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. RSA '11
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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. RSA '11
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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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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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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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Using the recurrence and simple algebraic
manipulation:
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Supervertex A
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Chung
Lu
Vu Mihail
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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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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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