Abstract

Co-authors: Mia Deijfen (Stockholm University), Gerard Hooghiemstra (Delft University of Technology)

We propose and study a random graph model on the hypercubic lattice that interpolates between models of scale-free random graphs and long-range percolation.

In our model, each vertex $x$ has a weight $W_x$, where the weights of different vertices are i.i.d. random variables. Given the weights, the edge between $x$ and $y$ is, independently of all other edges, occupied with probability $1-{mathrm{e}}^{{-lambda W_xW_y/|x-y|}{lpha}}$, where

(a) $lambda$ is the percolation parameter, (b) $|x-y|$ is the Euclidean distance between $x$ and $y$, and (c) $lpha$ is a long-range parameter.

The most interesting behavior can be observed when the random weights have a power-law distribution, i.e., when $mathbb{P}(W_x>w)$ is regularly varying with exponent $1- au$ for some $ au>1$. In this case, we see that the degrees are infinite a.s. when $gamma =lpha( au-1)/d leq 1$ or $lphaleq d$, while the degrees have a power-law distribution with exponent $gamma$ when $gamma>1$.

Our main results describe phase transitions in the positivity of the percolation critical value and in the graph distances in the percolation cluster as $gamma$ varies. Our results interpolate between those proved in inhomogeneous random graphs, where a wealth of further results is known, and those in long-range percolation. We also discuss many open problems, inspired both by recent work on long-range percolation (i.e., $W_x=1$ for every $x$), and on inhomogeneous random graphs (i.e., the model on the complete graph of size $n$ and where $|x-y|=n$ for every $x eq y$).