ID: cond-mat/0503316

High Dimensional Apollonian Networks

March 13, 2005

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Degrees and distances in random and evolving Apollonian networks

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István Kolossváry, Júlia Komjáthy, Lajos Vágó
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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 i...

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Zhongzhi Zhang, Francesc Comellas, Guillaume Fertin, André Raspaud, ... , Zhou Shuigeng
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We present a family of networks, expanded deterministic Apollonian networks, which are a generalization of the Apollonian networks and are simultaneously scale-free, small-world, and highly clustered. We introduce a labeling of their vertices that allows to determine a shortest path routing between any two vertices of the network based only on the labels.

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In this work we analyze basic properties of Random Apollonian Networks \cite{zhang,zhou}, a popular stochastic model which generates planar graphs with power law properties. Specifically, let $k$ be a constant and $\Delta_1 \geq \Delta_2 \geq .. \geq \Delta_k$ be the degrees of the $k$ highest degree vertices. We prove that at time $t$, for any function $f$ with $f(t) \rightarrow +\infty$ as $t \rightarrow +\infty$, $\frac{t^{1/2}}{f(t)} \leq \Delta_1 \leq f(t)t^{1/2}$ and fo...

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Maximal planar networks with large clustering coefficient and power-law degree distribution

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In this article, we propose a simple rule that generates scale-free networks with very large clustering coefficient and very small average distance. These networks are called {\bf Random Apollonian Networks}(RAN) as they can be considered as a variation of Apollonian networks. We obtain the analytic results of power-law exponent $\gamma =3$ and clustering coefficient $C={46/3}-36\texttt{ln}{3/2}\approx 0.74$, which agree very well with the simulation results. We prove that th...

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In this paper we introduce a model of spatial network growth in which nodes are placed at randomly selected locations on a unit square in $\mathbb{R}^2$, forming new connections to old nodes subject to the constraint that edges do not cross. The resulting network has a power law degree distribution, high clustering and the small world property. We argue that these characteristics are a consequence of the two defining features of the network formation procedure; growth and pla...

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Deterministic Small-World Networks

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Francesc Comellas, Michael Sampels
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Many real life networks, such as the World Wide Web, transportation systems, biological or social networks, achieve both a strong local clustering (nodes have many mutual neighbors) and a small diameter (maximum distance between any two nodes). These networks have been characterized as small-world networks and modeled by the addition of randomness to regular structures. We show that small-world networks can be constructed in a deterministic way. This exact approach permits a ...

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The Degree Sequence of Random Apollonian Networks

June 10, 2011

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Charalampos E. Tsourakakis
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We analyze the asymptotic behavior of the degree sequence of Random Apollonian Networks \cite{maximal}. For previous weaker results see \cite{comment,maximal}.

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Long paths in random Apollonian networks

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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 $c<2/3$.

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Deterministic multidimensional growth model for small-world networks

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Aoyuan Peng, Lianming Zhang
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We proposed a deterministic multidimensional growth model for small-world networks. The model can characterize the distinguishing properties of many real-life networks with geometric space structure. Our results show the model possesses small-world effect: larger clustering coefficient and smaller characteristic path length. We also obtain some accurate results for its properties including degree distribution, clustering coefficient and network diameter and discuss them. It i...

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Comment on "Maximal planar networks with large clustering coefficient and power-law degree distribution"

September 7, 2005

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Zhi-Xi Wu, Xin-Jian Xu, Ying-Hai Wang
Statistical Mechanics

This Comment corrects the error which appeared in the calculation of the degree distribution of random apollonian networks. As a result, the expression of $P(k)$, which gives the probability that a randomly selected node has exactly $k$ edges, has the form $P(k)\propto 1/[k(k+1)(k+2)]$.

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