Recurrence versus Transience for Weight-Dependent Random Connection Models

Abstract

We investigate random graphs on the points of a Poisson process in $d$-dimensional space, which combine scale-free degree distributions and long-range effects. Every Poisson point carries an independent random mark and given marks and positions of the points we form an edge between two points independently with a probability depending via a kernel on the two marks and the distance of the points. Different kernels allow the mark to play different roles, like weight, radius or birth time of a vertex. The kernels depend on a parameter~$\gamma$, which determines the power-law exponent of the degree distributions. A further independent parameter $\delta$ characterises the decay of the connection probabilities of vertices as their distance increases. We prove transience of the infinite cluster in the entire supercritical phase in regimes given by the parameters $\gamma$ and~$\delta$, and complement these results by recurrence results if $d=2$. Our results are particularly interesting for the soft Boolean graph model discussed in the preprint [arXiv:2108:11252] and the age-dependent random connection model recently introduced by Gracar et al.\ [Queueing Syst. 93.3-4 (2019)]}