Structures in supercritical scale-free percolation

Abstract

Scale-free percolation is a percolation model on ${\mathbb{Z}}^d$ which can be used to model real-world networks. We prove bounds for the graph distance in the regime where vertices have infinite degrees. We fully characterize transience vs. recurrence for dimension 1 and 2 and give sufficient conditions for transience in dimension 3 and higher. Finally, we show the existence of a hierarchical structure for parameters where vertices have degrees with infinite variance and obtain bounds on the cluster density.

Revised Definition 2.5 and an argument in Section 6, results are unchanged. Correction of minor typos. 29 pages, 7 figures

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• Recurrence and transience of symmetric random walks with long-range jumps
• Greedy Routing and the Algorithmic Small-World Phenomenom
• Arithmetic oscillations of the chemical distance in long-range percolation on ${\mathbb{Z}}^d$
• The random walk on the random connection model
• Stopping explosion by penalising transmission to hubs in scale-free spatial random graphs
• Scale-free percolation in continuum space: quenched degree and clustering coefficient
• Continuity of the critical value and a shape theorem for long-range percolation
• Inhomogeneous random graphs with infinite-mean fitness variables
• Cover and Hitting Times of Hyperbolic Random Graphs