On Certain Properties of Random Apollonian Networks

Frieze, Alan; Tsourakakis, Charalampos E.

doi:10.1007/978-3-642-30541-2_8

Cited by 19 publications

(30 citation statements)

References 33 publications

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“…Their high-dimensional generalizations were also proposed in [31]. The properties of random Apollonian networks were studied in [32] in the context of web graphs.…”

Section: Apollonian Networkmentioning

confidence: 99%

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Generalized Open Quantum Walks on Apollonian Networks

et al. 2015

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“…Their high-dimensional generalizations were also proposed in [31]. The properties of random Apollonian networks were studied in [32] in the context of web graphs.…”

Section: Apollonian Networkmentioning

confidence: 99%

“…Their high-dimensional generalizations were also proposed in [31]. The properties of random Apollonian networks were studied in [32] in the context of web graphs.The construction of a regular Apollonian network can be done by a recursive procedure. At first, a complete 3-vertex graph is created, we call it the 0 th generation Apollonian network.…”

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confidence: 99%

Generalized Open Quantum Walks on Apollonian Networks

et al. 2015

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“…The average distance between two vertices in a typical RAN was shown to be logarithmic by Albenque and Marckert [8]. The degree distribution, k largest degrees and k largest eigenvalues (for fixed k) and the diameter were studied in Frieze and Tsourakakis [9]. We continue this line of research by studying the asymptotic properties of the longest (simple) paths and cycles in RANs and giving sharp estimates for the diameter of a typical RAN.…”

Section: Introductionmentioning

confidence: 87%

“…. , 9 , it must go through a v i . So P does not contain vertices from more than one triangle between two consecutive occurrences of a v i .…”

Section: Upper Bound For a Longest Pathmentioning

confidence: 99%

On longest paths and diameter in random apollonian networks

Ebrahimzadeh

Farczadi

Gao

et al. 2014

Random Struct Algorithms

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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

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“…and so the degree distribution is (This rule is closely related to the one governing the evolution of random Apollonian networks [32][33][34]. The difference is that the manifold is closed here.…”

Section: Local Propertiesmentioning

confidence: 99%

Complex network view of evolving manifolds

et al. 2018

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We study complex networks formed by triangulations and higher-dimensional simplicial complexes representing closed evolving manifolds. In particular, for triangulations, the set of possible transformations of these networks is restricted by the condition that at each step, all the faces must be triangles. Stochastic application of these operations leads to random networks with different architectures. We perform extensive numerical simulations and explore the geometries of growing and equilibrium complex networks generated by these transformations and their local structural properties. This characterization includes the Hausdorff and spectral dimensions of the resulting networks, their degree distributions, and various structural correlations. Our results reveal a rich zoo of architectures and geometries of these networks, some of which appear to be small worlds while others are finite dimensional with Hausdorff dimension equal or higher than the original dimensionality of their simplices. The range of spectral dimensions of the evolving triangulations turns out to be from about 1.4 to infinity. Our models include simplicial complexes representing manifolds with evolving topologies, for example, an h-holed torus with a progressively growing number of holes. This evolving graph demonstrates features of a small-world network and has a particularly heavy-tailed degree distribution.

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On Certain Properties of Random Apollonian Networks

Cited by 19 publications

References 33 publications

Generalized Open Quantum Walks on Apollonian Networks

Generalized Open Quantum Walks on Apollonian Networks

On longest paths and diameter in random apollonian networks

Complex network view of evolving manifolds

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