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

Part of: Operations research and management science Special processes Time-dependent statistical mechanics (dynamic and nonequilibrium) Graph theory

Published online by Cambridge University Press:  16 November 2018

Alexey Medvedev  and
Gábor Pete
Alexey Medvedev*
Affiliation:
Université de Namur and Université Catholique de Louvain
Gábor Pete*
Affiliation:
Alfréd Rényi Institute of Mathematics and Budapest University of Technology and Economics
*
* Postal address: Namur Institute for Complex Networks (naXys), Université de Namur, Rempart de la Vierge, 8, Namur, 5000Belgium. Email address: [email protected]
** Postal address: Alfréd Rényi Institute of Mathematics, Reáltanoda u. 13-15., Budapest, 1053, Hungary. Email address: [email protected]
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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-α 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.

Keywords

Temporal networknear-critical random graphGalton‒Watson treeErdős‒Rényi graphPólya urn processspreading phenomenaSI modelfirst-passage percolationbursty time seriesnon-Markovian process

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory 60K37: Processes in random environments 05C80: Random graphs
Secondary: 82C99: None of the above, but in this section 90B18: Communication networks
Type
Original Article
Information
Advances in Applied Probability , Volume 50 , Issue 3 , September 2018 , pp. 858 - 886
Copyright
Copyright © Applied Probability Trust 2018 

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