We analyze the asymptotic behavior of the degree sequence of Random Apollonian Networks maximal . For previous weaker results see comment,maximal .
Bollob ' a s, Riordan, Spencer and Tusn ' a dy @cite_4 proved rigorously the power law distribution of the Barab ' a si-Albert model @cite_0 . Random Apollonian Networks were introduced in @cite_1 . Their degree sequence was analyzed inaccurately in @cite_1 (see comment in @cite_10 ) and subsequently in @cite_10 using physicist's methodology. Cooper & Uehara @cite_3 and Gao @cite_7 analyzed the degree distribution of random @math trees, a closely related model to RANs. In RANs --in contrast to random @math trees-- the random @math clique chosen at each step has never previously been selected. For example in the two dimensional case any chosen triangular face is being subdivided into three new triangular faces by connecting the incoming vertex to the vertices of the boundary. Darrasse and Soria analyzed the degree distribution of random Apollonian network structures in @cite_8 .
@cite_4 @cite_7 @cite_8 @cite_1 @cite_3 @cite_0 @cite_10
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Random Apollonian networks have been recently introduced for representing real graphs. In this paper we study a modified version: random Apollonian network structures (RANS), which preserve the interesting properties of real graphs and can be handled with powerful tools of random generation. We exhibit a bijection between RANS and ternary trees, that transforms the degree of nodes in a RANS into the size of particular subtrees. The distribution of degrees in RANS can thus be analysed within a bivariate Boltzmann model for the generation of random trees, and we show that it has a Catalan form which reduces to a power law with an exponential cutoff: @math , with @math . We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity. Recently, Barabasi and Albert [2] suggested modeling complex real-world networks such as the worldwide web as follows: consider a random graph process in which vertices are added to the graph one at a time and joined to a fixed number of earlier vertices, selected with probabilities proportional to their degrees. In [2] and, with Jeong, in [3], Barabasi and Albert suggested that after many steps the proportion P(d) of vertices with degree d should obey a power law P(d)αd−γ. They obtained γ=2.9±0.1 by experiment and gave a simple heuristic argument suggesting that γ=3. Here we obtain P(d) asymptotically for all d≤n1 15, where n is the number of vertices, proving as a consequence that γ=3. © 2001 John Wiley & Sons, Inc. Random Struct. Alg., 18, 279–290, 2001 We study an evolving spatial network in which sequentially arriving vertices are joined to existing vertices at random according to a rule that combines preference according to degree with preference according to spatial proximity. We investigate phase transitions in graph structure as the relative weighting of these two components of the attachment rule is varied. Previous work of one of the authors showed that when the geometric component is weak, the limiting degree sequence of the resulting graph coincides with that of the standard Barab 'asi--Albert preferential attachment model. We show that at the other extreme, in the case of a sufficiently strong geometric component, the limiting degree sequence coincides with that of a purely geometric model, the on-line nearest-neighbour graph, which is of interest in its own right and for which we prove some extensions of known results. We also show the presence of an intermediate regime, in which the behaviour differs significantly from both the on-line nearest-neighbour graph and the Barab 'asi--Albert model; in this regime, we obtain a stretched exponential upper bound on the degree sequence. Our results lend some mathematical support to simulation studies of Manna and Sen, while proving that the power law to stretched exponential phase transition occurs at a different point from the one conjectured by those authors. 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 random Apollonian networks (RANs) 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 ln (3 2) approx_equal 0.74, which agree with the simulation results very well. We prove that the increasing tendency of average distance of RANs is a little slower than the logarithm of the number of nodes in RANs. Since most real-life networks are both scale-free and small-world networks, RANs may perform well in mimicking the reality. The RANs possess hierarchical structure as C(k) approx k sup -1 that are in accord with the observations of many real-life networks. In addition, we prove that RANs are maximal planar networks, which are of particular practicability for layout of printed circuits and so on. The percolation and epidemic spreading process are also studied and the comparisons between RANs and Barabasi-Albert (BA) as well as Newman-Watts (NW) networks are shown. We find that, when the network order N (the total number of nodes) is relatively small (as N approx 10 sup 4 ), the performance of RANs under intentional attackmore » is not sensitive to N, while that of BA networks is much affected by N. And the diseases spread slower in RANs than BA networks in the early stage of the suseptible-infected process, indicating that the large clustering coefficient may slow the spreading velocity, especially in the outbreaks.« less