High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks
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Transcript of High Degree Vertices, Eigenvalues and Diameter of Random Apollonian Networks
Charalampos (Babis) E. Tsourakakis [email protected]
RSA 2011 24th May ‘11 RSA '11 1
Alan Frieze CMU
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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 modelling real-‐world planar graphs are Random Apollonian Networks.
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Apollonius (262-‐190 BC)
Construct circles that are tangent to three given circles οn the plane.
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Apollonian Gasket
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Higher Dimensional (3d) Apollonian Packing. From now on, we shall discuss the 2d case.
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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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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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:
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Diameter=2, Depth arbitrarily large
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Using the recurrence and simple algebraic manipulation:
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Supervertex A
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Chung Lu Vu
Mihail Papadimitriou
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S1 S2 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.
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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
Thanks a lot!
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