Skip to main content Accessibility help

Nonuniversality of weighted random graphs with infinite variance degree

  • Enrico Baroni (a1), Remco van der Hofstad (a1) and Júlia Komjáthy (a1)


We prove nonuniversality results for first-passage percolation on the configuration model with independent and identically distributed (i.i.d.) degrees having infinite variance. We focus on the weight of the optimal path between two uniform vertices. Depending on the properties of the weight distribution, we use an example-based approach and show that rather different behaviours are possible. When the weights are almost surely larger than a constant, the weight and number of edges in the graph grow proportionally to log log n, as for the graph distances. On the other hand, when the continuous-time branching process describing the first-passage percolation exploration through the graph reaches infinitely many vertices in finite time, the weight converges to the sum of two i.i.d. random variables representing the explosion times of the continuous-time processes started from the two sources. This nonuniversality is in sharp contrast to the setting where the degree sequence has a finite variance, Bhamidi et al. (2012).


Corresponding author

* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, PO Box 513, 5600 MB, Eindhoven,The Netherlands.
** Email address:


Hide All
[1] Albert, R. and Barabási, A.-L. (2002).Statistical mechanics of complex networks.Rev. Mod. Phys. 74,4797.
[2] Albert, R., Albert, I.and Nakarado, G. L. (2004).Structural vulnerability of the North American power grid.Phys. Rev. E 69, 125103.
[3] Amini, O., Devroye, L., Griffiths, S. and Olver, N. (2013).On explosions in heavy-tailed branching random walks.Ann. Prob. 41,18641899.
[4] Barabási, A. L., Réka, A. and Hawoong, J. (2000).Scale-free characteristics of random networks: the topology of the world-wide web.Physica A 281,6977.
[5] Baroni, E., van der Hofstad, R. and Komjáthy, J. (2015).Fixed speed competition on the configuration model with infinite variance degrees: unequal speeds.Electron. J. Prob. 20,148.
[6] 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.
[7] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2012).Universality for first passage percolation on sparse random graphs. To appear in Ann. Prob. Available at
[8] Bollobás, B. (1980).A probabilistic proof of an asymptotic formula for the number of labelled regular graphs.Europ. J. Combinatorics 1,311316.
[9] Bollobás, B. (2001). Random Graphs.Cambridge University Press.
[10] Bornholdt, S. and Schuster, H. G. (2003).Handbook of Graphs and Networks.John Wiley,New York.
[11] Chernoff, H. (1952).A measure of asymptotic efficiency for tests of a hypothesis based on the sum of observations.Ann. Math. Statist. 23,493507.
[12] Davies, P. L. (1978).The simple branching process: a note on convergence when the mean is infinite.J. Appl. Prob. 15,466480.
[13] Deijfen, M. and van der Hofstad, R. (2016).The winner takes it all.Ann. Appl. Prob. 26,24192453.
[14] Erdős, P. and Rényi, A. (1959).On random graphs I.Publ. Math. Debrecen 6,290297.
[15] Faloutsos, M., Faloutsos, P. and Faloutsos, C. (1999).On power-law relationships of the internet topology.Comp. Commun. Rev. 29,251262.
[16] Fountoulakis, N. (2007).Percolation on sparse random graphs with given degree sequence.Internet Math. 4,329356.
[17] Grey, D. R. (1974).Explosiveness of age-dependent branching processes.Z. Wahrscheinlichkeit. 28,129137.
[18] Harris, T. (1963).The Theory of Branching Processes.Springer,Berlin.
[19] Janson, S. (2009).On percolation in random graphs with given vertex degrees.Electron. J. Prob. 14,86118.
[20] Janson, S. and Luczak, M. J. (2009).A new approach to the giant component problem.Random Structures Algorithms 34,197216.
[21] Jasny, B. R., Zahn, L. M. and Marshall, E. (2009).Connections.Science 325, 405p.
[22] Kaluza, P., Kölzsch, A., Gastner, M. T. and Blasius, B. (2010).The complex network of global cargo ship movements.J. R. Soc. Interface. Available at
[23] Komjáthy, J. (2016).Explosive Crump–Mode–Jagers branching processes. Preprint. Available at
[24] Molloy, M. and Reed, B. (1995).A critical point for random graphs with a given degree sequence.Random Structures Algorithms 6,161180.
[25] Newman, M. E. J. (2003).The structure and function of complex networks.SIAM Rev. 45,167256.
[26] Radicchi, F., Fortunato, S. and Vespignani, A.(2012).Models of Science Dynamics. In Understanding Complex Systems, eds A. Scharnhost, K. Börner and P. van den Besselaar.Springer,Berlin, pp.233257.
[27] Van der Hofstad, R. (2016).Random Graphs and Complex Networks, Vol. 1.Cambridge University Press.
[28] Van der Hofstad, R. and Komjáthy, J. (2015).Fixed speed competition on the configuration model with infinite variance degrees: equal speeds.Electron. J. Prob. 20,148.
[29] Van der Hofstad, R. and Litvak, N. (2014).Degree-degree dependencies in random graphs with heavy-tailed degrees.Internet Math. 10,287334.
[30] Van der Hofstad, R., Hooghiemstra, G. and Znamenski, D. (2007).Distances in random graphs with finite mean and infinite variance degrees.Electron. J. Prob. 12,703766.
[31] Wu, J., Tse, C. K., Lau, F. C. M. and Ho, I. W. H. (2013).Analysis of communication network performance from a complex network perspective.IEEE T. Circuits Syst. 60,33033316.


MSC classification

Nonuniversality of weighted random graphs with infinite variance degree

  • Enrico Baroni (a1), Remco van der Hofstad (a1) and Júlia Komjáthy (a1)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed