Long paths in random Apollonian networks

We consider the length $L(n)$ of the longest path in a randomly generated Apollonian Network (ApN) ${\cal A}_n$. We show that w.h.p. $L(n)\leq ne^{-\log^cn}$ for any constant $c<2/3$

The height of random kk-trees and related branching processes

We consider the height of random k-trees and k-Apollonian networks. These random graphs are not really trees, but instead have a tree-like structure. The height will be the maximum distance of a vertex from the root. We show that w.h.p. the height of random k-trees and k-Apollonian networks is asymptotic to clog t, where t is the number of vertices, and c=c(k) is given as the solution to a transcendental equation. The equations are slightly different for the two types of process. In the limit as k-->oo the height of both processes is asymptotic to log t/(k log 2)

Degrees and distances in random and evolving Apollonian networks

This paper studies Random and Evolving Apollonian networks (RANs and EANs), in d dimension for any d>=2, i.e. dynamically evolving random d dimensional simplices looked 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 tau=(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 n-th step of the dynamics is q_n->0 and sum_{n=0}^infty q_n =infty. This result gives a rigorous proof for the conjecture of Zhang et al. that EANs tend to show similar behavior as RANs once the occupation parameter q->0. We also determine the asymptotic behavior of shortest paths in RANs and EANs for arbitrary d dimensions. For RANs we show that the shortest path between two uniformly chosen vertices (typical distance), the flooding time of a uniformly picked vertex and the diameter of the graph after n steps all scale as constant times log n. We determine the constants for all three cases and prove a central limit theorem for the typical distances. We prove a similar CLT for typical distances in EANs