High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks

Charalampos (Babis) E. Tsourakakis

ctsourak@math.cmu.edu

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Alan Frieze CMU RSA '11

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Internet Map [lumeta.com]

Friendship Network [Moody ’01] RSA '11

Food Web [Martinez ’91]

Protein Interactions [genomebiology.com] 3

 Modelling “real-­‐world” networks has

attracted a lot of attention. Common characteristics include:  Skewed degree distributions (e.g., power laws).  Large Clustering Coefficients  Small diameter

 A popular model for modelling real-­‐world

planar graphs are Random Apollonian Networks.

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Construct circles that are tangent to three given circles οn the plane.

Apollonius (262-­‐190 BC)

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Apollonian Gasket RSA '11

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Higher Dimensional (3d) Apollonian Packing. From now on, we shall discuss the 2d case.

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 Dual version of Apollonian Packing

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 Start with a triangle.  Until the network reaches the desired size  Pick a face F uniformly at random, insert a new

vertex in it and connect it with the three vertices of F

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

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Depth of a face (recursively): Let α be the initial face, then depth(α)=1. For a face β created by picking face γ depth(β) =depth(γ)+1. e.g.,

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

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Large Deviations for the Weighted Height of an Extended Class of Trees.

Algorithmica 2006 Broutin

Devroye

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 This cannot be used though to get a lower

bound:

Diameter=2, Depth arbitrarily large RSA '11

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

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

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 Using the recurrence and simple algebraic

manipulation:



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 Supervertex A

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

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

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

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

Chung

Lu

Vu Mihail

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Papadimitriou 31

S2

S1 ….

S3 ….

….

Star forest consisting of edges between S1 and S3-­‐S’3 where S’3 is the subset of vertices of S3 with two or more neighbors in S1. RSA '11

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

….

….

….

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

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

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 Diameter [Constant]  Smallest Separator: the Lipton-­‐Tarjan

theorem assures the existence of a good separator. Any smaller?  Robustness and Vulnerability: Random and malicious deletion of vertices, does the network remain connected?

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Alexis Darasse

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Thanks a lot!

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