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Degrees and distances in random and evolving apollonian networks

Part of: Markov processes Operations research and management science Graph theory

Published online by Cambridge University Press:  19 September 2016

István Kolossváry ,
Júlia Komjáthy;  and
Lajos Vágó
István Kolossváry*
Affiliation:
Budapest University of Technology and Economics and Inter-University Centre for Telecommunications and Informatics
Júlia Komjáthy;*
Affiliation:
Budapest University of Technology and Economics, Inter-University Centre for Telecommunications and Informatics and Eindhoven University of Technology
Lajos Vágó*
Affiliation:
Budapest University of Technology and Economics and Inter-University Centre for Telecommunications and Informatics
*
* Postal address: Budapest University of Technology and Economics, Inter-University Centre for Telecommunications and Informatics, 4028 Debrecen, Kassai út 26, Hungary.
*** Postal address: Eindhoven University of Technology, PO Box 513, 5600 MB Eindhoven, The Netherlands. Email address: j.komjathy@tue.nl
* Postal address: Budapest University of Technology and Economics, Inter-University Centre for Telecommunications and Informatics, 4028 Debrecen, Kassai út 26, Hungary.
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Abstract

In this paper we study random Apollonian networks (RANs) and evolving Apollonian networks (EANs), in d dimensions for any d≥2, i.e. dynamically evolving random d-dimensional simplices, looked at as graphs inside an initial d-dimensional simplex. We determine the limiting degree distribution in RANs and show that it follows a power-law tail with exponent τ=(2d-1)/(d-1). We further show that the degree distribution in EANs converges to the same degree distribution if the simplex-occupation parameter in the nth step of the dynamics tends to 0 but is not summable in n. This result gives a rigorous proof for the conjecture of Zhang et al. (2006) that EANs tend to exhibit similar behaviour as RANs once the occupation parameter tends to 0. We also determine the asymptotic behaviour of the shortest paths in RANs and EANs for any d≥2. For RANs we show that the shortest path between two vertices chosen u.a.r. (typical distance), the flooding time of a vertex chosen uniformly at random, and the diameter of the graph after n steps all scale as a constant multiplied by log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar central limit theorem for typical distances in EANs.

Keywords

Random graphrandom networktypical distancediameterhopcountdegree distribution

MSC classification

Primary: 05C80: Random graphs
Secondary: 05C82: Small world graphs, complex networks 05C12: Distance in graphs 90B15: Network models, stochastic 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)
Type
Research Article
Information
Advances in Applied Probability , Volume 48 , Issue 3 , September 2016 , pp. 865 - 902
Copyright
Copyright © Applied Probability Trust 2016 

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