Random Apollonian Networks@WAW12

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Alan Frieze af1p@random.math.cmu.edu Charalampos (Babis) E. Tsourakakis ctsourak@math.cmu.edu

WAW 2012 22 June ‘12 WAW '12

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Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

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Internet Map [lumeta.com]

Friendship Network [Moody ’01] WAW '12

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 modeling real-world planar graphs are Random Apollonian Networks. WAW '12

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

Apollonius (262-190 BC)

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Apollonian Gasket WAW '12

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Higher Dimensional (3d) Apollonian Packing. From now on, we shall discuss the 2d case. WAW '12

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ď‚Ą

Dual version of Apollonian Packing

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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

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For any đ?‘Ą ≼ 0 ď‚Ą Number of vertices nt =t+3 ď‚Ą Number of vertices mt=3t+3 ď‚Ą Number of faces Ft=2t+1 Note that a RAN is a maximal planar graph since for any planar graph đ?‘šđ?‘Ą ≤ 3đ?‘›đ?‘Ą − 6 = 3đ?‘Ą + 3 WAW '12

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Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

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ď‚Ą ď‚Ą

Let Nk(t)=E[Zk(t)]=expected #vertices of degree k at time t. Then: đ?‘ 3 đ?‘Ą + 1 = đ?‘ 3 đ?‘Ą

3đ?‘ 3 (đ?‘Ą) +1− 2đ?‘Ą+1 đ?‘˜ 1− + 2đ?‘Ą+1

đ?‘˜âˆ’1 2đ?‘Ą+1

đ?‘ đ?‘˜ đ?‘Ą + 1 = đ?‘ đ?‘˜ đ?‘Ą đ?‘ đ?‘˜âˆ’1 đ?‘Ą Solving the recurrence results in a power law with “slope 3â€?. ď‚Ą

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ď‚Ą ď‚Ą

Zk(t)=#of vertices of degree k at time t, đ?‘˜ ≼ 3 2 1 4 24 đ?‘?3 = , đ?‘?4 = , đ?‘?5 = , đ?‘?đ?‘˜ = đ?‘˜â‰Ľ6 5

5

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đ?‘˜(đ?‘˜+1)(đ?‘˜+2)

For t sufficiently large |đ??¸ đ?‘?đ?‘˜ đ?‘Ą − đ?‘?đ?‘˜ đ?‘Ą| ≤ 3.6 ď‚Ą Furthermore, for all possible degrees k Prob |Zk t − đ??¸ đ?‘?đ?‘˜ đ?‘Ą ≼ 10 đ?‘Ąđ?‘™đ?‘œđ?‘”(đ?‘Ą) = đ?‘œ(1) ď‚Ą

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Degree

Theorem

Simulation

3

0.4

0.3982

4

0.2

0.2017

5

0.1143

0.1143

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0.0714

0.0715

7

0.0476

0.0476

8

0.0333

0.0332

9

0.0242

0.0243

10

0.0182

0.0179

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0.0140

0.0137

12

0.0110

0.0111

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Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

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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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ď‚Ą

Note that if k* is the maximum depth of a face at time t, then diam(Gt)=O(k*).

ď‚Ą

Let Ft(k)=#faces of depth k at time t. Then, đ??¸ đ??šđ?‘Ą đ?‘˜ is equal to đ?‘˜

1≤đ?‘Ą1 <đ?‘Ą2 <..<đ?‘Ąđ?‘˜ ≤đ?‘Ą

đ?‘—=1

1 1 ≤ 2đ?‘Ąđ?‘— + 1 đ?‘˜!

đ?‘Ą

đ?‘—=1

đ?‘Ą

1 2đ?‘— + 1

đ?‘’đ?‘™đ?‘œđ?‘” đ?‘Ą ≤ 2đ?‘˜

đ?‘˜+1

Therefore by a first moment argument k*=O(log(t)) whp. WAW '12

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Large Deviations for the Weighted Height of an Extended Class of Trees. Algorithmica 2006 Broutin

Devroye

The depth of the random ternary tree T in probability is Ď /2 log(t) where 1/Ď =Ρ is the unique solution greater than 1 of the equation Ρ-1-log(Ρ)=log(3). Therefore we obtain an upper bound in probability đ?‘‘đ?‘–đ?‘Žđ?‘š đ??şđ?‘Ą ≤ đ?œŒlog(đ?‘Ą) WAW '12

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ď‚Ą

This cannot be used though to get a lower bound:

Diameter=2, Depth arbitrarily large WAW '12

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Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

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Let Δ1 ≼ Δ2 ≼ â‹Ż ≼ Δk be the k highest degrees of the RAN Gt where k=O(1). Also let f(t) be a function s.t. đ?‘“ đ?‘Ą + ∞. Then whp đ?‘Ąâ†’∞ đ?‘Ą ≤ Δ1 ≤ đ?‘Ąđ?‘“(đ?‘Ą) đ?‘“(đ?‘Ą) and for i=2,..,k đ?‘Ą đ?‘Ą ≤ Δđ?‘– ≤ Δđ?‘–−1 − đ?‘“(đ?‘Ą) đ?‘“(đ?‘Ą) WAW '12

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đ?‘Ą0 = log log(đ?‘“ đ?‘Ą )

đ?‘Ą1 = log(đ?‘“ đ?‘Ą )

• •

•

• WAW '12

đ?‘Ą

Break up time in periods Create appropriate supernodes according to their age. Let Xt be the degree of a supernode. Couple RAN process with a simpler process Y such that đ?‘‹đ?‘Ą ≼ đ?‘Œđ?‘Ą , đ?‘‹đ?‘Ą0 = đ?‘Œđ?‘Ą0 = đ?‘‘0 Upper bound the probability p*(r)=Pr đ?‘Œđ?‘Ą = đ?‘‘0 + đ?‘&#x; Union bound and k-th moment 28 arguments

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Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

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ď‚Ą

ď‚Ą

Let đ?œ†1 ≼ đ?œ†2 ≼ â‹Ż ≼ đ?œ†đ?‘˜ be the largest k eigenvalues of the adjacency matrix of Gt. Then đ?œ†đ?‘– = 1 Âą đ?‘œ 1 Δi whp. Proof comes for “freeâ€? from our previous theorem due to the work of two groups:

Chung

Lu

Vu Mihail

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Papadimitriou 30

đ?‘Ą1 = đ?‘Ą 1/8

đ?‘Ą0 = 0

đ?‘Ą2 = đ?‘Ą 9/16

S2

S1 ‌.

đ?‘Ą

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. WAW '12

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ď‚Ą ď‚Ą

Lemma: |� ′3 | ≤ � 1/6 This lemma allows us to prove that in F ‌.

‌.

‌.

Îťđ?‘– đ??š = 1 − đ?‘œ 1 WAW '12

Δi 32

Finally we prove that in H=G-F đ?œ†1 Η = o Îťk F Proof Sketch ď‚Ą First we prove a lemma. For any Îľ>0 and any f(t) s.t. đ?‘“ đ?‘Ą + ∞ the following holds đ?‘Ąâ†’∞

whp: for all s with đ?‘“ đ?‘Ą ≤ đ?‘ ≤ đ?‘Ą for all vertices đ?‘&#x; ≤ đ?‘ then đ?‘‘đ?‘ đ?‘&#x; ≤ đ?‘ WAW '12

1 đ?œ€+ 2

1 − 2

đ?‘&#x; .

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ď‚Ą

Consider six induced subgraphs Hi=H[Si] and Hij=H(Si,Sj). The following holds: đ?œ†1 đ??ť ≤

ď‚Ą

3

đ?‘–=1

đ?œ†1 đ??ťđ?‘– +

đ?œ†1 (đ??ťđ?‘– , đ??ťđ?‘— ) đ?‘–<đ?‘—

Bound each term in the summation using the lemma and the fact that the maximum eigenvalue is bounded by the maximum degree. WAW '12

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Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

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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) WAW '12

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Thank you!

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