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

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


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.


Corresponding author

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


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[1]Abraham, R., Delmas, J.-F. and Hoscheit, P. (2013). A note on the Gromov-Hausdorff-Prokhorov distance between (locally) compact metric measure spaces. Electron. J. Prob. 18, 14.
[2]Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2010). Critical random graphs: limiting constructions and distributional properties. Electron. J. Prob. 15, 741775.
[3]Addario-Berry, L., Broutin, N. and Goldschmidt, C. (2012). The continuum limit of critical random graphs. Prob. Theory Relat. Fields 152, 367406.
[4]Addario-Berry, L., Devroye, L. and Janson, S. (2013). Sub-Gaussian tail bounds for the width and height of conditioned Galton–Watson trees. Ann. Prob. 41, 10721087.
[5]Aldous, D. (1991). Asymptotic fringe distributions for general families of random trees. Ann. Appl. Prob. 1, 228266.
[6]Aldous, D. (1991). The continuum random tree. I. Ann. Prob. 19, 128.
[7]Aldous, D. (1991). The continuum random tree. II. An overview. In Stochastic Analysis, Cambridge University Press, pp. 2370.
[8]Aldous, D. (1993). The continuum random tree. III. Ann. Prob. 21, 248289.
[9]Aldous, D. (1997). Brownian excursions, critical random graphs and the multiplicative coalescent. Ann. Prob. 25, 812854.
[10]Aldous, D. and Lyons, R. (2007). Processes on unimodular random networks. Electron. J. Prob. 12, 14541508. (Errata: 22 (2017), 51.)
[11]Auffinger, A., Damron, M. and Hanson, J. (2017). 50 Years of First Passage Percolation. American Mathematical Society, Providence, RI.
[12]Baroni, E., van der Hofstad, R. and Komjáthy, J. (2017). Nonuniversality of weighted random graphs with infinite variance degree. J. Appl. Prob. 54, 146164.
[13]Benjamini, I. and Schramm, O. (2001). Recurrence of distributional limits of finite planar graphs. Electron. J. Prob. 6, 23.
[14]Benjamini, I., Kozma, G. and Wormald, N. (2014). The mixing time of the giant component of a random graph. Random Structures Algorithms 45, 383407.
[15]Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Prob. 20, 19071965.
[16]Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Prob. Comput. 20, 683707.
[17]Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2017). Universality for first passage percolation on sparse random graphs. Ann. Prob. 45, 25682630.
[18]Bhamidi, S., van der Hofstad, R. and Komjáthy, J. (2014). The front of the epidemic spread and first passage percolation. In Celebrating 50 Years of The Applied Probability Trust (J. Appl. Prob. Spec. Vol. 51A), pp. 101121.
[19]Bhowmick, A. K. et al. (2017). Temporal pattern of (re)tweets reveal cascade migration. In Proceedings of IEEE/ACM International Conference on Advances in Social Networks Analysis and Mining, ACM, New York, pp. 483488.
[20]Camarri, M. and Pitman, J. (2000). Limit distributions and random trees derived from the birthday problem with unequal probabilities. Electron. J. Prob. 5, 2.
[21]Ding, J., Kim, J. H., Lubetzky, E. and Peres, Y. (2011). Anatomy of a young giant component in the random graph. Random Structures Algorithms 39, 139178.
[22]Durrett, R. (2010). Probability: Theory and Examples, 4th edn. Cambridge University Press.
[23]Fill, J. A. and Pemantle, R. (1993). Percolation, first-passage percolation and covering times for Richardson's model on the n-cube. Ann. Appl. Prob. 3, 593629.
[24]Gandica, Y. et al. (2017). Stationarity of the inter-event power-law distributions. PLoS ONE 12, e0174509.
[25]Hammersley, J. M. and Welsh, D. J. A. (1965). First-passage percolation, subadditive processes, stochastic networks, and generalized renewal theory. In Proc. Internat. Res. Semin., Statist. Lab., Univ. California, Berkeley, Springer, New York, pp. 61110.
[26]Horváth, D. X. and Kertész, J. (2014). Spreading dynamics on networks: the role of burstiness, topology and non-stationarity. New J. Phys. 16, 073037.
[27]Iribarren, J. L. and Moro, E. (2009). Impact of human activity patterns on the dynamics of information diffusion. Phys. Rev. Lett. 103, 038702.
[28]Janson, S. (2012). Simply generated trees, conditioned Galton-Watson trees, random allocations and condensation. Prob. Surveys 9, 103252.
[29]Janson, S., Knuth, D. E., Łuczak, T. and Pittel, B. (1993). The birth of the giant component. Random Structures Algorithms 4, 231358.
[30]Jo, H.-H., Perotti, J. I., Kaski, K. and Kertész, J. (2014). Analytically solvable model of spreading dynamics with non-Poissonian processes. Phys. Rev. X 4, 011041.
[31]Karsai, M. et al. (2011). Small but slow world: how network topology and burstiness slow down spreading. Phys. Rev. E 83, 025102.
[32]Kersting, G. (2011). On the height profile of a conditioned Galton-Watson tree. Preprint. Available at
[33]Kesten, H. (1986). Subdiffusive behavior of random walk on a random cluster. Ann. Inst. H. Poincaré Prob. Statist. 22, 425487.
[34]Kesten, H. (1987). Percolation theory and first-passage percolation. Ann. Prob. 15, 12311271.
[35]Kesten, H., Ney, P. and Spitzer, F. (1966). The Galton-Watson process with mean one and finite variance. Theory Prob. Appl. 11, 513540.
[36]Kolchin, V. F. (1986). Random Mappings. Optimization Software, New York.
[37]Le Gall, J.-F. (2005). Random trees and applications. Prob. Surveys 2, 245311.
[38]Lindvall, T. (1999). On Strassen's theorem on stochastic domination. Electron. Commun. Prob. 4, 5159.
[39]Lyons, R. and Peres, Y. (2016). Probability on Trees and Networks. Cambridge University Press.
[40]Masuda, N. and Holme, P. (2013). Predicting and controlling infectious disease epidemics using temporal networks. F1000Prime Rep. 5, 6.
[41]Medvedev, A. and Pete, G. (2018). Speeding up non-Markovian first passage percolation with a few extra edges. Supplementary material. Available at
[42]Meir, A. and Moon, J. W. (1978). On the altitude of nodes in random trees. Canad. J. Math. 30, 9971015.
[43]Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45, 167256.
[44]Pastor-Satorras, R., Castellano, C., Van Mieghem, P. and Vespignani, A. (2015). Epidemic processes in complex networks. Rev. Mod. Phys. 87, 925979.
[45]Pemantle, R. (1990). A time-dependent version of Pólya's urn. J. Theoret. Prob. 3, 627637.
[46]Peres, Y. and Revelle, D. (2005). Scaling limits of the uniform spanning tree and loop-erased random walk on finite graphs. Preprint. Available at
[47]Pete, G. (2018). Probability and geometry on groups. In preparation. Available at
[48]Strassen, V. (1965). The existence of probability measures with given marginals. Ann. Math. Statist. 36, 423439.
[49]Stufler, B. (2016). Random enriched trees with applications to random graphs. Preprint. Available at
[50]Van der Hofstad, R. (2017). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press.
[51]Van der Hofstad, R. (2018). Random Graphs and Complex Networks, Vol. 2. In preparation. Available at
[52]Vespignani, A. (2012). Modelling dynamical processes in complex socio-technical systems. Nature Phys. 8, 3239.
[53]Wilson, D. B. (1996). Generating random spanning trees more quickly than the cover time. In Proceedings of the Twenty-Eighth Annual ACM Symposium on the Theory of Computing, ACM, New York, pp. 296303.
[54]Wormald, N. C. (1999). Models of random regular graphs. In Surveys in Combinatorics, Cambridge University Press, pp. 239298.
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