Degree distribution of random Apollonian network structures and Boltzmann sampling

Abstract

International audience 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: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity.

Characterizing several properties of high-dimensional random Apollonian networks

“…It was introduced in [17] that there exists an one-to-one relation between the evolution of HDRANs (of index k) and that of k-ary trees 4 . An illustrative example is presented in Figure 5.…”

Distances in random Apollonian network structures

“…The recursive definition of RANS shows a one-to-one correspondence with ternary trees. The degree distribution, which is a power law with an exponential cut-off, was studied in [1] by considering bivariate series marking the corresponding parameter in trees.…”

“…Proposition 2.1. [1] There is a bijection between random Apollonian network structures of order N and rooted plane ternary trees of size N (with N internal nodes).…”

“…The random Apollonian networks (RAN) proposed by [4] provides a very interesting model, with a power-law degree distribution, a mean distance of logarithmic order and a large clustering coefficient. We introduced in [1] a modified version, random Apollonian network structures (RANS), which preserves the interesting properties of real graphs concerning degree distribution (a power-law with an exponential cut-off) and large clustering. This paper is devoted to the analysis of distances in RANS, which is showed to be of square root order: we first characterize the distances from one special vertex to all the other vertices of the graph, and then work on the distances between pairs of vertices.…”

Boltzmann Samplers for Colored Combinatorial Objects

“…Efficient generation of extremely large objects is needed in many situations. For instance in statistical physics for observing limit behaviors [2], in biology for understanding and analysing genome properties [12], or in computer sciences for testing programs [6] or simulating and modelizing Network as Internet [7,3]. In 2003, Duchon, Flajolet, Louchard and Schaeffer [5] proposed a new model, called Boltzmann model, which leads to systematically construct samplers for random generation of objects in combinatorial constructible classes.…”