Random Apollonian Networks httpwwwmathcmueductsourakranhtml Alan Frieze af1prandommathcmuedu Charalampos Babis E Tsourakakis ID: 568851
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On Certain Properties of Random Apollonian Networks http://www.math.cmu.edu/~ctsourak/ran.html
Alan Frieze af1p@random.math.cmu.edu Charalampos (Babis) E. Tsourakakis ctsourak@math.cmu.edu
WAW 2012
22 June ‘12
WAW '12
1Slide2
OutlineIntroductionDegree DistributionDiameter
Highest Degrees Eigenvalues Open Problems WAW '122Slide3
MotivationWAW '12
3
Internet Map [lumeta.com]
Food Web [Martinez ’91]
Protein Interactions [genomebiology.com]
Friendship Network [Moody ’01]Slide4
MotivationModelling “real-world” networks has attracted a lot of attention. Common characteristics include:Skewed degree distributions (e.g., power laws).
Large Clustering Coefficients Small diameterA popular model for modeling real-world planar graphs are Random Apollonian Networks.WAW '124Slide5
Problem of Apollonius
WAW '125Apollonius(262-190 BC)
Construct circles that are tangent to three given circles οn the plane. Slide6
Apollonian PackingWAW '12
6 Apollonian GasketSlide7
Higher Dimensional Packings
WAW '127Higher Dimensional (3d) Apollonian Packing. From now on, we shall discuss the 2d case.Slide8
Apollonian NetworkDual version of Apollonian Packing
WAW '128Slide9
Random Apollonian NetworksStart 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 FWAW '129Slide10
Random Apollonian NetworksFor any
Number of vertices nt =t+3Number of vertices mt=3t+3Number of faces Ft=2t+1Note that a RAN is a maximal planar graph since for any planar graph
WAW '12
10Slide11
OutlineIntroductionDegree DistributionDiameter
Highest Degrees Eigenvalues Open Problems WAW '1211Slide12
Degree Distribution
Let Nk(t)=E[Zk(t)]=expected #vertices of degree k at time t. Then:
Solving the recurrence results
in a power law with “slope 3”.
WAW '12
12Slide13
Degree Distribution
Zk(t)=#of vertices of degree k at time t,
For t sufficiently large
Furthermore, for all possible degrees k
WAW '12
13Slide14
Simulation (10000 vertices, results averaged over 10 runs, 10 smallest degrees shown)
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
14Slide15
OutlineIntroductionDegree DistributionDiameter
Highest Degrees Eigenvalues Open Problems WAW '1215Slide16
Diameter
WAW '1216Depth 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.,Slide17
DiameterWAW '12
17Note that if k* is the maximum depth of a face at time t, then diam(Gt)=O(k*).Let Ft(k)=#faces of depth k at time t. Then,
is equal to
Therefore by a first moment argument k*=O(log(t))
whp
.
Slide18
Bijection with random ternary treesWAW '12
18Slide19
Bijection with random ternary treesWAW '12
19Slide20
Bijection with random ternary treesWAW '12
20Slide21
Bijection with random ternary treesWAW '12
21Slide22
Bijection with random ternary treesWAW '12
22Slide23
Bijection with random ternary treesWAW '12
23Slide24
Diameter WAW '12
24Broutin
DevroyeLarge Deviations for the Weighted Height of an Extended Class of Trees.
Algorithmica 2006
The depth of the random ternary tree T in probability is ρ/2 log(t)
where 1/ρ=η is the unique solution greater than 1
o
f the equation
η-1-
log(
η)=
log(3).
Therefore we obtain an upper bound in probability
Slide25
Diameter This cannot be used though to get a lower bound:
WAW '1225Diameter=2,
Depth arbitrarily largeSlide26
OutlineIntroductionDegree DistributionDiameter
Highest Degrees Eigenvalues Open Problems WAW '1226Slide27
Highest Degrees, Main Result
Let
be the k highest degrees of the RAN Gt where k=O(1). Also let f(t) be a function s.t.
Then
whp
and for i=2,..,k
WAW '12
27Slide28
Proof techniques
WAW '1228Break 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
Upper
bound the probability
p
*(r)=
Union bound
and
k-
th
moment arguments
Slide29
OutlineIntroductionDegree DistributionDiameter
Highest Degrees Eigenvalues Open Problems WAW '1229Slide30
Eigenvalues, Main ResultLet
be the largest k eigenvalues of the adjacency matrix of G
t. Then
whp.
Proof comes for “free” from our previous theorem due to the work of two groups:
WAW '12
30
Chung
Lu
Vu
Mihail
PapadimitriouSlide31
Eigenvalues, Proof SketchWAW '12
31
S
1
S
2
S
3
….
….
….
Star forest consisting of edges between S
1
and S
3
-S’
3
where S’
3
is the subset of vertices of S
3
with two or more
neighbors in S
1
.Slide32
Eigenvalues, Proof SketchLemma:
This lemma allows us to prove that in F
WAW '12
32
….
….
….Slide33
Eigenvalues, Proof SketchFinally we prove that in H=G-F
Proof Sketch
First we prove a lemma. For any ε>0
and any f(t) s.t.
the following holds
whp: for all
s
with
for all vertices
then
WAW '12
33Slide34
Eigenvalues, Proof SketchConsider six induced
subgraphs Hi=H[Si] and Hij=H(Si,Sj). The following holds:
Bound each term in the summation using the lemma and the fact that the maximum eigenvalue is bounded by the maximum degree.
WAW '12
34Slide35
OutlineIntroductionDegree DistributionDiameter
Highest Degrees Eigenvalues Open Problems WAW '1235Slide36
Open Problems
WAW '1236Conductance 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) Slide37
Thank you!WAW '1237