Random Apollonian Networks@WAW12

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

WAW 2012 22 June ‘12 WAW '12

1

  

  

Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

WAW '12

2

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

4

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

Apollonius (262-190 BC)

WAW '12

5

Apollonian Gasket WAW '12

6

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

WAW '12

7

ď‚Ą

Dual version of Apollonian Packing

WAW '12

8

 

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 '12

9

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

10

  

  

Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

WAW '12

11

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

WAW '12

12

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

35

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

WAW '12

13

Degree

Theorem

Simulation

3

0.4

0.3982

4

0.2

0.2017

5

0.1143

0.1143

6

0.0714

0.0715

7

0.0476

0.0476

8

0.0333

0.0332

9

0.0242

0.0243

10

0.0182

0.0179

11

0.0140

0.0137

12

0.0110

0.0111

WAW '12

14

  

  

Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

WAW '12

15

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.,

WAW '12

16

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?‘˜

1â&#x2030;¤đ?&#x2018;Ą1 <đ?&#x2018;Ą2 <..<đ?&#x2018;Ąđ?&#x2018;&#x2DC; â&#x2030;¤đ?&#x2018;Ą

đ?&#x2018;&#x2014;=1

1 1 â&#x2030;¤ 2đ?&#x2018;Ąđ?&#x2018;&#x2014; + 1 đ?&#x2018;&#x2DC;!

đ?&#x2018;Ą

đ?&#x2018;&#x2014;=1

đ?&#x2018;Ą

1 2đ?&#x2018;&#x2014; + 1

đ?&#x2018;&#x2019;đ?&#x2018;&#x2122;đ?&#x2018;&#x153;đ?&#x2018;&#x201D; đ?&#x2018;Ą â&#x2030;¤ 2đ?&#x2018;&#x2DC;

đ?&#x2018;&#x2DC;+1

Therefore by a first moment argument k*=O(log(t)) whp.

WAW '12

17

WAW '12

18

WAW '12

19

WAW '12

20

WAW '12

21

WAW '12

22

WAW '12

23

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?&#x2018;&#x2018;đ?&#x2018;&#x2013;đ?&#x2018;&#x17D;đ?&#x2018;&#x161; đ??şđ?&#x2018;Ą â&#x2030;¤ đ?&#x153;&#x152;log(đ?&#x2018;Ą) WAW '12

24

ď&#x201A;Ą

This cannot be used though to get a lower bound:

Diameter=2, Depth arbitrarily large WAW '12

25

  

  

Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

WAW '12

26

Let Î&#x201D;1 â&#x2030;Ľ Î&#x201D;2 â&#x2030;Ľ â&#x2039;Ż â&#x2030;Ľ Î&#x201D;k be the k highest degrees of the RAN Gt where k=O(1). Also let f(t) be a function s.t. đ?&#x2018;&#x201C; đ?&#x2018;Ą + â&#x2C6;&#x17E;. Then whp đ?&#x2018;Ąâ&#x2020;&#x2019;â&#x2C6;&#x17E; đ?&#x2018;Ą â&#x2030;¤ Î&#x201D;1 â&#x2030;¤ đ?&#x2018;Ąđ?&#x2018;&#x201C;(đ?&#x2018;Ą) đ?&#x2018;&#x201C;(đ?&#x2018;Ą) and for i=2,..,k đ?&#x2018;Ą đ?&#x2018;Ą â&#x2030;¤ Î&#x201D;đ?&#x2018;&#x2013; â&#x2030;¤ Î&#x201D;đ?&#x2018;&#x2013;â&#x2C6;&#x2019;1 â&#x2C6;&#x2019; đ?&#x2018;&#x201C;(đ?&#x2018;Ą) đ?&#x2018;&#x201C;(đ?&#x2018;Ą) WAW '12

27

đ?&#x2018;Ą0 = log log(đ?&#x2018;&#x201C; đ?&#x2018;Ą )

đ?&#x2018;Ą1 = log(đ?&#x2018;&#x201C; đ?&#x2018;Ą )

â&#x20AC;˘ â&#x20AC;˘

â&#x20AC;˘

â&#x20AC;˘ WAW '12

đ?&#x2018;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?&#x2018;&#x2039;đ?&#x2018;Ą â&#x2030;Ľ đ?&#x2018;&#x152;đ?&#x2018;Ą , đ?&#x2018;&#x2039;đ?&#x2018;Ą0 = đ?&#x2018;&#x152;đ?&#x2018;Ą0 = đ?&#x2018;&#x2018;0 Upper bound the probability p*(r)=Pr đ?&#x2018;&#x152;đ?&#x2018;Ą = đ?&#x2018;&#x2018;0 + đ?&#x2018;&#x; Union bound and k-th moment 28 arguments

  

  

Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

WAW '12

29

ď&#x201A;Ą

ď&#x201A;Ą

Let đ?&#x153;&#x2020;1 â&#x2030;Ľ đ?&#x153;&#x2020;2 â&#x2030;Ľ â&#x2039;Ż â&#x2030;Ľ đ?&#x153;&#x2020;đ?&#x2018;&#x2DC; be the largest k eigenvalues of the adjacency matrix of Gt. Then đ?&#x153;&#x2020;đ?&#x2018;&#x2013; = 1 Âą đ?&#x2018;&#x153; 1 Î&#x201D;i whp.

Proof comes for â&#x20AC;&#x153;freeâ&#x20AC;? from our previous theorem due to the work of two groups:

Chung

Lu

Vu Mihail

WAW '12

Papadimitriou 30

đ?&#x2018;Ą1 = đ?&#x2018;Ą 1/8

đ?&#x2018;Ą0 = 0

đ?&#x2018;Ą2 = đ?&#x2018;Ą 9/16

S2

S1 â&#x20AC;Ś.

đ?&#x2018;Ą

S3 â&#x20AC;Ś.

â&#x20AC;Ś.

Star forest consisting of edges between S1 and S3-Sâ&#x20AC;&#x2122;3 where Sâ&#x20AC;&#x2122;3 is the subset of vertices of S3 with two or more neighbors in S1. WAW '12

31

ď&#x201A;Ą ď&#x201A;Ą

Lemma: |đ?&#x2018;&#x2020; â&#x20AC;˛3 | â&#x2030;¤ đ?&#x2018;Ą 1/6 This lemma allows us to prove that in F â&#x20AC;Ś.

â&#x20AC;Ś.

â&#x20AC;Ś.

Îťđ?&#x2018;&#x2013; đ??š = 1 â&#x2C6;&#x2019; đ?&#x2018;&#x153; 1 WAW '12

Î&#x201D;i 32

Finally we prove that in H=G-F đ?&#x153;&#x2020;1 Î&#x2014; = o Îťk F Proof Sketch ď&#x201A;Ą First we prove a lemma. For any Îľ>0 and any f(t) s.t. đ?&#x2018;&#x201C; đ?&#x2018;Ą + â&#x2C6;&#x17E; the following holds đ?&#x2018;Ąâ&#x2020;&#x2019;â&#x2C6;&#x17E;

whp: for all s with đ?&#x2018;&#x201C; đ?&#x2018;Ą â&#x2030;¤ đ?&#x2018; â&#x2030;¤ đ?&#x2018;Ą for all vertices đ?&#x2018;&#x; â&#x2030;¤ đ?&#x2018; then đ?&#x2018;&#x2018;đ?&#x2018; đ?&#x2018;&#x; â&#x2030;¤ đ?&#x2018; WAW '12

1 đ?&#x153;&#x20AC;+ 2

1 â&#x2C6;&#x2019; 2

đ?&#x2018;&#x; .

33

ď&#x201A;Ą

Consider six induced subgraphs Hi=H[Si] and Hij=H(Si,Sj). The following holds: đ?&#x153;&#x2020;1 đ??ť â&#x2030;¤

ď&#x201A;Ą

3

đ?&#x2018;&#x2013;=1

đ?&#x153;&#x2020;1 đ??ťđ?&#x2018;&#x2013; +

đ?&#x153;&#x2020;1 (đ??ťđ?&#x2018;&#x2013; , đ??ťđ?&#x2018;&#x2014; ) đ?&#x2018;&#x2013;<đ?&#x2018;&#x2014;

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

WAW '12

34

  

  

Introduction Degree Distribution Diameter Highest Degrees Eigenvalues Open Problems

WAW '12

35

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

36

Thank you!

WAW '12

37