Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T16:50:18.596Z Has data issue: false hasContentIssue false
  • English
  • Français

From trees to graphs: collapsing continuous-time branching processes

Part of: Probability theory on algebraic and topological structures Graph theory

Published online by Cambridge University Press:  16 November 2018

A. Garavaglia  and
R. van der Hofstad
A. Garavaglia*
Affiliation:
Eindhoven University of Technology
R. van der Hofstad*
Affiliation:
Eindhoven University of Technology
*
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands.
* Postal address: Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, 5600 MB, The Netherlands.
Get access Rights & Permissions [Opens in a new window]

Abstract

Continuous-time branching processes (CTBPs) are powerful tools in random graph theory, but are not appropriate to describe real-world networks since they produce trees rather than (multi)graphs. In this paper we analyze collapsed branching processes (CBPs), obtained by a collapsing procedure on CTBPs, in order to define multigraphs where vertices have fixed out-degree m≥2. A key example consists of preferential attachment models (PAMs), as well as generalized PAMs where vertices are chosen according to their degree and age. We identify the degree distribution of CBPs, showing that it is closely related to the limiting distribution of the CTBP before collapsing. In particular, this is the first time that CTBPs are used to investigate the degree distribution of PAMs beyond the tree setting.

Keywords

Branching processpreferential attachmentageing

MSC classification

Primary: 05C80: Random graphs
Secondary: 60B05: Probability measures on topological spaces 60B10: Convergence of probability measures
Type
Research Papers
Information
Journal of Applied Probability , Volume 55 , Issue 3 , September 2018 , pp. 900 - 919
Copyright
Copyright © Applied Probability Trust 2018 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Albert, R. and Barabási, A.-L. (2002). Statistical mechanics of complex networks. Rev. Modern Phys. 74, 47–97.Google Scholar
[2]Athreya, K. B. (2007). Preferential attachment random graphs with general weight function. Internet Math. 4, 401–418.Google Scholar
[3]Athreya, K. B. and Ney, P. E. (2004). Branching Processes. Dover, Mineola, NY.Google Scholar
[4]Athreya, K. B., Ghosh, A. P. and Sethuraman, S. (2008). Growth of preferential attachment random graphs via continuous-time branching processes. Proc. Indian Acad. Sci. Math. Sci. 118, 473–494.Google Scholar
[5]Berger, N., Borgs, C., Chayes, J. T. and Saberi, A. (2014). Asymptotic behavior and distributional limits of preferential attachment graphs. Ann. Prob. 42, 1–40.Google Scholar
[6]Bhamidi, S. (2007). Universal techniques to analyze preferential attachment trees: global and local analysis. Preprint. Available at http://www.unc.edu/~bhamidi/preferent.pdf.Google Scholar
[7]Bianconi, G. and Barabási, A.-L. (2001). Competition and multiscaling in evolving networks. Europhys. Lett. 54, 436–442.Google Scholar
[8]Bollobás, B. and Riordan, O. (2004). The diameter of a scale-free random graph. Combinatorica 24, 5–34.Google Scholar
[9]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.Google Scholar
[10]Borgs, C., Chayes, J., Daskalakis, C. and Roch, S. (2007). First to market is not everything: an analysis of preferential attachment with fitness. In STOC'07–Proceedings of the 39th Annual ACM Symposium on Theory of Computing, ACM, New York, pp. 135–144.Google Scholar
[11]Caravenna, F., Garavaglia, A. and van der Hofstad, R. (2016). Diameter in ultra-small scale-free random graphs. To appear in Random Structures Algorithms.Google Scholar
[12]Csárdi, G. (2006). Dynamics of citation networks. In ICANN 2006 (Lecture Notes Comput. Sci. Artificial Neural Networks 4131), Springer, Berlin, pp. 698–709.Google Scholar
[13]Deijfen, M., van den Esker, H., van der Hofstad, R. and Hooghiemstra, G. (2009). A preferential attachment model with random initial degrees. Ark. Mat. 47, 41–72.Google Scholar
[14]Garavaglia, A., van der Hofstad, R. and Woeginger, G. (2017). The dynamics of power laws: fitness and aging in preferential attachment trees. J. Statist. Phys. 168, 1137–1179.Google Scholar
[15]Geng, X. and Wang, Y. (2009). Degree correlations in citation networks model with aging. EPL 88, 38002.Google Scholar
[16]Hagberg, O. and Wiuf, C. (2006). Convergence properties of the degree distribution of some growing network models. Bull. Math. Biol. 68, 1275–1291.Google Scholar
[17]Hajra, K. B. and Sen, P. (2005). Aging in citation networks. Physica A 346, 44–48.Google Scholar
[18]Hajra, K. B. and Sen, P. (2006). Modelling aging characteristics in citation networks. Physica A 368, 575–582.Google Scholar
[19]Jagers, P. (1975). Branching Processes with Biological Applications. John Wiley, London.Google Scholar
[20]Jagers, P. and Nerman, O. (1984). The growth and composition of branching populations. Adv. Appl. Prob. 16, 221–259.Google Scholar
[21]Janson, S. (2005). Asymptotic degree distribution in random recursive trees. Random Structures Algorithms 26, 69–83.Google Scholar
[22]Jordan, J. (2006). The degree sequences and spectra of scale-free random graphs. Random Structures Algorithms 29, 226–242.Google Scholar
[23]Nerman, O. (1981). On the convergence of supercritical general (C-M-J) branching processes. Z. Wahrscheinlichkeitsth. 57, 365–395.Google Scholar
[24]Rudas, A., Tóth, B. and Valkó, B. (2007). Random trees and general branching processes. Random Structures Algorithms 31, 186–202.Google Scholar
[25]Szymański, J. (2005). Concentration of vertex degrees in a scale-free random graph process. Random Structures Algorithms 26, 224–236.Google Scholar
[26]Van der Hofstad, R. (2017). Random Graphs and Complex Networks, Vol. 1. Cambridge University Press.Google Scholar
[27]Van der Hofstad, R., Hooghiemstra, G. and Van Mieghem, P. (2002). On the covariance of the level sizes in random recursive trees. Random Structures Algorithms 20, 519–539.Google Scholar
[28]Wang, M., Yu, G. and Yu, D. (2009). Effect of the age of papers on the preferential attachment in citation networks. Physica A 388, 4273–4276.Google Scholar
[29]Wu, Y., Fu, T. Z. J. and Chiu, D. M. (2014). Generalized preferential attachment considering aging. J. Informetrics 8, 650–658.Google Scholar