Long Paths in Random Apollonian Networks

Cooper, Colin; Frieze, Alan

doi:10.1080/15427951.2014.925524

Cited by 6 publications

(6 citation statements)

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“…After t steps, we obtain a random triangulated plane graph with t + 3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ < 1, such that eventually every path in this graph has length less than t δ , which verifies a conjecture of Cooper and Frieze (2015). …”

supporting

confidence: 51%

“…Frieze and Tsourakakis [7] conjectured that there exists a fixed δ > 0 such that P(δt ≤ L t < t) → 1. Ebrahimzadeh et al [6] refuted this conjecture and showed that there exists a fixed δ > 0 such that P(L t < t/(log t) δ ) → 1. Cooper and Frieze [3] improved this result by showing that, for every constant c < 2 3 , we have P(L t ≤ t exp(− log c t)) → 1, and conjectured that there exists a fixed δ < 1 such that P(L t ≤ t δ ) → 1. The second main result of this paper is the following, which in particular confirms this conjecture.…”

Section: Theorem 1 There Exists a Fixed δ < 1 Such That S T < T δ Evmentioning

confidence: 99%

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Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

2016

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Let r and d be positive integers with r < d. Consider a random d-ary tree constructed as follows. Start with a single vertex, and in each time-step choose a uniformly random leaf and give it d newly created offspring. Let T d,t be the tree produced after t steps. We show that there exists a fixed δ < 1 depending on d and r such that almost surely for all large t, every r-ary subtree of T d,t has less than t δ vertices. The proof involves analysis that also yields a related result. 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. In this way, one face is destroyed and three new faces are created. After t steps, we obtain a random triangulated plane graph with t + 3 vertices, which is called a random Apollonian network. We prove that there exists a fixed δ < 1, such that eventually every path in this graph has length less than t δ , which verifies a conjecture of Cooper and Frieze (2015).

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supporting

confidence: 51%

Section: Theorem 1 There Exists a Fixed δ < 1 Such That S T < T δ Evmentioning

confidence: 99%

Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

2016

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“…Some aspects of Apollonian networks have been investigated in [4]- [6] and [9]. We were inspired by these very recent works to investigate additional properties.…”

Section: Scope and Resultsmentioning

confidence: 99%

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

Zhang

Mahmoud

2016

Adv. Appl. Probab.

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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.

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“…We also numerically study the Wiener index, and conjecture that its limiting distribution is not normal based on simulation results. The diameter of HDRANs is retrieved from [14].…”

Section: Discussionmentioning

confidence: 99%

“…The study of RAN first appeared in [55], where the power-law and the clustering coefficient were investigated. From then on, many more properties of RANs were uncovered by applied mathematicians and probablists: The degree distribution was characterized by [24]; the diameter was calculated by [21,24]; the length of the longest path in RANs was determined by [12,14,21]. All these resources, however, only focused on planar RANs, the evolution of which is based on continuing triangulation.…”

Section: Introductionmentioning

confidence: 99%

Characterizing several properties of high-dimensional random Apollonian networks

Zhang

2019

Preprint

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In this article, we investigate several properties of high-dimensional random Apollonian networks (HDRANs), including two types of degree profiles, the small-world effect (clustering property), sparsity, and several distance-based properties. The methods that we use to characterize the degree profiles are a twodimensional mathematical induction, analytic combinatorics, and Pólya urns, etc. The small-world property and sparsity are respectively measured by the local clustering coefficient and a proposed Gini index. Finally, we look into three distancebased properties, which are total depth, diameter and the Wiener index.

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Long Paths in Random Apollonian Networks

Abstract: We consider the length L(n) of the longest path in a randomly generated Apollonian Network (ApN) A n . We show that with high probability L(n) ≤ ne − log c n for any constant c < 2/3.

Cited by 6 publications

References 5 publications

Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

Longest paths in random Apollonian networks and largest r-ary subtrees of random d-ary recursive trees

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

Characterizing several properties of high-dimensional random Apollonian networks

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