On longest paths and diameter in random apollonian networks

Abstract

We consider the following iterative construction of a random planar triangulation.Start with a triangle embedded in the plane. In each step, choose a bounded face uniformly at random, add a vertex inside that face and join it to the vertices of the face. After n -3 steps, we obtain a random triangulated plane graph with n vertices, which is called a Random Apollonian Network (RAN). We show that asymptotically almost surely (a.a.s.) a longest path in a RAN has length o(n), refuting a conjecture of Frieze and Tsourakakis. We also show that a RAN always has a cycle (and thus a path) of length (2n − 5) log 2/ log 3 , and that the expected length of its longest cycles (and paths) is n 0.88 . Finally, we prove that a.a.s. the diameter of a RAN is asymptotic to c log n, where c ≈ 1.668 is the solution of an explicit equation.and we say a.a.s. X = o(f (n)) if for every fixed ε > 0, lim n→∞ P X ≤ εf (n) = 1.The authors of [9] conjecture in their concluding remarks that a.a.s. a RAN has a path of length (n). We refute this conjecture by showing the following theorem. Let L m be a random variable denoting the number of vertices in a longest path in a RAN with m faces. Theorem 1.1. A.a.s. we have L m = o(m).Recall that a RAN on n vertices has 2n − 5 faces, so Theorem 1.1 implies that a.a.s. a RAN does not have a path of length (n).We also prove lower bounds for L m deterministically, and its expected value in a RAN. In fact, we will prove lower bounds for C m and E [C m ], where C m denotes the number of Random Structures and Algorithms

The height of random k‐trees and related branching processes

“…Recently, and independently of this work results for random Apollonian networks were obtained by Ebrahimzadeh et al [9] and Kolossváry [16]. The work of [9] adapted the results of Broutin and Devroye [3] to derive the height of random Apollonian triangulations. The value of c = 0.8342... they obtained is the solution to (1), (2) with k = 3 and corresponds to a value of a = 2.0683.…”

“…For simplicity of notation, put α = (k − )/k and β = ( + 1)/k. Then iterating the main variable backwards in recurrences (7) - (9), and recalling that W i, (t) is non-decreasing in t gives…”

Long Paths in Random Apollonian Networks

“…They determined properties of its degree sequence, properties of the spectra of its adjacency matrix, and its diameter. Cooper and Frieze [2], Ebrahimzadeh, Farczadi, Gao, Mehrabian, Sato, Wormald and Zung [3] improved the diameter result and determine the diameter asymptotically. The paper [3] proves the following result concerning the length of the longest path in A n :…”

“…for some positive ε > 0. For lower bounds, [3] shows that L(n) ≥ n log 3 2 + 2 always and E(L(n)) = Ω(n 0.8 ). Chen and Yu [1] have proved an Ω(n log 3 2 ) lower bound for arbitrary 3-connected planar graphs.…”

Scalable Betweenness Centrality Maximization via Sampling

“…Also, when a new node t chooses its face (i, j, k) it is embedded in the barycenter of the corresponding triangle and connects to i, j, k via the straight lines: (i, t), (j, t), and (k, t). It has been shown that the diameter of a RAN is O(log(n)) with high probability [18,14].…”