The degree profile and weight in Apollonian networks and k-trees

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 MittagLeffler-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.

On Several Properties of a Class of Preferential Attachment Trees—plane-Oriented Recursive Trees

“…Each index tends to capture certain features of the graphs, such as sparseness, regularity, and centrality. Examples of indices that have been introduced for random graphs include the Zagreb index [13], the Randíc index [16], the Wiener index [17,31], the Gini index [4,40], and a topological index measuring graph weight [41].…”

Maximum matchings and minimum dominating sets in Apollonian networks and extended Tower of Hanoi graphs

“…Their mean shortest distance of all pairs of vertices increases logarithmically with the number of vertices [39]. Thus, the Apollonian networks are good models mimicking real networks, and have received considerable attention from the scientific community [40,41,42].…”

Distributions in a class of Poissonized urns with an application to Apollonian networks

“…We insert a new vertex in the graph, connect it with k edges to the k vertices of the chosen clique, and we then discard the recruiting clique. For more motivational information about Apollonian networks, we refer the reader to Zhang and Mahmoud (2016).…”

“…Proposition 4.1 (Zhang and Mahmoud, 2016). Let X (k) n be the number terminal nodes in an Apollonian network of index k at age n. We then have…”