Geodesic
Degree distribution of random Apollonian network structures and Boltzmann sampling
Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007).

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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: $α ^k k^{-3/2}$, with $α = 8/9$. We also show analogous distributions for the degree in RANS of higher dimension, related to trees of higher arity. 
@article{DMTCS_2007_special_253_a3,
     author = {Darrasse, Alexis and Soria, Mich\`ele},
     title = {Degree distribution of random {Apollonian} network structures and {Boltzmann} sampling},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07)},
     year = {2007},
     doi = {10.46298/dmtcs.3521},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3521/}
}
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Darrasse, Alexis; Soria, Michèle. Degree distribution of random Apollonian network structures and Boltzmann sampling. Discrete mathematics & theoretical computer science, DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07), DMTCS Proceedings vol. AH, 2007 Conference on Analysis of Algorithms (AofA 07) (2007). doi : 10.46298/dmtcs.3521. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.3521/

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