Alan Frieze Charalampos (Babis) E. Tsourakakis WAW 2012 22 June ‘12 WAW '121.

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Presentation transcript:

Alan Frieze Charalampos (Babis) E. Tsourakakis WAW June ‘12 WAW '121

 Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '122

3 Internet Map [lumeta.com] Food Web [Martinez ’91] Protein Interactions [genomebiology.com] Friendship Network [Moody ’01]

 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 modeling real-world planar graphs are Random Apollonian Networks. WAW '124

5 Apollonius ( BC ) Construct circles that are tangent to three given circles οn the plane.

WAW '126 Apollonian Gasket

WAW '127 Higher Dimensional (3d) Apollonian Packing. From now on, we shall discuss the 2d case.

 Dual version of Apollonian Packing WAW '128

 Start with a triangle (t=0).  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 WAW '129

10

 Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1211

WAW '1212

WAW '1213

DegreeTheoremSimulation WAW '1214

 Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1215

WAW '1216 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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WAW '1224 Broutin Devroye Large Deviations for the Weighted Height of an Extended Class of Trees. Algorithmica 2006

 This cannot be used though to get a lower bound: WAW '1225 Diameter=2, Depth arbitrarily large

 Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1226

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WAW '1228

 Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1229

WAW '1230 Chung LuVu Mihail Papadimitriou

WAW '12 31 S1S1 S2S2 S3S3 …. Star forest consisting of edges between S 1 and S 3 -S’ 3 where S’ 3 is the subset of vertices of S 3 with two or more neighbors in S 1.

WAW '1232 ….

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WAW '1234

 Introduction  Degree Distribution  Diameter  Highest Degrees  Eigenvalues  Open Problems WAW '1235

WAW '1236 Conductance Φ is at most t -1/2. Conjecture: Φ= Θ(t -1/2 ) Are RANs Hamiltonian? Conjecture: No Length of the longest path? Conjecture: Θ(n)

Thank you! WAW '1237