The degree profile and weight in Apollonian networks and k-trees. (March 2016)

• Record Type: Journal Article
• Authors: Zhang, Panpan; Mahmoud, Hosam
• Is Part Of: Advances in applied probability. Volume 48:Number 1(2016)
• Page Start: 163
• Page End: 175
• Publication Date: 2016-03
• Subjects: Random structure, -- network, -- random graph, -- self-similarity, -- degree profile, -- phase transition, -- stochastic recurrence, -- Pólya urn, -- martingale; Primary 05082, -- 90B15, -- Secondary 60C05; Probabilities -- Periodicals; Stochastic models -- Periodicals; Electronic journals; Periodicals; 519.2
• DOI: 10.1017/apr.2015.11
• Languages: English
• ISSNs: 0001-8678

Abstract

Abstract: We investigate the degree profile and total weight in Apollonian networks. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi–Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k -trees, with minor modification in the proof methods, done mutatis mutandis.