Cambridge, UK Cambridge University Press 01.12.2021; Applied Probability Trust
• Subjects: Apexes; Boolean; Graph theory; Graphs; Mathematical models; Original Article; Original Articles; Parameters; Percolation; Phase transitions; Poisson density functions; Probability; Statistical analysis; Boolean model; long-range percolation; scale-free percolation; random geometric graph; spatial network; robustness; 05C80; 60K35; Subcritical regime; age-based spatial preferential attachment
• ISSN: 0001-8678; 1475-6064
• DOI: 10.1017/apr.2021.13

We investigate spatial random graphs defined on the points of a Poisson process in d-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point is assigned an independent weight. Given the weight and position of the points, we form an edge between any pair of points independently with a probability depending on the two weights of the points and their distance. Preference is given to short edges and connections to vertices with large weights. We characterize the parameter regime where there is a non-trivial percolation phase transition and show that it depends not only on the power-law exponent of the degree distribution but also on a geometric model parameter. We apply this result to characterize robustness of age-based spatial preferential attachment networks.