Barabási-Albert random graphs, scale-free distributions and bounds for approximation through ...

196 Pages·2010·1.33 MB·English

Barab´asi-Albert random graphs,
scale-free distributions
and bounds for approximation
through Stein’s method
Elizabeth Ford
University of Oxford
A thesis submitted for the degree
of Doctor of Philosophy.
Trinity Term 2009
Barab´asi-Albert random graphs, scale-free distributions and bounds for
approximation through Stein’s method
Elizabeth Ford, Jesus College
A thesis submitted for the degree of Doctor of Philosophy.
Trinity Term 2009
Abstract
Baraba´si-Albert random graph models are a class of evolving random graphs that are
frequently used to model social networks with scale-free degree distributions. It has been
shown that Baraba´si-Albert random graph models have asymptotic scale-free degree dis-
tributions as the size of the graph tends to infinity. Real world networks, however, have
finitesizesoitis importanttoknowhow closethedegreedistributionofaBaraba´si-Albert
random graph of a given size is to its asymptotic distribution.
Stein’smethodischosenasonemainmethodforobtainingexplicitboundsforthedistance
between distributions. We derive a new version of Stein’s method for a class of scale-free
distributions and apply the method to a Baraba´si-Albert random graph.
WecomparetheevolutionofasequenceofBaraba´si-Albertrandomgraphswithacontinuous-
time stochastic processes motivated by Yule’s model for evolution. Through a coupling
of the models we bound the total variation distance between their degree distributions.
Using these bounds,we extend degree distribution boundsthat we findfor specific models
within the scheme to find bounds for every member of the scheme.
We apply the Azuma-Hoeffding inequality and Chernoff bounds to find bounds between
the degree sequences of the random graph models and the given scale-free distribution.
These bounds prove that the degree sequences converge completely (and therefore also
converge almost surely) to our scale-free distribution.
We discuss the relationship between the random graph processes and the Chinese restau-
rant process. Aided by the construction of an inhomogeneous Markov chain, we apply
our results for the degree distribution in a Baraba´si-Albert random graph to a particular
statistic of the Chinese restaurant process.
Finally, we explore how our methods can be adapted and extended to other evolving
random graph processes. We study a Bernoulli evolving random graph process, for which
weboundthedistancebetween its degreedistributionandageometric distributionandwe
boundthedistancebetweenthenumberoftrianglesinthegraphandanormaldistribution.
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Acknowledgements
I am pleased to acknowledge the many sources of assistance and support that I have
received during the course of this work.
To my supervisor Prof. Gesine Reinert I express my sincere thanks for her guidance,
patience and enthusiastic encouragement.
I thank the Institute for Mathematical Sciences of the National University of Singapore
for enabling my participation in the Progress in Stein’s Method programme. I thank all
those with whom I have had useful discussions both on this programme and elsewhere.
My thanks go to Sophie Schbath and to the Statistics for Systems Biology research group
in Paris for inviting me to talk about my work as well as financing my visit.
Finally I thank the Engineering and Physical Sciences Research Council for its financial
support.
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Contents
1 Introduction 1
1.1 Social network models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Scale-free distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 Degree distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 The scale-free order (γ) . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.3 Log-log plots . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Outline of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Barab´asi-Albert models 10
2.1 A subclass of Baraba´si-Albert models . . . . . . . . . . . . . . . . . . . . . 12
2.2 This thesis in context: a discussion of previous results from the literature . 16
2.3 The approach in this thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3 Yule’s model for evolution and a general power law distribution 24
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
3.2 Power law distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.2.1 The Yule-Simon distribution . . . . . . . . . . . . . . . . . . . . . . 25
3.2.2 A more general power law distribution . . . . . . . . . . . . . . . . . 26
3.2.3 Distance from a power law . . . . . . . . . . . . . . . . . . . . . . . 28
3.2.4 The value of ρ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.3 Yule’s model for evolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.3.1 The requirements of Yule’s model for evolution . . . . . . . . . . . . 33
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3.3.2 The Yule process as a branching process . . . . . . . . . . . . . . . . 34
3.3.3 The Yule process as a pure birth process . . . . . . . . . . . . . . . . 35
3.3.4 The mathematical model . . . . . . . . . . . . . . . . . . . . . . . . 35
3.3.5 A class of continuous-time Baraba´si-Albert random graph models . . 36
3.4 A generalisation of Yule’s model for evolution . . . . . . . . . . . . . . . . . 38
3.5 Some new results at a sequence of stopping times . . . . . . . . . . . . . . . 39
3.6 Bounds for convergence in Yule’s model for evolution . . . . . . . . . . . . . 45
4 Stein’s method 53
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
4.1.1 The general idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.2 Stein’s method for discrete probability distributions . . . . . . . . . 55
4.2 Stein’s method for the geometric distribution . . . . . . . . . . . . . . . . . 58
4.3 Stein’s method for the normal distribution . . . . . . . . . . . . . . . . . . . 61
5 Stein’s method for the µα-distribution 62
ρ
5.1 A Stein equation for the µα-distribution . . . . . . . . . . . . . . . . . . . . 62
ρ
5.2 Stein factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
5.2.1 Stein factors for Kolmogorov distance approximation . . . . . . . . . 67
5.2.2 Stein factors for total variation distance approximation . . . . . . . 70
5.2.3 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Coupling the degree evolution in the discrete-time and continuous-time
models 82
6.1 Baraba´si-Albert models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
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6.1.1 Model BA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.1.2 Model I: more independence . . . . . . . . . . . . . . . . . . . . . . . 84
6.1.3 Model I: starting with one vertex . . . . . . . . . . . . . . . . . . . . 86
6.1.4 Model Y: motivated by Yule’s model for evolution . . . . . . . . . . 88
6.2 Coupling Model Y and Model I . . . . . . . . . . . . . . . . . . . . . . . . 89
7 Bounds for convergence using negative binomial approximation 104
8 Bounds for convergence using recurrence equations 115
8.1 The recurrence equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
8.2 Bounding the error term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9 Convergence of the degree sequence 121
9.1 The Azuma-Hoeffding inequality and Model BA . . . . . . . . . . . . . . . 121
9.2 Chernoff bounds and models with independent vertices . . . . . . . . . . . . 127
10 A connection with the Chinese restaurant process 131
10.1 An inhomogeneous Markov chain . . . . . . . . . . . . . . . . . . . . . . . . 132
10.1.1 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.1.2 The Markov chain . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
10.2 The Chinese restaurant process . . . . . . . . . . . . . . . . . . . . . . . . . 135
10.2.1 Simple random seating plan . . . . . . . . . . . . . . . . . . . . . . . 135
10.2.2 Random seating plan with departure . . . . . . . . . . . . . . . . . . 136
10.3 Associated homogeneous Markov chains . . . . . . . . . . . . . . . . . . . . 141
10.3.1 The stationary distribution . . . . . . . . . . . . . . . . . . . . . . . 143
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10.3.2 Rate of convergence to stationarity . . . . . . . . . . . . . . . . . . . 145
11 Applying Stein’s method for the µα-distribution 149
ρ
12 Evolving Bernoulli random graph models 154
12.1 A discrete-time evolving Bernoulli random graph . . . . . . . . . . . . . . . 155
12.1.1 Bounds for the degree distribution . . . . . . . . . . . . . . . . . . . 155
12.1.2 Bounds for the degree sequence . . . . . . . . . . . . . . . . . . . . . 161
12.1.3 Bounds for the number of triangles . . . . . . . . . . . . . . . . . . . 163
12.2 An analogous continuous-time evolving Bernoulli random graph . . . . . . . 171
12.2.1 The model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
12.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
13 Concluding remarks 174
A Gamma and beta functions 177
B A list of random graphs 178
B.1 Discrete-time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
B.1.1 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
B.1.2 Model I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
B.2 Continuous-time models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
B.2.1 Graph based on Yule’s model for evolution . . . . . . . . . . . . . . 180
B.2.2 Model Y . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
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Notation
The following provides a summary of common notation and conventions used throughout
this thesis.
The natural numbers are defined not to include zero.
•
N := 1,2,3,...
{ }
Z+ := 0,1,2,...
0 { }
The forward difference is denoted by ∆. For a function f on Z,
•
∆f(k):= f(k+1) f(k).
−
We use the usual order notation:
•
f(k)
f(k)= o(g(k)) as k 0 as k ;
→ ∞ ⇐⇒ g(k) → → ∞
f(k) =O(g(k)) as k C,k 0, f(k) C g(k) k k .
0 0
→ ∞ ⇐⇒ ∃ ≥ | | ≤ | | ∀ ≥
We write f(k)<< g(k) to mean f(k) = o(g(k)).
•
The supremum norm is denoted by . For a function f,
• k·k∞
f := sup f(x).
k k∞ x | |
For random variables X and Y, we write X Y to mean that X and Y are inde-
• ⊥
pendent.
We use the usual notation for the gamma and beta functions (defined on C+ and
•
C+ C+ respectively),
×
Γ(z) := ∞tz 1e tdt;
− −
Z0
1 Γ(x)Γ(y)
B(x,y) := tx 1(1 t)y 1dt = .
− −
− Γ(x+y)
Z0
A graph = ( , ) is described by its vertex set and edge set .
• G V E V E
If is a directed graph then an edge from vertex i to vertex j is denoted
G ∈ V ∈ V
by the ordered pair (i,j) . If is an undirected graph then an edge between
∈ E G
vertices i,j is denoted by the set i,j .
∈ V { } ∈ E
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Following the definition by Hsu and Robbins [29] we say that a sequence of random
•
variables (X ) converges completely to a constant c if the series
n n
∞
P(X c > ε)
n
| − |
n=1
X
converges for every ε > 0. We write X C c.
n
→
Distributions
GeometricN(p) denotes the geometric distribution with parameter p (0,1] and
• ∈
state space N; µ(k) = p(1 p)k 1 for k N.
−
− ∈
Geometric (p) denotes the geometric distribution with parameter p (0,1] and
Z+
• 0 ∈
state space Z+; µ(k) = p(1 p)k for k Z+.
0 − ∈ 0
Yule Simon(ρ) denotes the Yule-Simon distribution with parameter ρ > 0. For a
• −
random variable V,
V Yule Simon(ρ) P(V = k) = ρB(k,ρ+1) k N .
∼ − ⇐⇒ ∀ ∈
(cid:8) (cid:9)
Negbinom(r,p) denotes the negative binomial distribution with parameters r > 0
•
and p (0,1). For a random variable V,
∈
Γ(k+r)
V Negbinom(r,p) P(V = k) = pr(1 p)k k Z+ .
∼ ⇐⇒ k!Γ(r) − ∀ ∈ 0
( )
Equivalently,
psr 1(1 s)kds
V Negbinom(r,p) P(V k) = 0 − − k Z+ .
∼ ⇐⇒ ≤ B(r;k+1) ∀ ∈ 0
( R )
For r N, if U ,U ,...,U are independent identically distributed random variables
1 2 r
∈
with (U ) = Geometric (p), equivalently we can say,
1 Z+
L 0
r
V Negbinom(r,p) (V)= U .
i
∼ ⇐⇒ L L
!
i=1
X
For a random variable R that takes values in [0,1], we use V Negbinom(α;R)
• ∼
to mean that the random variable V is defined conditionally on R, such that given
R = r, V Negbinom(α;r). The notation is extended to other such mixtures of
∼
distributions.
For α > 0,ρ > 1 we define µα to be the measure on Z+ given by
• ρ 0
B(k+α;ρ+1)
µα(k) := .
ρ B(α;ρ)
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Distances
Thetotal variation distancebetween two randomvariables, X andY, definedjointly
•
on a state space is defined by,
S
d (X,Y) := sup P(X A) P(Y A)
TV
| ∈ − ∈ |
A
⊂S
= sup E(h(X)) E(h(Y)),
| − |
h
∈H
where := 1 :A , A measurable .
A
H { ⊂ S }
If is countable then, equivalently,
S
1
d (X,Y) = P(X = k) P(Y = k).
TV
2 | − |
k
X∈S
If is Hausdorff separable then, equivalently (see [8] p. 254),
S
d (X,Y) = min P(X = Y ).
TV ′ ′
X′=DX,Y′=DY 6
There are many other equivalent definitions, but these will be adequate for our
purposes.
The Kolmogorov distance between two random variables, X and Y, taking values in
•
a subset of the real line is defined by,
d (X,Y) := sup P(X x) P(Y x)
K
x R| ≤ − ≤ |
∈
= sup E(h(X)) E(h(Y)),
| − |
h
∈H
where := 1 :x R, h measurable .
( ,x]
H { −∞ ∈ }
The Wasserstein distance between random variables, X and Y, defined jointly on a
•
state space is defined by,
S
d (X,Y) := sup E(h(X)) E(h(Y)),
W
| − |
h
∈H
where := h :R R : h differentiable h 1 .
′
H { → || ||∞ ≤ }
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