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 Wx, 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−λWxWy/|x−y|α, where (a) λ is the percolation parameter, (b) |x−y| is the Euclidean distance between x and y, and (c) α 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(Wx>w) is regularly varying with exponent 1−τ for some τ>1. In this case, we see that the degrees are infinite a.s.\ when γ=α(τ−1)/d≤1 or α≤d, while the degrees have a power-law distribution with exponent γ when γ>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 γ 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., Wx=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≠y).
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