Advances in Applied Probability

Degrees and distances in random and evolving apollonian networks

István Kolossváry, Júlia Komjáthy, and Lajos Vágó

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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.

Article information

Adv. in Appl. Probab., Volume 48, Number 3 (2016), 865-902.

First available in Project Euclid: 19 September 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C80: Random graphs [See also 60B20]
Secondary: 05C82: Small world graphs, complex networks [See also 90Bxx, 91D30] 05C12: Distance in graphs 90B15: Network models, stochastic 60J80: Branching processes (Galton-Watson, birth-and-death, etc.)

Random graph random network typical distance diameter hopcount degree distribution


Kolossváry, István; Komjáthy, Júlia; Vágó, Lajos. Degrees and distances in random and evolving apollonian networks. Adv. in Appl. Probab. 48 (2016), no. 3, 865--902.

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