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Physica D 214 (2006) 132–143
www.elsevier.com/locate/physd
Asymptotic behavior of connecting-nearest-neighbor models for growing
networks
David Juher, Joan Salda˜
na∗, Jaume Soler
Departament d’Inform`
atica i Matem`
atica Aplicada, Universitat de Girona - Campus de Montilivi, 17071 Girona, Spain
Received 9 November 2005; accepted 2 January 2006
Available online 30 January 2006
Communicated by A. Mikhailov
Abstract
This paper deals with the mathematical description of the asymptotic behavior of the solutions of a couple of models for the dynamics of
growing networks based on connecting, with a higher probability, nodes that have a neighbor in common. The first model, proposed by A. V´
azquez,
is nonlinear and, in general, the long-time behavior of the solutions differs from the one predicted by the linear reduction proposed in its original
treatment. A second model is specifically derived from the rules defining an in silico model also proposed by V´
azquez to simulate the growth of a
network under the mechanism of connecting nearest neighbors. The two analytical models lead to very different predictions for the configuration
of the network that are tested using the simulations of the in silico model.
c
2006 Elsevier B.V. All rights reserved.
Keywords: Growing networks; Continuum approach; Asymptotic behavior of network models
1. Introduction
The study of complex networks started in the 1950s
and was motivated by the analysis of the spread of
information/infectious diseases through populations with a
certain contact structure. This process was, for instance, one
of the main motivations for the first percolation model [2]
and for the development of the theory of random and biased
networks [3–5]. In the 1960s and 1970s the study of social
networks received a wave of interest, especially from Milgram’s
empirical work [6]. This interest was focused on the structure
of the network itself and, in particular, on the so-called
small-world property, defined by a low average distance
(“handshakes”) between any pair of persons in the network.
A social network is a set of people with some pattern
of interactions (friendship, acquaintances, sexual contacts,
business relationships) among them. One of the relevant
features of this sort of network is the presence of a high degree
of transitivity, i.e., a high probability that one’s acquaintances
∗Corresponding author. Tel.: +34 972 41 8834; fax: +34 972 41 8792.
E-mail addresses: juher@ima.udg.es (D. Juher), jsaldana@ima.udg.es
(J. Salda˜
na), jaume@ima.udg.es (J. Soler).
will also be acquainted with each other [7,8]. This fact as
well as the presence of “strong” and “weak” ties in the
relationships between individuals have been discussed and their
consequences have been analyzed for a long time in social
network research. At the beginning, such a research pursued
the statistical description of networks (average shortest path
length, clustering, degree distribution). During the 1980s, much
effort was invested in developing geometrical methods in order
to represent and compare complex social networks [9].
However, more recently there has been a change in the
philosophy of network modeling. In the former models, the
goal was to capture correctly the topological aspects of complex
networks by constructing graph models based on the classical
Erd¨
os–R´
enyi random graph [5,10]. In contrast to this static
approach, since the end of the 1990s, the modeling emphasis
has lain on capturing processes governing the growth or
evolution of complex networks, the network topology being
considered only a by-product of this dynamics [11,12]. In other
words, such a modeling approach assumes the idea that a
suitable growing mechanism has to lead to a correct prediction
of its topology. One such growing mechanism is preferential
linking. Introduced in [13], it simply assumes that, in growing
networks, the new nodes are attached preferentially to those in
0167-2789/$ - see front matter c
2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2006.01.001
D. Juher et al. / Physica D 214 (2006) 132–143 133
the network with a high connectivity (degree). The importance
of this mechanism lies in the fact that it explains the origin of
a basic feature of real growing networks: the power-law degree
distribution [7,11,12].
In recent years, several models offering a more realistic
description of the growth of real networks have been proposed.
Some of these models, for instance, consider node duplication
and divergence in evolving protein interaction networks [14]
and other biological networks [15], edge redirection, or the
addition of new nodes based on an “attaching to edges”
mechanism (see [11] for a review of alternative mechanisms).
The unifying point of these works is that preferential attachment
does not arise because of some rule explicitly introduced
in terms of a preference function, but is induced by the
mechanisms used to place nodes and links during the network
growth. An illustrative example of this modeling philosophy is
given by [1]. In that paper, V´
azquez presents several network
models based on local rules defining how links between pairs
of nodes appear during the growth of the network. From these
local rules, global properties of the corresponding growing
networks emanate, for instance, the degree distribution,
degree correlations, and clustering spectrum. For instance, in
the so-called connecting-nearest-neighbor model, the closure
occurring in acquaintance networks [16] is taken into account
by means of the concept of potential link between two nodes
and, hence, by considering two kinds of edges in the graph: the
actual and the potential edges.
The first goal of the present paper is to give a complete
description of the asymptotic behavior of the solutions to
this model. The model is described by a couple of nonlinear
mean-field equations for the evolution of the degree and
the potential degree, respectively, of an arbitrary node. In
Section 2 the model is introduced and the asymptotic behaviors
of the solutions are obtained by transforming the original
non-autonomous model into an asymptotically autonomous
differential system. Section 3 is devoted to the second goal
of the paper, namely, the analysis of an alternative model
also based on the mechanism of connecting nearest neighbors.
The model is derived for the study of the dynamics of an in
silico model proposed in [1] to simulate the one presented
in Section 2, with the aim of improving the predictions
of V´
azquez’s model as regards the network configuration.
Finally, in Section 4, the predictions of the derived model are
checked against the output of networks of size 106generated
according to the connecting-nearest-neighbor mechanism.
These predictions are also compared with those of V´
azquez’s
model.
2. V´
azquez’s connecting-nearest-neighbor model
2.1. The model
The basic idea of the mechanism of connecting nearest
neighbors (CNN), which is already present in [8], is that
social networks evolve mainly following connection rules
based on transitive linking, i.e., rules that favor the creation
of new connections between pairs of individuals (nodes)
with a common acquaintance. This assumption leads to the
introduction of the concept of a potential link. According to [1],
we say that a pair of nodes is connected by a potential link if
(1) they are not connected by a link and (2) they have at least
one common neighbor. As usual in such models, the growth of
the network is due to the addition, at each timestep, of a new
node. So, if we start with the n0=1 node, the total number of
nodes Nin the network at time tis N(t)=t. In particular, this
regular addition of nodes implies that any transition rate νcan
be considered either per unit of time or, equivalently, per unit
of added node.
The evolution of the network will be described by focusing
on the dynamics of node degrees and using the continuum
approach introduced in [13]. More precisely, at any time t, each
node in the network is characterized by two quantities that are
assumed to be continuous: its degree, denoted here by x(t),
and the number of potential links incident to it or its potential
degree, denoted by y(t). These quantities define three possible
states of any pair of nodes: disconnected (d), connected (c)
and potentially connected (p), i.e., connected by a potential
link. With this continuum approach in time and degrees and
with deletion of edges neglected, V´
azquez proposed in [1]
the following system of ordinary differential equations for the
dynamics of the degree xsand potential degree ysof a node s
for t≥1:
dxs
dt=µ0
t2(t−xs−ys)+µ1
tys
dys
dt=µ0
t2xs(t−xs−ys)−µ1
tys,
(1)
where µ0/t2(µ0>0) is the rate of transitions νd→cfrom the
disconnected state of a pair of nodes to the connected one, µ1/t
(µ1>0) is the rate of transitions νp→cfrom the potential state
to the connected one, and the difference t−xs−ysgives the
number of nodes in the network that are neither neighbors nor
potential neighbors of s.
The two main assumptions introduced in the derivation of
Eq. (1) are the following:
(a) By definition of a potential link, the rate of transitions
νs
d→pfrom disconnected nodes to new pairs of potentially
connected nodes having the node sas one of its ends is
equal to νd→ctimes the number of neighbors (degree) of
this node: νs
d→p=νd→cxs. Note that νd→cis the same for
any pair of nodes in the network.
(b) The transitive linking hypothesis is translated into the
assumption νs
p→cνd→c. In fact, since the transition
d→cinvolves two arbitrarily chosen nodes of the network
and the number of pairs of disconnected nodes without a
common neighbor is approximately proportional to t2for
large t, a natural hypothesis to consider is νd→c∼1/t2.
Conversely, since given a node sthe transition p→c
involves the neighbors of sand, at most, they can be as
much as t−1, then νs
p→c∼1/t.
In order to obtain a simple but useful preliminary result
as regards the trajectories of the non-autonomous system (1),
134 D. Juher et al. / Physica D 214 (2006) 132–143
we consider its autonomous version in R3, that is,
dxs
dτ=µ0
t2(t−xs−ys)+µ1
tys,
dys
dτ=µ0
t2xs(t−xs−ys)−µ1
tys,dt
dτ=1.(2)
From a first analysis of the vector field it follows that
R=n(x,y,t)∈R3
+:0≤x+y≤to
is a positively invariant region under the flow defined by (2). In
particular, at any section Rt0= {(x,y,t)∈R3:0≤x+y≤
t0,t=t0}, the vector field (dxs/dτ, dys/dτ ) on the boundary
x+y=t0points along it while, on the axes, it points inwards
except at the point (t0,0)where (dxs/dτ, dys/dτ ) =(0,0).
Hence the invariance of Rfollows and, using that dxs/dt>0
inside Rt0, there also follows immediately the non-existence of
a steady state in R, as has to be the case in a growing network
such as the one described above. This means that a deeper
understanding of the dynamics is needed if we want to give
an answer to questions like: “Is there any final configuration
of the growing network?” and, if the answer is yes, “Does this
configuration depend on the initial conditions?”.
Note that the invariance of Ris also a necessary (and
natural) requirement for the suitability of (1) as a model for the
growth of a social network. In words, it says that the sum of the
degree of a node plus its potential degree cannot be larger than
the size of the network. Clearly, this feature of the solutions
has to also be fulfilled by the solutions to any simpler system
that is going to be used as a convenient approximation to (1). In
particular, one has to be careful when neglecting a priori some
terms in the RHS of (1), as was done in the original treatment of
the model in [1], because the absence of these terms can cause,
under certain election of the parameter values, the solutions of
the resulting system to violate this requirement.
2.2. The rescaled model
In order to obtain a more accurate description of the
dynamics of the CNN model and since we know that, for any
positive time, xs(t)+ys(t)≤t, let us introduce the following
change of variables in the system (1):
¯x=xs
t,¯y=ys
t.
Since the size of the network is equal to t, these new variables
are the normalized degree and the normalized potential degree
of the node s, respectively. Under this transformation, the
system becomes
d¯x
dt=1
thµ0
t(1− ¯x− ¯y)+µ1¯y− ¯xi
d¯y
dt=1
t[µ0¯x(1− ¯x− ¯y)−(µ1+1)¯y],
t≥1.(3)
It is easily seen that, at any time, the corresponding vector field
on the boundary of the region
R0=n(x,y)∈R2
+:0≤x+y≤1o(4)
always points inwards, except at the point (0,0)where it points
along the x-axis. So, R0is a positively invariant region for this
system and, hence, any solution to (3) with initial condition
inside R0is bounded for any positive time. Moreover, for this
system, the null cline d ¯y/dt=0 is a humped curve through the
origin and the point (1,0)which does not depend on t, and the
null cline d ¯x/dt=0 is a straight line that rotates, as tincreases,
around the point (µ1
µ1+1,1
µ1+1)of the boundary ¯x+ ¯y=1 and
tends to the straight line that also passes through the origin,
namely, ¯y= ¯x/µ1.
From this preliminary analysis and taking into account the
direction of the vector field on the null clines and on the
boundary of R0, it is clear that, as long as the null cline
d¯y/dt=0 intersects the straight line ¯y= ¯x/µ1at an interior
point (¯x∗,¯y∗)of R0, the solutions to (3) with initial conditions
inside R0will tend to this point as t→ ∞. Otherwise, (0,0)
will be the only intersection point of the previous curves and
the unique limit of the phase trajectories of the system (3) as
t→ ∞.
To prove this claim rigorously let us start by rescaling the
time variable in (3) with the following change of variables:
¯
ξ(ln t)= ¯x(t)and ¯η(ln t)= ¯y(t). With this transformation
the system becomes
d¯
ξ
dt=µ0
et(1−¯
ξ− ¯η) +µ1¯η−¯
ξ,
d¯η
dt=µ0¯
ξ (1−¯
ξ− ¯η) −(µ1+1)¯η,
t≥1.(5)
A simple analysis of the vector field given by (5) shows that it
has the same properties as that of (3), that is, the same invariant
region R0and the same null clines. In particular, the limit of
the null cline d¯
ξ/dt=0 as t→ ∞ is also the same straight
line ¯η=¯
ξ/µ1passing through the origin and through the point
(µ1
µ1+1,1
µ1+1). However, the main feature of this system is that
it is asymptotically autonomous (see [17] for definitions) with
a limit system given by
dξ
dt=µ1η−ξ,
dη
dt=µ0ξ (1−ξ−η) −(µ1+1)η,
(6)
which has also R0as a positively invariant region. Note that,
without rescaling the time variable, the corresponding limit
system of (3) is simply given by d¯x/dt=0,d¯y/dt=0 which
does not give us any knowledge about the asymptotic behavior
of the solutions of the original system.
2.3. The dynamics of the asymptotically autonomous system
Let us begin with a general result that relates the long time
behavior of the solutions of (5) to that of (6):
Lemma 1. Let R0be the invariant region for the system (5)
given by (4). Then, the ω-limit set W of the trajectories of
this system with initial condition inside R0is the same as
that of the trajectories of (6) inside R0, W∞. Moreover, if
¯
Φ(t;t0,Φ0)denotes the solution to (5) with initial condition
D. Juher et al. / Physica D 214 (2006) 132–143 135
(t0,Φ0)∈ [0,t] × R0and Φ(t;Φ0)is the solution to (6)
satisfying Φ(0,Φ0)=Φ0∈R0, then
¯
Φ(tj+sj;sj,Φj)→Φ(t;Φ0), j→ ∞,
for any three sequences tj→t , s j→ ∞,Φj→Φ0as j → ∞
with 0≤t,tj<∞, s j≥0, and Φj∈R0.
Proof. The proof of this lemma follows from the boundedness
of the forward solutions of (5), which implies that Wis compact
and non-empty, and from the uniform convergence of the
RHS of (5) towards the RHS of (6) on R0when t→ ∞,
which guarantees that W=W∞(see [17] for details). The
second part of the statement corresponds to the embedding
of an asymptotically autonomous continuous semiflow into an
autonomous continuous one. Such an embedding is guaranteed
by the boundedness of the forward solutions of (5) and the
uniform convergence of the RHS of (5) towards the RHS of
(6) on R0as t→ ∞ (see [18]).
From Lemma 1 it follows that if we know the dynamics of
the limit system we will know the asymptotic dynamics of (3)
and, hence, that of (1).
Theorem 2. At µ0=1+1/µ1, the system (6) undergoes a
supercritical bifurcation. In particular, if µ0>1+1/µ1then
the system (6) has a nontrivial equilibrium inside R0which
is asymptotically stable and attracts every trajectory in this
region, while the origin is an unstable equilibrium. Conversely,
if µ0<1+1/µ1then the origin is the only equilibrium of (6)
in R0and attracts every trajectory in this region.
Proof. The equilibrium points of (6) are P1=(0,0)and
P2=µ1(µ0−1)−1
µ0(1+µ1),µ0−(1+1/µ1)
µ0(1+µ1).
Hence, µ0>1+1/µ1is a necessary and sufficient condition
for the existence of a positive equilibrium, which always lies
inside R0.
Linearizing (6) around P1=(0,0), we get the Jacobian
matrix
JP1=−1µ1
µ0−(µ1+1),(7)
whose eigenvalues are real and equal to λ1= −[(µ1+
2)+qµ2
1+4µ0µ1]/2<0 and λ2= [−(µ1+2)+
qµ2
1+4µ0µ1]/2. Hence, λ2<0 if and only if µ0<1+1/µ1.
That is, P1will be asymptotically stable whenever it is the
unique non-negative equilibrium. In this case, from the phase
portrait of the system (6) (see Fig. 1), it immediately follows
that P1attracts every trajectory inside R0.
When µ0>1+1/µ1, the matrix of the linearized system
around P2=(ξ∗, η∗)∈R0is
JP2=−1µ1
µ0[1−(2µ1+1)η∗] −µ0µ1η∗−(µ1+1)(8)
with eigenvalues
Fig. 1. Phase portrait of the system (6) in the region R0for the case µ0<
1+1/µ1. The numerical integration of this system has been performed with
µ0=1.95 and µ1=1. The dashed lines correspond to the null clines of the
system. The unique equilibrium of the system in this region is the trivial one
and it is represented by a circle.
λ±= −1
2µ0µ1¯η∗+µ1+2∓
q(µ0µ1¯η+µ1+2)2−4(µ1(1−µ0)+1+2µ0µ1(µ1+1)¯η∗),(9)
which are both negative as long as µ1(1−µ0)+1+2µ0µ1(µ1+
1)¯η∗>0 or, equivalently, as long as µ0>1+1/µ1, just
the condition on the parameters we are assuming to hold.
In this case, from the phase portrait of (6) (see Fig. 2) it
follows that no periodic orbit is possible. Moreover, from the
Poincar´
e–Bendixson theorem, P2attracts every trajectory in
R0.
From Theorem 2 it follows that the ω-limit set of the
trajectories of the autonomous limit system of (5) is W∞=
{(0,0)}if µ0<1+1/µ1and W∞= {P2}otherwise. Hence,
from Lemma 1 we have the following:
Theorem 3. The trajectories in R0of the system (5) tend to
µ1(µ0−1)−1
µ0(1+µ1),µ0−(1+1/µ1)
µ0(1+µ1)
as t → ∞ if µ0>1+1/µ1, and to (0,0)if µ0≤1+1/µ1.
Finally, since (¯x(t), ¯y(t)) is obtained from (¯
ξ(t), ¯η(t)) by
rescaling the time, the previous results imply that, for any initial
condition in R0, the corresponding trajectory of (3) satisfies, as
t→ ∞,
(¯x(t), ¯y(t)) −→
µ1(µ0−1)−1
µ0(1+µ1),µ0−(1+1/µ1)
µ0(1+µ1)if µ0>1+1
µ1
,
(0,0)if µ0≤1+1
µ1
.
136 D. Juher et al. / Physica D 214 (2006) 132–143
Fig. 2. Phase portrait of the system (6) in the region R0for the case µ0>
1+1/µ1. The numerical integration of this system has been performed with
µ0=4 and µ1=1. The dashed lines correspond to the null clines of the
system. The two equilibria of the system are represented by circles.
2.4. The long time behavior of V´
azquez’s CNN model
Case µ0>1+1/µ1: It is clear that, as long as the
normalized degree ¯x(t)of node sand the normalized potential
degree ¯y(t)of the same node tend asymptotically to strictly
positive constant values, its degree xs(t)and potential degree
ys(t)will increase linearly with tas t→ ∞. More precisely,
for any initial degree x0
sand potential degree y0
s, we have that
(xs(t), ys(t)) ∼µ1(µ0−1)−1
µ0(1+µ1)t,µ0−(1+1/µ1)
µ0(1+µ1)t
as t→ ∞.
In other words, there is a trajectory (in R) that attracts any other
trajectory in Rregardless of its initial condition.
Case µ0≤1+1/µ1: Up to now, the only thing we know
about it is that the growth of the degree and the potential degree
of a node sis sublinear in time. So, more information about
the asymptotic dynamics of (5) is needed. By analyzing how
(¯x(t), ¯y(t)) reaches the origin (0,0)we will obtain a deeper
insight into its dynamics.
Computing the eigenvectors corresponding to the eigenval-
ues of (7), one easily realizes that the only eigenvector that
lies in the first quadrant is Ev2, the one associated with λ2.
In particular, it follows that (ξ(t), η(t)) approaches the origin
along the direction given by Ev2=(1,√1/4+µ0/µ1−1/2).
Note that this direction lies in the region between the null cline
dξ/dt=0, which is given by η=ξ/µ1, and the straight line
η=µ0ξ/(µ1+1)which is tangent to the null cline dη/dt=0
at the origin. Therefore, we obtain that, as t→ ∞,
ξ(t)
η(t)∼eλ2t
1
s1
4+µ0
µ1−1
2
.
Hence, from Lemma 1 and rescaling back to the original
time variable, it follows that
¯x(t)
¯y(t)∼tλ2
1
s1
4+µ0
µ1−1
2
.
Finally, since xs=t¯xand ys=t¯y, the asymptotic growth of
the degree xsand the potential degree ysof a node sfor any
initial degree x0
sand potential degree y0
sis
xs(t)
ys(t)∼t1+λ2
1
s1
4+µ0
µ1−1
2
=tµ1−1
2+q1
4+µ0
µ1
1
s1
4+µ0
µ1−1
2
.(10)
According to these results, what are the consequences of
being in each one of the previous cases for the asymptotic
network configuration?
Once we know the evolution in time of the (expected) degree
of a given node, an important feature of the network that
can be obtained from it is the degree distribution P(k,t)=
∂P(xs(t) < k)/∂k. If nodes are added to the network at
equal time intervals and tsis the time of the introduction of
node s, then P(xs(t;ts) < k)=P(ts>T(k;t)) with ts
being a random variable with a uniform probability distribution
between 0 and t, i.e., P(ts≤a)=a/t, since we are assuming
that we start with the n0=1 node. Here T(k;t)is the solution
of the equation xs(t;T)=k. Hence P(ts>T(k;t)) =
1−T(k;t)/tand the cumulative degree distribution P(k,t)
is given by (see, for instance, [11,12])
P(k,t)= −1
t
∂T(k;t)
∂k.(11)
Clearly, we need an expression of xs(t;ts)in order to
compute P(k,t)but, since the original system is nonlinear, this
expression will be only approximate.
In the case µ0≤1+1/µ1and due to the sublinear growth
of xsand ysfor tts, an approximated expression of xs(t;ts)
is available thanks to the fact that t−xs−ys≈tand, then,
the nonlinear system (1) can be approximated by the following
linear system:
dx
dt=µ0
t+µ1
ty,
dy
dt=µ0
tx−µ1
ty,
(12)
which is the one considered in [1] as a reduction of the system
(1). In this case, for ttsthe solution to (1) with initial
condition (x(ts), y(ts)) =(x0
s,y0
s)can be approximated by the
solution to (12) with the same initial condition, i.e.,
xs(t)
ys(t)≈C1Ev1t
tsβ1
+C2Ev2t
tsβ2
∼ Ev2t
tsβ2
D. Juher et al. / Physica D 214 (2006) 132–143 137
where C1and C2are constants depending on x0
sand y0
s, and
βi=1+λi(i=1,2) are the eigenvalues of the matrix of
the system (12) with λithe eigenvalues of the matrix given by
(7). Note that β1<0, 0 ≤β2≤1 (cf. (10)), and that the
eigenvectors Ev1,Ev2are equal for both matrices. Taking the RHS
of the previous approximation of the solution to (1) one obtains,
as in [1], that the degree distribution is approximately given by
the stationary potential law P(k)∼k−γwith γ=1+1/β2∈
[2,∞).
One refers to a degree distribution of the form P(k)∼k−γ
as a scale-free distribution, usually with 2 < γ ≤3. Although
this degree distribution does not have a finite variance for this
range of values of γ, the average degree is finite. For γ > 3,
the degree distribution progressively approaches an exponential
law (the first and second moments are now both finite), which
is the typical situation when there is no preferential attachment
in the growth of the network.
When µ0>1+1/µ1the previous expression for β2is no
longer valid since the asymptotic growth of x2(t;ts)is always
linear with time (i.e., β2=1). This corresponds to the limit
case P(k)∼k−2for which even the average degree in the
network is not finite. Note that this condition is fulfilled when
both µ0and µ1are large enough. This means a high conversion
(per time unit) of disconnected and potential pairs of nodes into
connected pairs which leads to a linear increase in time of xs(t)
and ys(t).
What is this telling us about the network configuration?
An answer is easily obtained by realizing that the potential
neighbors of a node are, indeed, the second neighbors of
this node. Then, it follows that the fraction of the nodes in
the network that are first or second neighbors of a node s,
(xs(t)+ys(t))/tis always strictly positive, even when the
number of nodes N(t)=t→ ∞. This implies a strongly
connected network with the degree of all its old nodes tending
to infinity as t→ ∞, i.e., a network with an infinite average
degree. Conversely, when µ0≤1+1/µ1, this fraction of nodes
tends to zero as t→ ∞ for any node, which implies a degree
distribution with a finite expectation value.
3. An alternative CNN model
3.1. The evolution rules for an in silico model
A simulation approach to the dynamics of a growing
network based on the CNN mechanism was also suggested by
V´
azquez in [1] from a similar model for a non-growing network
presented in [8]. Here we consider the following extension of
such a computational or in silico model: at each time step t,
(i) either with probability 1 −uintroduce a new node in the
network and create m≥1 links from this new node to
randomly selected nodes (i.e. without any preference),
(ii) or with probability uselect at random a potential link of the
network and convert it into a link.
The model proposed in [1] considers the case m=1.
Observe that, when (i) occurs, a potential link is created
between the new node and each neighbor of the selected nodes.
We remark that the probability that (i) gives two different new
links pointing at the same node is negligible when the size of
the network is big. When (ii) occurs, new potential links appear
between each end of the selected link and some neighbors of
the other end. The expected numbers of added vertices and
added links after tsteps are, respectively, (1−u)tand ¯mt, with
¯m=m(1−u)+u. Finally, we assume 0 ≤u<1 in order to
guarantee the growth of the network.
According to V´
azquez [1], this growing network is a
particular realization of the situation modelled by the system
(1). To see this, take µ0=m(the rate of conversion from
disconnected to linked pairs of nodes is m(1−u)per unit of
time, and m(1−u)/(1−u)=mper unit of added node),
and µ1=u/(1−u)(the rate of conversion from potential to
connected pairs of nodes is uper unit of time, and u/(1−u)per
unit of added node). Hence, in terms of rates per unit of added
node, the system (1) becomes
dxs
dN=m
N2(N−xs−ys)+u
(1−u)Nys,
dys
dN=m
N2xs(N−xs−ys)−u
(1−u)Nys.
(13)
Note that, when m≤1/u, we are in the case µ0≤1+1/µ1of
the system (1) (but now with Ninstead of tas the independent
variable). Otherwise, µ0>1+1/µ1which leads to a highly
connected network.
Since in [1] only m=1 is considered, µ0=1≤1+1/µ1
for any u∈ [0,1)and the previous system can always be
approximated by (12). More precisely, using that the expected
number of nodes in the network at time t N (t)=(1−u)t+
N(0)≈(1−u)tfor t0, we can write down (13) in terms of
rates per unit of time and, afterwards, approximate the resulting
system. Finally, one obtains the following version of the system
(12) for the growing network model described by rules (i) and
(ii):
dxs
dt=1
t+u
(1−u)tys,
dys
dt=xs
t−u
(1−u)tys.
(14)
However, as was already noted by V´
azquez in [1], the value of
the exponent γas a function of the probability upredicted by
(14) is not in agreement with the simulations of the in silico
model for different values of u(see [1], inset of Fig. 8, for
details). The explanation of this disagreement between data
and predictions given by the author was that the presence of
large fluctuations is not correctly reflected by the mean-field
description of the network.
Nevertheless, the existence of a systematic bias between
simulations and the output of the model (the predicted
values of γ (u)are always lower than those observed in the
simulations) suggests that something is missing when modeling
the dynamics of the in silico model by means of (14). To
support this claim, we will derive an analytical model based
only on the evolution rules (i) and (ii) defining the in silico
model and compare, for different values of the probability u,
138 D. Juher et al. / Physica D 214 (2006) 132–143
the predictions of both models for another feature of the
network, namely, the mean potential degree.
3.2. Derivation of an analytical CNN model
Let l(t),l∗(t)and N(t)denote respectively the numbers
of links, potential links and nodes at time t. Label each node
with an integer s≥0 according to the rule that s0>sif
and only if the node s0was added later than the node s. Let
k(s,t)and k∗(s,t)be, respectively, the degree and the potential
degree of the node sat time t. Then the mean degree and the
mean potential degree of the network at time tare given by
k(t)=2l(t)/N(t)and k∗(t)=2l∗(t)/ N(t), respectively.
According to the rules (i) and (ii), if we start with l(0)edges
and N(0)nodes then l(t)=l(0)+ ¯mt and N(t)=N(0)+(1−
u)t. In order to simplify the analysis of the model, from now on
we will assume that l(t)= ¯mt and N(t)=(1−u)t, which is a
good approximation for t0. Consequently, the mean degree
of the network is constant for t0 and is given by
k(t)=2¯m
1−u.(15)
The dynamics of some features of the growing network can
be easily obtained for large times. We start by proposing an
equation for the time evolution of k(s,t). Following the so-
called continuum approach (see for instance [11] or [12]), we
can write
∂k(s,t)
∂t=m(1−u)Π1(s)+uΠ2(s), (16)
where Π1(s)is the probability that, when choosing at random a
node in the network, this node is s. That is, Π1(s)=1/(1−u)t.
On the other hand, Π2(s)is the probability that, when choosing
at random a potential link α,sis an end of a α. That is,
Π2(s)=k∗(s,t)/l∗(t). Summarizing, (16) becomes
∂k(s,t)
∂t=m
t+uk∗(s,t)
l∗(t).(17)
Now let us briefly examine the consistency of Eq. (17).
Integrating both sides with respect to sfrom 0 to (1−u)twe
obtain
d
dtZ(1−u)t
0
k(s,t)ds=(1−u)k((1−u)t,t)+Z(1−u)t
0
m
tds
+u
l∗(t)Z(1−u)t
0
k∗(s,t)ds.(18)
Since the degree of the node which is just being added at
time tis m, it follows that k((1−u)t,t)=m. Taking this
into account, together with the fact that R(1−u)t
0k∗(s,t)dsis the
total sum of the potential degrees, which coincides with 2l∗(t),
we get
d
dtZ(1−u)t
0
k(s,t)ds=m(1−u)+m(1−u)+2u=2¯m,
which is consistent with the fact that the increase of the sum of
the degrees of the nodes in the network is 2 ¯mat each time step
(equivalently, the increase of the number of links is ¯mat each
time step).
Now we have tested the consistency of (17), let us derive
an analogous equation for the evolution of the potential degree.
There are three ways to get a variation on the potential degree
k∗(s,t). With probability 1 −u, we attach mnew links to m
nodes selected at random. The node swill be a neighbor of
them with probability m k(s,t)/(1−u)t, and, in this case, the
potential degree of swill increase by one.
On the other hand, with probability uwe choose at random
a potential link αto convert it into an actual link. The node s
will be an end of αwith probability k∗(s,t)/l∗(t), and in this
case the potential degree of swill decrease by one (since α
becomes an actual link). In addition, a new potential link will
emerge between sand each neighbor of the other end of αbeing
not a neighbor of s. Since αis a potential link, there is at least
one common neighbor of both ends. Neglecting the existence
of other common neighbors, the potential degree of swill
approximately increase by π(t)−1 with π (t)the average degree
of a node at the end of a randomly chosen potential link. Note
that, since we are not choosing at random a node of the network
but a potential link, in general π(t)will be different from k.
Finally, let us call Π3(s)the probability that sis a neighbor
of an end of αbut not a neighbor (either actual or potential) of
the other end. In this case, the potential degree of swill increase
by one. Summarizing, we have that
∂k∗(s,t)
∂t=(1−u)m k(s,t)
(1−u)t+u(π(t)−2)k∗(s,t)
l∗(t)
+uΠ3(s). (19)
Let us estimate Π3(s). Observe that Π3(s)is the fraction
of all the potential links which join a neighbor of sand a
node which is not a neighbor (actual or potential) of s. Let us
call such a potential link an ns-link. Next we will obtain an
approximation of the number of ns-links. Let N(s)denote the
set of neighbors of s. Let {v, v0}be an ns-link with v∈N(s)
and v06∈ N(s). By definition of a potential link, there is a node
wsuch that {v, w}and {w , v0}are links. If w6∈ N(s), then there
exists a potential link between sand w. If w∈N(s), then there
exists a potential link between sand v0in contradiction with
the fact that {v, v0}is an n s-link. Therefore, the existence of an
ns-link forces the existence of a potential link emanating from
s. Summarizing, we take k∗(s,t)/l∗(t)as an approximation of
Π3(s). Hence, from (19) we get
∂k∗(s,t)
∂t=m
tk(s,t)+u(π(t)−1)k∗(s,t)
l∗(t).(20)
To close the system of differential equations for k(s,t)
and k∗(s,t), we need an expression for l∗(t), the number of
potential links in the network at time t, which is obtained in a
similar way to how we tested the consistency of (17) and using
that R(1−u)t
0k∗(s,t)ds=2l∗(t). So, integrating both sides of
(20) with respect to sfrom 0 to (1−u)t, it follows that
2d
dtl∗(t)=(1−u)k∗((1−u)t,t)+(1−u)m¯
k
+2u(π(t)−1).
D. Juher et al. / Physica D 214 (2006) 132–143 139
Since the potential degree of the node which has just been added
to the network equals the sum of the degrees of the nodes
selected at random for the attachment, we can approximate
k∗((1−u)t,t)≈m¯
k. On the other hand, from (15) and
assuming l∗(0)=0, we get that, for t0,
l∗(t)=(2m¯m−u)t+uZt
0
π(ξ ) dξ. (21)
As a final step in the model derivation, let us look for an
expression for π(t)which, recall, is defined as the expected
value of the degree distribution for nodes reached by following
apotential link chosen at random. It is not immediately obvious
how to guess an analytical expression for this average since the
relationship between the degree kand the potential degree k∗of
a node will depend on the assortative or disassortative mixing
of degrees in the network, both types of mixing being present in
growing networks [19]. Nevertheless, under a CNN mechanism
of connecting nodes, one has to expect a positive assortative
mixing, as is the case in many social networks [7,8,19,20]. In
fact, this is also what follows from the asymptotic behavior
of the solutions to (13) because, according to (10), a positive
correlation between kand k∗is always expected for ts.
It is important to realize that the predictions of the model
will strongly depend on the form of π(t). To see this, note that
from (21) it follows that, for t0, the number of potential
links l∗(t)will increase linearly with tas long as π(t)does not
depend on t. This is equivalent to saying that π(t)=α > 0
implies a constant mean potential degree k∗during the network
growth for t0. On the other hand, if π(t)=αtβ,β > 0,
or π(t)=αln t, then l∗(t)is proportional to tβ+1or to tln t,
respectively, for t0, and, consequently, k∗∼tβor k∗∼ln t.
Therefore, let us consider an assortative mixing by assuming
k∗(s,t)∝tβk(s,t),β≥0, which implies that k∗(t)∼
tβ(or k∗(t)∼ln tif we alternatively assume k∗(s,t)∝
ln t k(s,t)) because kis constant for any u∈ [0,1). In both
cases, the expected value of the potential degree distribution
for nodes reached by following a randomly chosen potential
link, k∗2(t)/k∗(t), will be proportional to tβk2(t)/kor to
ln t k2(t)/k, with k2(t)/kbeing the mean degree of a node at the
end of a randomly chosen link [10,20]. Hence, π(t)=k2(t)/k
according to the definition of π. Note that this relationship
between the degree of a node and its potential degree is exact for
tswhen u=0 because, in this limit case, k(s,t)≈mln t
and k∗(s,t)≈k2(s,t)/2, that is, k∗(s,t)∝ln t k(s,t), as can
be easily seen by solving Eqs. (17) and (20) with u=0.
To obtain an expression for k2(t)multiply (17) by k(s,t)
and, afterwards, introduce the assumption k∗(s,t)∝tβk(s,t)
and, hence, l∗(t)∝tβl(t). Then, integrating both sides
of the resulting expression with respect to sfrom 0 to
(1−u)twe obtain the following differential equation for
R(1−u)t
0k2(s,t)ds:
d
dtZ(1−u)t
0
k2(s,t)ds=(1−u)m(2¯
k+m)
+2u
¯mt Z(1−u)t
0
k2(s,t)ds.
Solving it with an arbitrary initial condition at t=1,
R1−u
0k2(s,1)ds=φ0, one obtains that
Z(1−u)t
0
k2(s,t)ds=
(1−u)m¯m(2¯
k+m)
¯m−2ut1−t2u/¯m−1+φ0t2u/¯m,2u6= ¯m
(1−u)m(2¯
k+m)tln t+φ0t,2u= ¯m.
Finally, since π(t)=k2(t)/k, from (15) we have
π(t)=
m(5m(1−u)+4u)
2(m(1−u)−u)1−t2u/¯m−1+φ0
2¯mt2u/¯m−1,u<m
m+1
m(5m(1−u)+4u)
2(m(1−u)+u)ln t+φ0
2¯m,u=m
m+1
m(5m(1−u)+4u)
2(u−m(1−u)) t2u/¯m−1−1+φ0
2¯mt2u/¯m−1,u>m
m+1
which, for t0, is approximately equal to
π(t)=
m(5m(1−u)+4u)
2(m(1−u)−u),u<m
m+1
m(5m(1−u)+4u)
2(m(1−u)+u)ln t,u=m
m+1
m(5m(1−u)+4u)
2(u−m(1−u)) +φ0
2¯mt2u/¯m−1,u>m
m+1.
(22)
The previous expression shows that the mean value π(t)
will be constant for t0 when degree fluctuations in the
network are small, i.e., under exponential regimes of the degree
distribution. Of course, this situation is the one that corresponds
to small values of usince, then, the dominant process is the
addition of new nodes to the network without any preference in
the attachment. However, when the conversion of a potential
link into an actual link becomes the dominant process, the
dynamics of the networks enters into the scale-free regime, in
which degree fluctuations are so important that π(t)tends to
infinity with the size of the network (see Fig. 3).
In summary, the model we propose for the dynamics of the
network with a growth described by the rules (i) and (ii) is given
by the Eqs. (17),(20) and (21) with π(t)given by (22).
3.3. Predictions of the model
First, since k∗(t)∼tβif π(t)∼tβwith β≥0, from the
approximation of π(t)given by (22) it follows that k∗(t)∼tβ
with 0 < β < 1 for any m
m+1<u<1; otherwise k∗(t)is con-
stant. Hence, the first prediction of the model is that the mean
potential degree of the network will be an increasing and un-
bounded sublinear function of time for large enough values of
u,while it will remain constant for small values of u and t 0.
Note that this prediction is completely different from the
one corresponding to V´
azquez’s model. Since in his model
k∗(s,t)∝k(s,t), it immediately follows from (15) that,
140 D. Juher et al. / Physica D 214 (2006) 132–143
Fig. 3. Non-normalized cumulative degree distributions in networks of 106
nodes generated from the rules (i) and (ii) with m=2. For values of uclose to
0, the dominant process during the network growth is the random addition of
new nodes to the already existing nodes. For higher values of u, the dominant
process is a sort of preferential attachment arising from the conversion of
potential links into real links.
regardless of the value of u,k∗(t)will remain constant as
the size of the network increases. In particular, for m=1,
integrating (14)2with respect to sfrom 0 to (1−u)t, it follows
that the number of potential links at time tis l∗(t)≈2(1−u)t
for t0. Hence, according to this model, k∗(t)=4 for any
u∈ [0,1)and t0.
A second prediction of the model has to do with the
heterogeneity of the network. According to the assumption
π(t)=k2/kand since k(t)is constant, our model forecasts
that the variance of the degree distribution will tend to infinity
with the number of iterations (i.e., with the size of the network)
whenever π(t)→ ∞,which occurs at least for u ≥m
m+1. This
prediction can be improved if we integrate our model in the
simplest case, namely, when π(t)is constant, and, afterwards,
we derive the corresponding stationary degree distribution to
obtain under which conditions its variance is not finite.
It is clear from (22) that, for u<m/(m+1), the model
we propose reduces to a system similar to the one analyzed in
Section 2, i.e., a system of the form
d
dtEx(t)=1
tAEx(t)
with Aa constant matrix. For u≥m/(m+1),Adepends on t.
More precisely, according to (22), we have that π(t)=
m(5m(1−u)+4u)
2(m(1−u)−u)for u∈ [0,m
m+1)and, for this range of u-values,
the model given by (17),(20) and (21) reads
∂k(s,t)
∂t=1
tm+2u(m(1−u)−u)
g(u)k∗(s,t)
∂k∗(s,t)
∂t=1
t m k(s,t)+u(5m2(1−u)+2m(3u−1)+2u)
g(u)k∗(s,t)!(23)
with g(u):= 4m3(1−u)2+mu (1−u)(5m−2)+2u2. As
in Section 2.2, once we have solved (23) by means of rescaling
the time variable with the change of variables ¯x(ln t)=k(s,t),
¯y(ln t)=k∗(s,t), we finally get that, at large times, the
solutions of (23) are potential functions of time. More precisely,
k(s,t)
k∗(s,t)∼ Evtλm(u),(24)
where Evis the eigenvector associated with the only
positive eigenvalue λm(u)of the matrix of the corresponding
autonomous linear system,
λm(u)=
qu2(5m2(1−u)+2m(3u−1)+2u)2+8mu (m(1−u)−u)g(u)
+u(5m2(1−u)+2m(3u−1)+2u)2g(u).
Note that, from (24),k∗(s,t)∝k(s,t)for t0 which is
compatible with our assumption k∗(s,t)∝tβk(s,t),β≥0,
introduced in the derivation of the model taking β=0.
Finally, since the system (23) is linear, one can proceed
along the same lines as in Section 2.4 in order to obtain the
exponent γof the stationary scale-free degree distribution, now
given by γ (u)=1+1/λm(u)for u∈ [0,m
m+1). What follows
from this computation is that, for m=1, the degree distribution
will have finite variance (i.e., γ (u) > 3) for u∈ [0,0.26)while,
for m=2, γ (u) > 3 for u∈ [0,0.4). In fact, for this range of
values of u, the degree distribution will be described either by
a potential law with γ > 3 or, when uis close to 0, by an
exponential law (see Fig. 3).
D. Juher et al. / Physica D 214 (2006) 132–143 141
Fig. 4. Evolution of the mean potential degree k∗as a function of the number of iterations (t) according to the rules (i) and (ii) of the in silico model for low values
of the probability uand for m=1,2. The final size of the network is always 106. In all the cases, k∗(t)is either constant for t0 or slightly increasing. The solid
line corresponds to the predicted value of k∗.
From the definition of π(t), this result implies that a constant
k∗has to be expected for u<0.26 when m=1, and for
u<0.40 when m=2, which is closer to the output of
the simulations presented in Figs. 4 and 5than the condition
u<m/(m+1).
4. Discussion and conclusions
In the first part of the paper we presented a mathemat-
ical analysis of a nonlinear model presented by V´
azquez
in [1] for modeling the dynamics of networks based on the
142 D. Juher et al. / Physica D 214 (2006) 132–143
Fig. 5. Evolution of the mean potential degree k∗as a function of the number of iterations (t) according to the rules (i) and (ii) of the in silico model with u=0.3,0.5
and m=1,2. The final size of the network is always 106. We plot the log–log representation of the evolution of k∗(t), which clearly shows that k∗(t)∼tβwith
0< β < 1.
connecting-nearest-neighbor mechanism. Two rates of transi-
tions between states of pairs of nodes define the model: the
rate of transitions from the disconnected to the connected state,
given by µ0/t2, and the rate of transitions from the potential
state to the connected one, given by µ1/t.
The main conclusion of this analysis is that only when
µ0≤1+1/µ1is the asymptotic growth rate of the degree
and potential degree of a node s, which is given by
µ1 −1
2+s1
4+µ0
µ1!,
equal to the one predicted in the original treatment of the system
(1) (see Eqs. (32) and (33) in [1]), where the nonlinear model
is reduced to a linear one by neglecting those terms in the
RHS of (1) with a factor 1/t2. Conversely, when the relation
between µ0and µ1is the opposite one, i.e., µ0>1+1/µ1,
the asymptotic growth rate is equal to 1. In this case, the linear
reduction of (1) given by Eq. (31) in [1] is not correct since it
neglects the contribution of nonlinear terms that now becomes
essential for controlling the growth of the degrees, predicting
growth rates of xiand yigreater than 1, which clearly are not
admissible in the context of the growing networks considered
here.
The consequences on the network configuration of the
previous situations become clear if one realizes the potential
neighbors of a node defined as in [1] are, in fact, the second
neighbors of this node. When µ0>1+1/µ1, the degree and
the potential degree of a node increase linearly with time for
ts. Hence, the fraction of nodes that are first or second
neighbors of an old node (i.e., (xs+ys)/t) is always strictly
positive—even when the size of the network tends to infinity,
which implies a strongly connected network. Conversely, when
µ0≤1+1/µ1, this fraction tends to zero due to the sublinear
growth of xs+ys, leading to a less connected network with a
finite expected value of the degree distribution.
In the second part of the paper we derived the model given
by Eqs. (17),(20) and (21) with π(t)given by (22). The aim
of this model was to analyze the dynamics of the network
growth based on the connecting-nearest-neighbor mechanism
according to a simulation model introduced in [1]. The crucial
point in the derivation is how to estimate π(t), the mean degree
D. Juher et al. / Physica D 214 (2006) 132–143 143
of a node at the end of a potential link chosen at random. Here
we have assumed an assortative mixing, as is usually the case in
social networks, to justify a proportionality between the degree
and the potential degree of a node.
The derived model predicts a change of the behavior of the
mean potential degree k∗(t)when the value of u, the probability
of converting a potential link into an actual link, increases (see
rules (i) and (ii) at the beginning of Section 3). This change is
not forecast by the model used in [1] for analyzing the network
growth of this simulation model, which predicts, for m=1 and
t0, k∗(t)=4 for any value of u∈ [0,1).
To check which of the previous predictions as regards the
behavior of k∗(t)is the correct one, networks of size 106were
generated for different values of the probability u. In Figs. 4
and 5we show the evolution in time of k∗(t)for some of these
networks. From the simulations it clearly follows that
•k∗(t)remains constant for small values of u,
•for larger values of u,k∗(t)is a potential function of twith
an exponent lower than 1.
Therefore, there is an overall good agreement between the
observed and the theoretically expected behavior of k∗(t),
although the transition from a constant to an increasing k∗(t)
occurs at values of uthat are smaller than those predicted by our
model. More precisely, from (21) and (22), and the stationary
degree distribution obtained at the end of the previous section,
it follows that k∗(t)will be constant and equal to 3u2−5u+4
(1−2u)(1−u)
for any u<0.26 if m=1, while the critical value of useems
to be around 0.1 in the simulations. Similarly, for m=2, k∗(t)
is expected to be constant and equal to 16(2−u)(1−u)+2u2
(2−3u)(1−u)for any
u<0.4, while the observed critical value of uis around 0.2.
In both cases, the predicted values of k∗are close to values
observed in networks generated with small values of u(Fig. 4).
For higher values of u, the accuracy of the prediction decreases.
This is probably because of the occurrence of higher order
correlations between the degrees of connected nodes, which
have not been considered in the model derivation.
Moreover, comparing Figs. 3 and 4, it becomes clear that the
change in the behavior of k∗(t)takes place when the degree
distribution enters into a scale-free regime, which is again in
agreement with the predictions of the model since, from (21)
and the definition of π(t), it follows that k∗(t)∼k2(t)for
t0.
Such a discrepancy between the predictions of the two
models as regards the number of potential links in the network,
that is, as regards the topology of the resulting network,
strongly suggests that one has to look for network properties
other than the degree distribution in order to validate a growing
network model. In fact, it is clear that any linear system of
differential equations for the evolution of the degree of a
node and of another feature of the same node (such as the
potential degree) will amount to scale-free distributions under
the assumption of a regular addition of nodes to the network.
So, such a property cannot be used alone to test the validity of
alternative models.
Acknowledgement
This work was partially supported by the grant MTM2005-
07660-C02-02 of the Spanish government.
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azquez, Growing network with local rules: Preferential attachment,
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