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Speeding up non-Markovian first-passage percolation with a few extra edges

  • Alexey Medvedev (a1) and Gábor Pete (a2)

Abstract

One model of real-life spreading processes is that of first-passage percolation (also called the SI model) on random graphs. Social interactions often follow bursty patterns, which are usually modelled with independent and identically distributed heavy-tailed passage times on edges. On the other hand, random graphs are often locally tree-like, and spreading on trees with leaves might be very slow due to bottleneck edges with huge passage times. Here we consider the SI model with passage times following a power-law distribution ℙ(ξ>t)∼t with infinite mean. For any finite connected graph G with a root s, we find the largest number of vertices κ(G,s) that are infected in finite expected time, and prove that for every k≤κ(G,s), the expected time to infect k vertices is at most O(k1/α). Then we show that adding a single edge from s to a random vertex in a random tree 𝒯 typically increases κ(𝒯,s) from a bounded variable to a fraction of the size of 𝒯, thus severely accelerating the process. We examine this acceleration effect on some natural models of random graphs: critical Galton--Watson trees conditioned to be large, uniform spanning trees of the complete graph, and on the largest cluster of near-critical Erdős‒Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

Copyright

Corresponding author

* Postal address: Namur Institute for Complex Networks (naXys), Université de Namur, Rempart de la Vierge, 8, Namur, 5000Belgium. Email address: an_medvedev@yahoo.com
** Postal address: Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15., Budapest, 1053, Hungary. Email address: robagetep@gmail.com

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