Strict inequalities of critical values in continuum percolation

TL;DR

This paper investigates the supercritical continuum percolation model, establishing strict inequalities between critical probabilities and showing how modifications to the connection function affect the critical Poisson intensity, with implications for real networks.

Contribution

It proves strict inequalities between site and bond percolation thresholds and shows how changing the connection function influences the critical Poisson intensity.

Findings

Critical probabilities satisfy p_c^{site} > p_c^{bond}.

Reducing the connection function f increases the critical Poisson intensity.

Spreading transformations on f decrease the critical Poisson intensity.

Abstract

We consider the supercritical finite-range random connection model where the points $x,y$ of a homogeneous planar Poisson process are connected with probability $f(|y-x|)$ for a given $f$. Performing percolation on the resulting graph, we show that the critical probabilities for site and bond percolation satisfy the strict inequality $p_c^{\rm site} > p_c^{\rm bond}$. We also show that reducing the connection function $f$ strictly increases the critical Poisson intensity. Finally, we deduce that performing a spreading transformation on $f$ (thereby allowing connections over greater distances but with lower probabilities, leaving average degrees unchanged) {\em strictly} reduces the critical Poisson intensity. This is of practical relevance, indicating that in many real networks it is in principle possible to exploit the presence of spread-out, long range connections, to achieve connectivity at a strictly lower density value.