Speeding up non-Markovian First Passage Percolation with a few extra edges

Alexey Medvedev, Gábor Pete

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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(k 1/α). Then we show that adding a single edge from s to a random vertex in a random tree typically increases κ(T,s) from a bounded variable to a fraction of the size of , 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 Erdos-Rényi graphs. In particular, at the upper end of the critical window, the process is already much faster than exactly at criticality.

Original languageEnglish
Pages (from-to)858-886
Number of pages29
JournalAdvances in Applied Probability
Issue number3
Early online date31 Aug 2017
Publication statusPublished - 1 Sept 2018


  • temporal networks
  • near-critical random graphs
  • Galton-Watson tree
  • Erdős-Rényi graph
  • Pólya urn process
  • First Passage Percolation
  • SI model
  • bursty time series
  • non-Markovian processes
  • non-Markovian process
  • Erdos-Rényi graph
  • spreading phenomena
  • Temporal network
  • first-passage percolation
  • near-critical random graph


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