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
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Alan Frieze CMU
RSA '11 2
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RSA '11 3
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Protein Interactions [genomebiology.com]
Friendship Network [Moody ’01]
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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.
RSA '11 4
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RSA '11 5
Apollonius (262-‐190 BC)
Construct circles that are tangent to three given circles οn the plane.
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RSA '11 6
Apollonian Gasket
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RSA '11 7
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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RSA '11 10
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RSA '11 11
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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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RSA '11 14
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RSA '11 17
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RSA '11 18
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RSA '11 19
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Broutin Devroye
Large Deviations for the Weighted Height of an Extended Class of Trees. Algorithmica 2006
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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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RSA '11 25
Supervertex A
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RSA '11 30
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RSA '11 31
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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RSA '11 34
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RSA '11 35
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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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RSA '11 37
Alexis Darasse
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Thanks a lot!
RSA '11 38