Finiteness and fluctuations in growing networks
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2002 J. Phys. A: Math. Gen. 35 9517
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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS A: MAT H E M ATI CAL AND GENERAL
J. Phys. A: Math. Gen. 35 (2002) 9517–9534 PII: S0305-4470(02)39251-5
Finiteness and fluctuations in growing networks
PLKrapivskyand S Redner
Center for BioDynamics, Center for Polymer Studies, and Department of Physics,
Boston University, Boston, MA 02215, USA
Received 10 July 2002
Published 29 October 2002
Online at stacks.iop.org/JPhysA/35/9517
Abstract
We study the role of finiteness and fluctuations about average quantities for basic
structural properties of growing networks. We first determine the exact degree
distribution of finite networks by generating function approaches. The resulting
distributions exhibit an unusual finite-size scaling behaviour and they are also
sensitive to the initial conditions. We argue that fluctuations in the number of
nodes of degree k become Gaussian for fixed degree as the size of the network
diverges. We also characterize the fluctuations between different realizations
of the network in terms of higher moments of the degree distribution.
PACS numbers: 02.50.Cw, 05.40.−a, 05.50.+q, 87.18.Sn
1. Introduction
Networks such as the Internet and the World Wide Web do not grow in an orderly manner. For
example, the Web is created by the uncoordinated effort of millions of users and thus lacks an
engineered architecture. Although such networks are complex in structure [1, 2], their large
size is a simplifying feature, and for infinitely large networks the rate equation approach
1
provides analytical predictions for basic network characteristics. Nevertheless, social and
technological networks are not large in a thermodynamic sense (e.g., the number of molecules
in a glass of water vastly exceeds the number of routers in the Internet). Thus fluctuations
in network properties can be expected to play a more prominent role than in thermodynamic
systems
2
.Additionally, extreme properties, such as the degree of the node with the most
links in a network [5, 6], the website with the most hyperlinks, or the wealth of the richest
person in a society are important characteristics of finite systems. The size dependence of
these properties or their distribution is difficult to treat within a rate equation approach.
In this paper, we examine the role of finiteness and the nature of fluctuations about
mean values for large, but finite growing networks. We shall focus primarily on the degree
distribution N
k
(N),the number of nodes that are linked to k other nodes in a network of N links,
1
Ashort review of applications of the rate equation approach to network growth is given in [3].
2
It was also suggested that fluctuations can affect the growth of the Web itself (see, e.g., [4]).
0305-4470/02/459517+18$30.00 © 2002 IOP Publishing Ltd Printed in the UK 9517
9518 PLKrapivskyand S Redner
as well as related local structural characteristics. We shall argue that self-averaging holds for
thedegree distribution, so that the random variables N
k
(N) become sharply peaked about their
average values in the N →∞limit. We shall also argue that the probability distribution for
the number of nodes of fixed degree, P(N
k
,N),isgenerally a Gaussian, with fluctuations that
vanish as N →∞.Ontheother hand, higher moments of the degree distribution do not self
average. This loss of self-averaging ultimately stems from the power-law tail in the degree
distribution itself.
In the next section, we define the growing network model and briefly review the behaviour
of the average degree distribution in the thermodynamic N →∞limit. We also discuss how
the average degree distribution can naturally be expected to attain a finite-size scaling form
for large but finite N.Wethen describe our general strategy for studying fluctuations in these
growing networks. In section 3, we outline our simulational approach and present data for the
average degree distribution. In the following two sections, we examine the role of finiteness
on the degree distribution, both within a continuous formulation based on the rate equations
(section 4), and an exact discrete approach (section 5). The former approach is the one that is
conventionally applied to study the kinetics of evolving systems, such as growing networks.
While this approach has the advantage of simplicity and it provides an accurate description
for the degree distribution in an appropriate degree range, it is quantitatively inaccurate in the
large degree limit. This is the domain where discreteness effects play an important role and
the exact discrete recursion relations for the evolution of the degree distribution are needed to
fully account its properties. In section 6, we discuss the implications of our results for higher
moments of the degree distribution and their associated fluctuations. Section 7 provides
conclusions and some perspectives. Calculational details are given in the appendices.
2. Statement oftheproblem
The growing networks considered in this work are built by adding nodes to the network one
at a time according to the rule that each new node attaches to a single previous node with a
rate proportional to A
k
,wherek is thedegree of the target node. We investigate the class of
models in which A
k
= k + λ,whereλ>−1, but is otherwise arbitrary. The general situation
of −1 <λ<∞ corresponds to linear preferential attachment, but with an additive shift λ in
therate. This model was originally introduced by Simon to account for the word frequency
distribution [7]. The case λ = 0corresponds to the Barab
´
asi–Albert model [8], while the
limit λ →∞corresponds to random attachment in which each node has an equal probability
of attracting a connection from the new node. Thus by varying λ,wecan tune the relative
importance of popularity in the attachment rate.
Previous work on the structure of such networks was primarily concerned with the
configuration-averaged degree distribution N
k
(N),wheretheangle brackets denote an
average over all realizations of the growth process for an ensemble of networks with the
same initial condition. Additionally, most studies focused on the tail region where k is much
smaller than any other scale in the system. For attachment rate A
k
= k + λ,thisaverage degree
distribution has a power-law tail [7, 9],
N
k
(N)=Nn
k
with n
k
∝ k
−(3+λ)
(1)
as N →∞.Inthespecific case of A
k
= k,the average degree distribution explicitly is [7–11]
N
k
(N)
= Nn
k
with n
k
=
4
k(k +1)(k +2)
. (2)
For finite N,however,thedegree distribution must eventually deviate from these
predictions because the maximal degree cannot exceed N.Toestablish the range of applicability
Finiteness and fluctuations in growing networks 9519
of equation (1), we estimate the magnitude of the largest degree in the network, k
max
by the
extreme statistics criterion
kk
max
N
k
(N)
≈ 1. This yields k
max
∝ N
1/(2+λ)
.We,therefore,
anticipate that the average degree distribution will deviate from equation (1)whenk becomes
of the order of k
max
.The existence of a maximal degree also suggests that the average degree
distribution should attain a finite-size scaling form
N
k
(N)Nn
k
F(ξ) ξ = k/k
max
. (3)
Some aspects of these finite-size corrections were recently studied in [12–15]. One basic result
of our work is that we can compute the scaling function explicitly. We find that this function
is peaked for k of the order of k
max
and that it depends substantially on the initial condition. In
contrast, the small-degree tail of the distribution—the reason why such networks were dubbed
scale-free—is independent of N and the initial condition.
To study finite networks where fluctuations can be significant, we need a stochastic
approach rather than a deterministic rate equation formulation. For finite N,thestateofa
network is generally characterized by the set N ={N
1
,N
2
, ...} that occurs with probability
P(N).Thenetworkstate N evolves by the following processes:
(N
1
,N
2
) → (N
1
,N
2
+1)(N
1
,N
k
,N
k+1
) → (N
1
+1,N
k
− 1,N
k+1
+1).
The first process corresponds to the new node attaching to an existing node of degree 1; in this
case, the number of nodes of degree 1 does not change while the number of nodes of degree
2increases by 1. The second line accounts for the new node attaching to a node of degree
k>1.
From these processes, it is straightforward, in principle, to write the master equation for
the joint probability distribution P(N).Itturns out that correlation functions of a given order
are coupled only to correlation functions of the same and lower orders. Thus, we do not
need to invoke factorization (as in the kinetic theory) and we could, in principle, solve for
correlation functions recursively. However, this would provide much more information than
is of practical interest. Typically, we are interested in the degree distribution, or perhaps two-
body correlation functions of the form N
i
N
j
.Eventhough straightforward in principle,
it is difficult to compute even the two-point correlation functions N
i
N
j
for general i
and j .Inthiswork, we shall restrict ourselves to the specific (and simpler) examples of
N
2
1
, N
1
N
2
and
N
2
2
.Wewill use these results to help characterize fluctuations in finite
networks.
3. Simulation method and data
To simulate a network with attachment rate A
k
= k + λ efficiently, we exploit an equivalence
to thegrowing network with re-direction (GNR) [9]. In the GNR, a newly-introduced node n
selects an earlier ‘target’ node x uniformly.With probability 1−r,alink from n to x is created.
However, with probability r,thelink is re-directed to theancestor node y of node x (figure 1).
As discussed in [9], the GNR is equivalent to a growing network with the attachment rate
A
k
= k + λ,with λ = r
−1
− 2. Thus, for example, the GNR with r = 1/2corresponds to
the growing network with linear preferential attachment, A
k
= k.Simulation of the GNR
is extremely simple because theselection of the initial target node is purely random and the
ensuing re-direction step is local.
There is, however, an important subtlety about this equivalence that was not discussed
previously in [9]. Namely, the redirection process does not apply when a node has no ancestor.
By construction, every node that is added to the network does have a single ancestor, but some
9520 PLKrapivskyand S Redner
probability r
n
probability 1-r
y
x
Figure 1. There-direction process. The new node n selects a random target node x.With
probability 1 − r a link is established to this target node (dashed), while with probability r the link
is established to y,theancestor of x (solid).
primordial nodes may have none. For example, forthe very natural ‘dimer’ initial condition
◦←−◦,the seed node on the left has no ancestor and the GNR construction for this node is
ambiguous. One way to resolve this dilemma is to adopt the ‘triangle’ initial condition in
whichthere are three nodes in a triangle with cyclic connections between nodes. This leads
to the correct attachment rate for each node for any value of λ.Wetherefore typically use
this initial state to generate degree distribution data. On the other hand, theoretical analysis is
simpler for the dimer initial condition. This state can also be simulated in a simple manner
(for the case λ = 0) by a slightly modified GNR construction in which direct attachment to
the seed node is not allowed. It is straightforwardtocheck that this additional rule leads to
thecorrect attachment rates for all the nodes in the network.
Figure 2 shows the average degree distribution for attachment rates A
k
= k and A
k
= k+λ
with λ =−0.9for the triangle initial condition. This latter value of λ gives results that are
representative for values of λ close to −1. The data exhibit a shoulder at k ≈ k
max
that is much
more pronounced when λ<0(figure 2(b)). This shoulder is also at odds with the natural
expectation that the average degree distribution should exhibit a monotonic cutoff when k
becomes of the order of k
max
.This shoulder turns into a clearly-resolved peak that exhibits
relatively good data collapse when the degree distribution is re-expressed in the scaling form
of equation (3)(figure 3). Conversely, the magnitude of the peak diminishes rapidly when λ
is positive and becomesimperceptible for λ 0.5.
In the following two sections, we will attempt to understand this anomalous feature of the
degree distribution by studying the rate equations for the node degrees of finite networks.
4. Continuum formulation
We fo cus on the case of the linear attachment rate A
k
= k and briefly quote corresponding
results for other attachment rates. In the continuum approach, N is treated as continuously
varying. Then the change in the average degree distribution satisfies the rate equation
dN
k
(N)
dN
=
(k − 1)N
k−1
(N) − kN
k
(N)
2N
+ δ
k,1
. (4)
We assume the dimer initial state—two nodes connected by a single link so that N
k
(N = 1) =
2δ
k,1
.
Equations (4)arerecursive and can be solved sequentially, starting with N
1
.Explicit
results for N
k
,k 4, are given in appendix A. These expressions show that the dominant
contribution in the N →∞limit is linear in N and this corresponds to the solution in
equation (2). Indeed, if we substitute
N
k
(N)
= n
k
N into equations (4), we obtain the
recursion n
k
= n
k−1
(k − 1)/(k +2),whose solution is equation (2). From the first few N
k
,
![Figure 2. Normalized degree distributions for the triangle initial condition for networks of 102, 103, . . . links (upper left to lower right), with 105 realizations for each N, for (a) Ak = k (up to 106 links) and (b) Ak = k + λ, with λ = −0.9 (up to 105 links). In (a), the dashed line is the asymptotic result nk = 4/[k(k + 1)(k + 2)]; the last three data sets were averaged over 3, 9 and 27 points, respectively. In (b), the last two data sets were averaged over 10 and 100 points, respectively.](/figures/figure-2-normalized-degree-distributions-for-the-triangle-2wl8esxx.webp)
