###### Remco van der Hofstad

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-e^{-\lambda W_xW_y/|x-y|^{\alpha}}, where

(a) \lambda is the percolation parameter,

(b) |x-y| is the Euclidean distance between x and y, and

(c) \alpha is a long-range parameter.

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

The 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. We discuss open problems, inspired both by 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\neq y).

We also discuss relations to recent results on geometric inhomogeneous random graphs, which is a close of models alike scale-free percolation, but then on a torus with a finite number of nodes.

[This is joint work with Mia Deijfen and Gerard Hooghiemstra.]

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###### Remco van der Hofstad

****

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-e^{-\lambda W_xW_y/|x-y|^{\alpha}}, where

(a) \lambda is the percolation parameter,

(b) |x-y| is the Euclidean distance between x and y, and

(c) \alpha is a long-range parameter.

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

The 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. We discuss open problems, inspired both by 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\neq y).

We also discuss relations to recent results on geometric inhomogeneous random graphs, which is a close of models alike scale-free percolation, but then on a torus with a finite number of nodes.

[This is joint work with Mia Deijfen and Gerard Hooghiemstra.]