The Annals of Applied Probability

Degree distribution of shortest path trees and bias of network sampling algorithms

Shankar Bhamidi, Jesse Goodman, Remco van der Hofstad, and Júlia Komjáthy

Full-text: Open access


In this article, we explicitly derive the limiting degree distribution of the shortest path tree from a single source on various random network models with edge weights. We determine the asymptotics of the degree distribution for large degrees of this tree and compare it to the degree distribution of the original graph. We perform this analysis for the complete graph with edge weights that are powers of exponential random variables (weak disorder in the stochastic mean-field model of distance), as well as on the configuration model with edge-weights drawn according to any continuous distribution. In the latter, the focus is on settings where the degrees obey a power law, and we show that the shortest path tree again obeys a power law with the same degree power-law exponent. We also consider random $r$-regular graphs for large $r$, and show that the degree distribution of the shortest path tree is closely related to the shortest path tree for the stochastic mean-field model of distance. We use our results to shed light on an empirically observed bias in network sampling methods.

This is part of a general program initiated in previous works by Bhamidi, van der Hofstad and Hooghiemstra [ Ann. Appl. Probab. 20 (2010) 1907–1965], [ Combin. Probab. Comput. 20 (2011) 683–707], [ Adv. in Appl. Probab. 42 (2010) 706–738] of analyzing the effect of attaching random edge lengths on the geometry of random network models.

Article information

Ann. Appl. Probab., Volume 25, Number 4 (2015), 1780-1826.

Received: October 2013
Revised: April 2014
First available in Project Euclid: 21 May 2015

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60C05: Combinatorial probability 05C80: Random graphs [See also 60B20] 90B15: Network models, stochastic

Flows random graph random network first passage percolation hopcount Bellman–Harris processes stable-age distribution bias network algorithms power law mean-field model of distance weak disorder


Bhamidi, Shankar; Goodman, Jesse; van der Hofstad, Remco; Komjáthy, Júlia. Degree distribution of shortest path trees and bias of network sampling algorithms. Ann. Appl. Probab. 25 (2015), no. 4, 1780--1826. doi:10.1214/14-AAP1036.

Export citation


  • [1] Achlioptas, D., Clauset, A., Kempe, D. and Moore, C. (2005). On the bias of traceroute sampling. In STOC’05, May 22–24, 2005. ACM, Baltimore, MD.
  • [2] Aldous, D. (1992). Asymptotics in the random assignment problem. Probab. Theory Related Fields 93 507–534.
  • [3] Aldous, D. and Steele, J. M. (2004). The objective method: Probabilistic combinatorial optimization and local weak convergence. In Probability on Discrete Structures. Encyclopaedia Math. Sci. 110 1–72. Springer, Berlin.
  • [4] Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.
  • [5] Barabási, A.-L. and Albert, R. (1999). Emergence of scaling in random networks. Science 286 509–512.
  • [6] Bhamidi, S. and van der Hofstad, R. (2012). Weak disorder asymptotics in the stochastic mean-field model of distance. Ann. Appl. Probab. 22 29–69.
  • [7] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). Extreme value theory, Poisson–Dirichlet distributions, and first passage percolation on random networks. Adv. in Appl. Probab. 42 706–738.
  • [8] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2010). First passage percolation on random graphs with finite mean degrees. Ann. Appl. Probab. 20 1907–1965.
  • [9] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2011). First passage percolation on the Erdős–Rényi random graph. Combin. Probab. Comput. 20 683–707.
  • [10] Bhamidi, S., van der Hofstad, R. and Hooghiemstra, G. (2012). Universality for first passage percolation on sparse random graphs. Available at arXiv:1210.6839.
  • [11] Bollobás, B. (2001). Random Graphs, 2nd ed. Cambridge Studies in Advanced Mathematics 73. Cambridge Univ. Press, Cambridge.
  • [12] Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24 5–34.
  • [13] Bollobás, B., Riordan, O., Spencer, J. and Tusnády, G. (2001). The degree sequence of a scale-free random graph process. Random Structures Algorithms 18 279–290.
  • [14] Braunstein, L. A., Buldyrev, S. V., Cohen, R., Havlin, S. and Stanley, H. E. (2003). Optimal paths in disordered complex networks. Phys. Rev. Lett. 91 168701.
  • [15] Broido, A. and Claffy, K. (2001). Internet topology: Connectivity of IP graphs. In SPIE International Symposium on Convergence of IT and Communication, 172–187. Denver, CO.
  • [16] Chung, F. and Lu, L. (2006). Complex Graphs and Networks. CBMS Regional Conference Series in Mathematics 107. Published for the Conference Board of the Mathematical Sciences, Washington, DC.
  • [17] Dorogovtsev, S. N. and Mendes, J. F. F. (2003). Evolution of Networks. Oxford Univ. Press, Oxford.
  • [18] Durrett, R. (2007). Random Graph Dynamics. Cambridge Univ. Press, Cambridge.
  • [19] Eckhoff, M., Goodman, J., van der Hofstad, R. and Nardi, F. R. (2013). Short paths for first passage percolation on the complete graph. J. Stat. Phys. 151 1056–1088.
  • [20] Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events: For Insurance and Finance. Applications of Mathematics (New York) 33. Springer, Berlin.
  • [21] Faloutsos, M., Faloutsos, P. and Faloutsos, C. (1999). On power-law relationships of the internet topology. SIGCOMM Comput. Commun. Rev. 29 251–262.
  • [22] Govindan, R. and Tangmunarunkit, H. (2000). Heuristics for internet map discovery. In INFOCOM 2000. Nineteenth Annual Joint Conference of the IEEE Computer and Communications Societies. Proceedings. IEEE 3 1371–1380. IEEE, New York.
  • [23] Grey, D. R. (1973/74). Explosiveness of age-dependent branching processes. Z. Wahrsch. Verw. Gebiete 28 129–137.
  • [24] Jagers, P. (1975). Branching Processes with Biological Applications. Wiley, London.
  • [25] Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. in Appl. Probab. 16 221–259.
  • [26] Janson, S. (1999). One, two and three times $\log n/n$ for paths in a complete graph with random weights. Combin. Probab. Comput. 8 347–361.
  • [27] Janson, S. (2009). The probability that a random multigraph is simple. Combin. Probab. Comput. 18 205–225.
  • [28] Lakhina, A., Byers, J., Crovella, M. and Xie, P. (2003). Sampling biases in ip topology measurements. In INFOCOM. Twenty-Second Annual Joint Conference of the IEEE Computer and Communications. IEEE Societies 1 332–341. IEEE, New York.
  • [29] Levinson, N. (1960). Limiting theorems for age-dependent branching processes. Illinois J. Math. 4 100–118.
  • [30] Molloy, M. and Reed, B. (1995). A critical point for random graphs with a given degree sequence. In Proceedings of the Sixth International Seminar on Random Graphs and Probabilistic Methods in Combinatorics and Computer Science, “Random Graphs ’93 (Poznań, 1993) 6 161–179.
  • [31] Newman, M., Barabási, A.-L. and Watts, D. J., eds. (2006). The Structure and Dynamics of Networks. Princeton Univ. Press, Princeton, NJ.
  • [32] Newman, M. E. J. (2003). The structure and function of complex networks. SIAM Rev. 45 167–256 (electronic).
  • [33] Pansiot, J. J. and Grad, D. (1998). On routes and multicast trees in the internet. ACM SIGCOMM Computer Communication Review 28 41–50.
  • [34] Resnick, S. I. (1986). Point processes, regular variation and weak convergence. Adv. in Appl. Probab. 18 66–138.
  • [35] Salez, J. (2013). Joint distribution of distances in large random regular networks. J. Appl. Probab. 50 861–870.
  • [36] van der Hofstad, R. (2014). Random graphs and complex networks. Lecture notes in preparation. Available at
  • [37] Van Mieghem, P. (2009). Performance Analysis of Communications Networks and Systems. Cambridge Univ. Press, Cambridge.
  • [38] Wästlund, J. (2006). Random assignment and shortest path problems. In Fourth Colloquium on Mathematics and Computer Science Algorithms, Trees, Combinatorics and Probabilities. Discrete Math. Theor. Comput. Sci. Proc., AG 31–38. Assoc. Discrete Math. Theor. Comput. Sci., Nancy.