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Microdynamics in stationary complex networks
Aurelien Gautreau,1 Alain Barrat,1, 2, 3 and Marc Barthe´lemy4, 5
1Laboratoire de Physique The´orique (CNRS UMR 8627), Universite´ de Paris-Sud, 91405 Orsay, France
2Centre de Physique The´orique (CNRS UMR 6207),
Luminy Case 907, 13288 Marseille Cedex 9, France
3Complex Networks Lagrange Laboratory, ISI Foundation, Torino, Italy
4CEA-De´partement de Physique The´orique et Applique´e, 91680 Bruyeres-Le-Chatel, France
5Centre d’Analyse et de Mathe´matique Sociales (CAMS,
UMR 8557 CNRS-EHESS), Ecole des Hautes Etudes en Sciences Sociales,
54 bd. Raspail, F-75270 Paris Cedex 06, France
(Dated: November 6, 2008)
Many complex systems, including networks, are not static but can display strong fluctuations
at various time scales. Characterizing the dynamics in complex networks is thus of the utmost
importance in the understanding of these networks and of the dynamical processes taking place on
them. In this article, we study the example of the US airport network in the time period 1990−2000.
We show that even if the statistical distributions of most indicators are stationary, an intense activity
takes place at the local (‘microscopic’) level, with many disappearing/appearing connections (links)
between airports. We find that connections have a very broad distribution of lifetimes, and we
introduce a set of metrics to characterize the links’ dynamics. We observe in particular that the
links which disappear have essentially the same properties as the ones which appear, and that links
which connect airports with very different traffic are very volatile. Motivated by this empirical
study, we propose a model of dynamical networks, inspired from previous studies on firm growth,
which reproduces most of the empirical observations both for the stationary statistical distributions
and for the dynamical properties.
PACS numbers:
INTRODUCTION
Despite the presence of stable statistical regularities at
the global level, many systems exhibit an intense activity
at the level of individual components, i.e. at the ‘micro-
scopic’ level. An important illustration of this fact was
recently put forward by Batty [1] in the case of city popu-
lations. Indeed, even if the population Zipf plots display
negligible changes in time, the same city can have very
different ranks in the course of history. Similarly, many
other systems, in particular occurring in human dynam-
ics studies, present simultaneously stationary statistical
distributions and strong time fluctuations at the micro-
scopic level, with activity bursts separated by very het-
erogeneous time intervals [2, 3, 4, 5]. These systems thus
challenge us with the fundamental puzzle which consists
in reconciling an important dynamical activity occurring
at the local level on many timescales and the emergence
of stable distributions at a macroscopic level which can be
maintained even when the external conditions are highly
non-stationary [6]. For instance, the dynamics of the
rank is not consistent with processes such as preferential
attachment [7] where the rank is essentially constant in
time.
These issues naturally apply to the case where complex
systems are structured under the form of large networks.
In most recent studies, these networks have been consid-
ered as static objects with a fixed topology. However,
their structure may in principle evolve, links may ap-
pear and disappear. Such topological fluctuations have
important consequences: many dynamical processes take
place on complex networks [8, 9, 10, 11], and a non trivial
interplay can occur between the evolutions of the topol-
ogy and of these dynamical processes. The structure of
the network strongly influences the characteristics of the
dynamical processes [11], and the topology of the net-
work can simultaneously be modified as a consequence of
the process itself. In this framework, recent studies have
been devoted to simple models of coevolution and adap-
tive networks [12, 13, 14, 15, 16]. Another illustration of
the importance of taking into account the dynamics of
the network is given by concurrency effects in epidemi-
ology [17]. Indeed, while a contact network is usually
measured at a certain instant or aggregated over a cer-
tain period, the actual spread of epidemics depends on
the instantaneous contacts. In such contexts, it is thus
crucial to gain insights into the dynamics of the network,
possibly by putting forward convenient new measures and
to propose possible models for it.
These considerations emphasize the need for empiri-
cal observations and models for the dynamics of complex
networks, which are up to now quite scarce. In this pa-
per, we study the case of the US airport network (USAN)
where nodes are airports and links represent direct con-
nections between them. It is indeed possible to gather
data on the time evolution of this network [18] (see also
[19] for a study of the yearly evolution of the Brazilian
airport network) which represents an important indica-
tor of human activity and economy. Moreover, air trans-

2portation has a crucial impact on the spread of infectious
diseases [20, 21], and it would be interesting to include its
dynamical variations in large-scale epidemiological mod-
eling. We first present empirical measures on the dynam-
ics of the USAN. In particular, we provide evidence for
the large-scale statistical regularity of many indicators,
and we also define convenient metrics that enable us to
characterize the small-scale dynamical activity. We then
propose a model, based on simple but realistic mecha-
nisms, which reproduces most empirical observations.
EMPIRICAL OBSERVATIONS: STABLE
STATISTICAL DISTRIBUTIONS IN A
FLUCTUATING SYSTEM
We analyze data available from the Bureau of Trans-
portation Statistics [18]. These data give the number
of passengers per month on every direct connection be-
tween the US airports in the period 1990 − 2007. We
limit ourselves to the period 1990 − 2000 during which
the data collection technique is consistent. We obtain
12× 11 = 132 weighted, undirected [22] networks where
the nodes are airports, the links are direct connections,
and the weights represent the number of passengers on
a given link during a given month. We denote by w the
weight of a link, by k the degree (number of neighbors)
of a node, and by s its strength, equal to the sum of the
weights of the issuing links [23] (in the air-transportation
case, the strength gives thus the total traffic handled by
each airport).
Figure 1 displays the cumulative distributions of de-
grees, weights and strengths at four different times.
These distributions are broad, as already shown in pre-
vious studies [23, 24, 25], highlighting the strong hetero-
geneities present in the air transportation network, for
both the topology and the traffic. Figure 1D moreover
shows the dependence of the strength s of an airport on
its number of connections k, with a clear non-linear be-
havior denoting a strong correlation between weights and
topology [23, 25]. Interestingly, Figure 1 clearly shows
that the distributions of degrees, weights, and strengths
measured at different times are identical (we have ob-
tained the same distributions at other dates). These dis-
tributions are therefore stationary even if, as we will show
later, non trivial dynamics occur continuously.
The first and simplest evidence for the presence of a
dynamical evolution in the network is displayed in Fig.
2A, which shows the total traffic T (t) (equal to the sum
of the weights of all links) as a function of time. When
the seasonal effects are averaged out, the data can be
fitted, as often assumed in economics, by an exponen-
tial growth T (t) = T (0) exp(t/δ) with δ ≈ 312 months
(δ ≈ 25 years). Note that the data can also be fitted
linearly, due to the large value of δ. We also observe
similar growth (Figure 3) with seasonal fluctuations of
100 101 102k
10-3
10-2
10-1
100
P >
(k)
100 102 104w
10-4
10-3
10-2
10-1
100
P >
(w
)
01/1992
06/1996
12/1998
09/2000
100 102 104 106
s
10-3
10-2
10-1
100
P >
(s)
100 101 102
k
100
102
104
106
108
s
A B
DC
FIG. 1: Characteristics of the US airport network mea-
sured at four different dates: 01/1992, 06/1996, 12/1998, and
09/2000. (A) Cumulative distribution of degrees; (B) Cu-
mulative distribution of weights; (C) Cumulative distribution
of strengths; (D) Strengths versus degrees for the year 2000
(circles: raw data, squares: average strength for each degree
value). The dashed line is a power law with exponent 1.6.
the total number N(t) of connected airports, the total
number L(t) of links, the average weight and the node
strength. The fits give the same growth rate for N(t)
and L(t) (roughly half the growth rate of T (t)), and the
average degree 〈k〉 = 2L(t)/N(t) has small fluctuations
(±3) around a constant value (≈ 15), as shown in Fig.
2B, over the 10 years period under study, while the aver-
age weight 〈w〉 grows exponentially with a typical time
of order 2δ.
THE DYNAMICS AT THE MICROSCOPIC
LEVEL
We now study in detail the dynamics at the micro-
scopic level, i.e. the evolution of single links. We de-
note by wij(t) the weight of the link between nodes i
and j at time t (in months) and by ηij(t) = [wij(t +
1)− wij(t)]/wij(t) the relative variation of wij from one
month to the next. Figure 2C shows the distributions of
ηij(t) for all links present in the network both at t and
t+ 1, for all months in the 10 years dataset under study
(period January 1990-December 2000), as well as for a
single month (t = May 1995). The fact that the distri-
butions can be superimposed leads to the conclusion that
the weights’ evolution can be modeled by the form
wij(t + 1) = wij(t) (1 + η) (1)
where the multiplicative noise η is a random variable
whose distribution does not depend on the link (i, j) nor
on the time t. The inset of Fig. 2C moreover shows that

301/90 01/92 01/94 01/96 01/98 01/00
Month
4×107
5×107
T(
t)
01/90 01/92 01/94 01/96 01/98 01/00
14
16
〈k〉
0 5 10
η
10-3
10-2
10-1
100
P(
η)
1 10
10-2
100
A
B
C
FIG. 2: (A) Total traffic on the US airport network versus
time. The symbols represent the annual traffic and the dashed
line is an exponential fit with time scale δ ≈ 312 months. (B)
Average degree 〈k〉, approximately constant and of order 15
(dashed line). (C) Distribution of the relative weights incre-
ments η = (w(t + 1) − w(t))/w(t). The full line corresponds
to the distribution obtained over the 11 years under study.
Circles correspond to one single month (May 1995). In the
inset, we show the tail of the distribution of η, with a power
law fit P (η) ∼ η−ν , giving ν = 1.9 ± 0.1 (dashed line).
01/90 01/94 01/98
320
360
400
N(
t)
01/90 01/94 01/98 1.0×10
4
2.0×104
〈w
〉(t)
01/90 01/94 01/98
2500
3000
E(
t)
01/90 01/94 01/98
2×105
3×105
〈s〉
(t)
B
CA
D
FIG. 3: Time evolution of (A) the number of nodes N(t), (B)
the number of links L(t), (C) the average link weight 〈w〉(t)
and (D) the average node strength 〈s〉(t) in the US airport
network from January 1990 to December 2000. Dashed lines
are exponential fits.
the distribution of η is broad, with a power law behav-
ior P (η) ∝ η−ν (with ν ≈ 1.9 ± 0.1) for η > 0. The
broadness of this distribution indicates that most rela-
tive increments are small but that sudden and large vari-
ations of the weights can be observed with a small but
non negligible probability.
The distribution of η is truncated at −1, since this cor-
responds to a weight going to 0, i.e. to the disappearance
of a link. Links indeed can be created or suppressed be-
tween airports, and in fact the number of link creation
events is 4 × 104 for the 11 years period under study,
for a total number of links in the 132 networks close to
3 × 105. This result immediately raises the question of
the lifetime τ of links. As shown in Fig. 4A, the dis-
tribution of τ is very broad, with a power law behavior
P (τ) ∼ τ−α with α = 2.0 ± 0.1. Some comments are
100 102 104 106
s
max /smin
10-8
10-4
100
P(s
m
a
x
/s m
in
)
1 month 1/2 year 1 year 2 years 10 yearsτ
10-4
10-2
100
P(
τ)
Slope -2
10-1
100
f d ,

f a fd
f
a
A
B
FIG. 4: (A) Lifetime distribution of links (in months). Links
which were existing at the beginning of the measure (Jan-
uary 1990) and still present at the end (December 2000) are
discarded. The full line is a power law fit with an exponent
−2.0±0.1. (B) Fraction fd (open circles) of disappearing links
and fa (pluses) of appearing links as a function of the ratio
smax/smin of the strengths of their extremities. The scale
is on the left-hand y-axis. We also show the logarithmically
binned reference distribution P (smax/smin) (line above the
shaded area, scale on the right-hand y-axis).
in order. First, we consider in this distribution only the
links which appear and disappear during the period un-
der study. This is necessary since we cannot know the
real lifetime of a link which is already present at the start
of the period or still present at the end. Second, while
the most probable value for τ is small, which implies
that new links are the most fragile, the distribution ex-
tends over all available timescales: links of an arbitrary
age may disappear. This indicates a non trivial dynam-
ics with appearance/disappearance of both ‘young’ and
‘old’ links. This strong heterogeneity of lifetimes is in
line with other results about human activity [3], where
it has been shown to have a strong impact on dynamical
processes [26]. It is therefore important to characterize
and incorporate it into models of dynamically evolving
complex networks.
These results show that, behind the stability of the
statistical characteristics of the USAN, incessant micro-
scopic rearrangements occur. We now propose a system-

4atic way to characterize the corresponding fluctuating
connections, whose importance stem from the fact that
they induce changes in the topology of the network. Each
link (i, j) can be characterized by a certain number of
quantities such as its weight wij , the strengths of its ex-
tremities si and sj , etc. It is usual to consider the distri-
butions of these quantities over the whole network, and
we will consider these distributions as reference (see Fig.
1). In addition, we propose to focus at each time t on the
links which appear (or disappear), to study the distribu-
tions of these links’ characteristics, and to compare them
with the reference distributions. For instance, if Nt(w) is
the number of links with weight w at time t, and Ndt (w)
is the number of such links that disappear between t and
t + 1, we measure the fraction of links of weight w that
disappear at time t,
fd(w) =
Ndt (w)
Nt(w)
. (2)
We also define the number Nat (w) and fraction fa(w) of
links of weight w that appear at t. Similarly, fd and fa
can be measured for other links characteristics as we will
investigate. A priori, all these quantities depend on the
measurement time t. We have already seen that the ref-
erence distributions are stationary (Fig. 1). Strikingly,
we observe that the fractions fd and fa display as well a
stationary behaviour (Fig. 5), even if they clearly high-
light a strong dynamical evolution. In the following, we
will therefore drop any t index and measure fd and fa
averaged over the whole period under study.
100 102 104 106
s
max
/s
min
10-1
100
f a 1992
1995
1998
all years
FIG. 5: Fraction fa of appearing links in the USAN as a func-
tion of the ratio smax/smin of the strengths of their extremi-
ties. Circles, squares and diamonds correspond to the data of
three distinct years, while the pluses represent the data aver-
aged over the whole 10-years time period. This figure clearly
illustrates the stationarity of fa.
The measure of fd(w) and fa(w) indicate that most
links have a small weight just before they disappear or
just after their birth, which is not a surprise. How-
ever, fd and fa are broad, extending on several orders
of magnitude of w values: appearing and disappearing
connections occur with non negligible probabilities even
at strong weights. We also note a strong similarity be-
tween fd and fa, due to the large number of links with
lifetime of order a few months, during which no strong
evolution of w occurs. A more detailed analysis shows the
presence of two regimes in fd(a): for links with w < 102
passengers per month, fd and fa present rather large val-
ues close to 0.8. For w > 102, these fractions decrease
slowly: also links with large weights can appear or dis-
appear. We also observe that for w > 103 there are more
links which appear than which disappear, an effect which
is consistent with the increase of the total traffic.
As previously mentioned, a similar analysis can be
carried out for various links’ characteristics; particu-
larly relevant quantities include the traffic of the air-
ports located at both ends of the link. In the fol-
lowing we denote by smax(l) = maxl=(i,j) (si, sj) and
smin(l) = minl=(i,j) (si, sj) the larger and smaller traf-
fic of the extremities of a link l = (i, j). A measure of
the importance of the link for i and j is given by w/smin
and w/smax. For instance, if w/smin is small, the link
carries only a small fraction of i’s and j’s traffic; on the
contrary, a large w/smax indicates that the link is impor-
tant for both its extremities. The study of fd(a)(w/smin)
and fd(a)(w/smax) shows that most links which disap-
pear/appear display small values of these ratios, of order
w/smin < 10−3 and w/smax < 10−4. This means that
most of these links have a small importance for the air-
ports to which they are attached. For larger values of
w/smin(max), the ratios fd and fa decrease, from ∼ 0.7
to ∼ 10−2, and surprisingly increase again (from ∼ 10−2
to ∼ 10−1) for w/smin > 10−1 and w/smax > 10−2. This
phenomenon corresponds to links which are very impor-
tant for some airports, the extreme case being airports
with a single connection (these airports have thus usually
a small strength).
Finally, we also consider the ratio smax/smin of the
traffic of the links extremities. This quantity indicates
indeed how similar the airports connected by the link
are, in terms of traffic. We plot in Fig. 4B the fractions
fd(a)(smax/smin) of links which disappear (appear) as a
function of smax/smin. On this figure we also show the
reference probability distribution P (smax/smin) which
displays a broad behavior: most links connect airports
of similar importance, but the ratio smax/smin varies
over 6 orders of magnitude, and a non negligible frac-
tion of links connect very different airports. Interest-
ingly, fd(a)(smax/smin) displays two different regimes.
For smax/smin < 103, small values of fd are obtained:
links which connect airports of similar, or not too dissim-
ilar, sizes, are rather stable. In the opposite case when
smax/smin > 103, the fraction fd increases rapidly to
reach another plateau, at values of order 0.7− 0.8. This

5last regime corresponds to links connecting airports with
very different traffic, which turn out to be the most frag-
ile and to have a short lifetime.
We can now summarize the results of our empirical
observations, obtained through the analysis of the tools
introduced in Eq. 2: (i) The links which disappear
have essentially the same properties as the ones which
appear. (ii) The disappearing/appearing links have a
weight which is low on average but broadly distributed:
large weights links may appear or disappear with a non
negligible probability. (iii) Most disappearing links have
small weights with respect to the traffic of their extremi-
ties, but links appear or disappear in the whole range of
w/s. (iv) Links which connect airports with very differ-
ent traffic are very volatile. (v) The lifetime of links is
broadly distributed and covers all available timescales.
The set of measures we have presented, while not ex-
haustive, is able to give a clear characterization of the
dynamics of the network under study [32]. They are also
easily applicable to any network undergoing topological
changes, and can be generalized to include other links
characteristics.
The results of the empirical analysis may moreover
serve as guidelines in the elaboration of a model for dy-
namically fluctuating networks. In particular and in con-
trast with most models found in the literature, topolog-
ical modifications of the network result here from the
stochastic evolution of weights.
A MODEL FOR DYNAMICAL NETWORKS
Using the results of the empirical analysis of the air-
port network as guidelines, we now propose a model for
dynamically fluctuating networks able to reproduce the
main features observed for the USAN, and which high-
lights important features of dynamical networks model-
ing. We consider simple ingredients that can easily be ex-
tended with more detailed rules, and can therefore serve
as a modeling basis in many other fields where the dy-
namics of weights and links is essential. In this model,
topological modifications of the network result from the
stochastic evolution of weights.
We start from ideas developed in [27, 28, 29] to model
firm growth through a process based on multiplicative
growth of subunits together with fusion/creation rules.
In our framework, we consider airports (nodes) and con-
nections (links) instead of firms and subunits. The equiv-
alent of a firm’s size is then given by the traffic of the air-
port as measured by its strength, and the subunits sizes
correspond to the traffic on each link. An essential differ-
ence distinguishes our model from the firm growth model
where the various firms undergo independent evolutions:
here, each node is connected to many others by links
whose weights evolve randomly, so that the evolution of
the airports sizes are correlated.
Let us present the details of the modeling framework.
We start (at time t = 0) from an initial network com-
posed of N0 and L0 links with L0 ≈ N0 (we have checked
that the initial conditions do not influence the results).
At each time step t, we first compute for each link (i, j)
with weight wij(t) a random increment
δwij(t) = wij(t)η , (3)
where η is a random variable drawn from a distribution
independent from time and from the pair (i, j), and which
may a priori take values in ] − 1,+∞[. For 〈η〉 > 0, the
total traffic will on average grow exponentially. For the
sake of simplicity we will choose for η a Gaussian dis-
tribution (truncated at −1), with variance σ2 [34]. The
weights’ increments govern the evolution of the network’s
topology: depending on the values of δwij , the nodes i
and j can either update the weight of (i, j), delete it or
create new links towards other nodes. More precisely,
each airport i has a threshold value ss(i) which sets a
criterium of viability for a connection: if a link’s weights
drops below this threshold, the airport i does not con-
sider the link anymore as interesting and removes it. For
simplicity, we take thresholds independent from time and
uniform: ss(i) = 1.0 for all i. The detailed evolution rules
are as follows:
• (1) If δwij(t) < 0, i and j test each the vi-
ability of the connection (i, j). If wij(t) +
δwij(t) < max (ss(i), ss(j)), the link disappears
and its weight is uniformly redistributed over the
other connections of i and j. In the opposite
case, wij(t)+δwij(t) > max (ss(i), ss(j)), the link’s
weight is simply updated: wij(t + 1) = wij(t) +
δwij(t).
• (2) If the weight increment δwij(t) is positive, we
assume that i and j have contributed equally to it
and can decide each on how half of it should be
used: If δwij(t) > ss(i), with probability pf node
i will use its part δwij(t)/2 of the increment to
create a new link (i, ℓ) with weight wiℓ = δwij(t)/2.
With probability 1 − pd, ℓ is an existing airport
chosen at random, and with probability pd it is a
new node. pd therefore governs the rate of growth
of the number of nodes. With probability 1 − pf ,
node i simply increases the weight wij of δwij(t)/2.
Node j then chooses independently either to create
a new link (j, k), or to increase the weight wij by
an amount equal to δwij(t)/2.
• (3) If 0 < δwij(t) < ss(i), node i increases the
weight of (i, j) of δwij(t)/2. The same procedure is
applied to node j.
The rules (1)-(3) express the concept that the evolu-
tion of the traffic governs the topological modifications

6of the network. If a weight becomes too small, the corre-
sponding connection will be stopped. On the other hand
if it grows too fast, new connections can be created. The
quantity pf determines the rate of new connections. If
pf is close to one, as soon as an increment δw is large
enough a new link will be created, which in turn will
limit the growth of weights since they are used to create
new connections. In the opposite case of small pf , the
number of links will grow very slowly but the weights
will reach more easily large values. At each time step,
the total traffic T (t) is multiplied on average by 1 + 〈η〉
leading to an exponential growth T (t) ∝ exp(t/d) with
d = 1/〈ln(1 + η)〉. The number of nodes and links also
grow in time, and their simultaneous growth, controlled
by pf and pd, results in an average degree 〈k〉 which
fluctuates around a constant value, function of the pa-
rameters pf , σ, 〈η〉, and pd. For instance, for larger pd,
N(t) grows faster and 〈k〉 is smaller.
The model rules can easily be modified to incorporate
other elements, such as preferential attachment mech-
anisms or random distributions of the threshold values
ss(i). While we will focus here on the simplest ver-
sion as described above, we have also considered vari-
ants (i) in which the link’s relevance is tested if δwij <
max (ss(i), ss(j)) (instead of the condition δwij < 0), or
(ii) where the weight of deleted links is re-distributed at
random, or (iii) only one new link can be created, ei-
ther from i or j. The conclusion is that the qualitative
features are not modified, showing that the simulation
results presented below are robust with respect to such
changes. We have also simulated the case pd = 0 in which
no new nodes are inserted, N(t) = N0. In this case, the
global increase of traffic leads at large time to a fully con-
nected network, but during a long time, it remains sparse
(〈k〉 ≪ N(t)) and the same results are again obtained in
this regime.
Figures 6 and 7 summarize some results of our numer-
ical simulations of the dynamical network model. Al-
though the network evolves with many links creations
and deletions, the distributions of degrees, weights, and
strengths display a remarkable stability, as shown in Fig-
ure 6 for N(t) growing from 104 to 105. All these distribu-
tions are broad, consistently with empirical observations,
and the non-linear behavior of the strength versus de-
gree is reproduced as well. Interestingly, this behavior (a
power law with an exponent of the order 1.4, see Fig. 6D)
emerges here as a result of the stochastic dynamics with-
out any reference to preferential attachment mechanisms
combined with spatial constraints [25] or with link addi-
tions between nodes [30, 31].
While many network models are able to produce broad
degree and strength distributions, the focus of this paper
lies in the small-scale dynamical aspects. We show in Fig.
7A that the lifetime distribution of the links is broad, as
in the USAN case, and we report in Fig. 7B the behavior
of fd(smax/smin). Strikingly, our model reproduces the
10-1 100 101 102k/〈k〉
10-4
10-2
100
P >
(k/
〈k〉
)
10-2 100 102 104w/〈w〉
10-4
10-2
100
P >
(w
/〈w
〉)
N=104
N=5.104
N=105
N=1.5.105
10-1 100 101 102 103
s/〈s〉
10-4
10-2
100
P >
(s/
〈s〉
)
10-1 100 101 102
k/〈k〉
10-2
100
102
104
s/〈
s〉
A
C
B
D
FIG. 6: Model simulation obtained with the parameters
pd = 0.01, pf = 0, 1, σ = 0.0225 and 〈η〉 = 10−3. Cumu-
lative distributions obtained at different times of the network
growth (N = 104, 5.104, 105 and 1.5.105) of: (A) normal-
ized degrees P (k/〈k〉), the dashed line represents a power law
with exponent −2.5; (B) normalized weights P (w/〈w〉), the
dashed line represents a power law of exponent −1.5; (C) nor-
malized strength P (s/〈s〉), the dashed line represents a power
law with exponent −1.5. (D) Strength versus degree. Circles
represent s(k) for each node; full squares represent the same
data, averaged for each k, and binned logarithmically. This
measure is done for N = 105 and the dashed line represents
a power law with exponent 1.4.
empirical behavior shown in Fig. 4B, with two different
plateaus at small and large smax/smin. Other properties
of the appearing or disappearing links coincide in the
model with the empirical results, such as the fact that
most disappearing links have a small weight, or the non-
trivial shape of fd(a)(w/smin(max)), with a decreasing fd
for w/s > 0.01, and an increase at w/s > 0.1.
In summary, the simple assumptions on which our
model is based yields stationary non-trivial emergent
properties such as broad distributions and nonlinearities,
together with an active local dynamics of links occur-
ing on all time scales, and whose characteristics repro-
duce the empirical findings concerning the USAN’s mi-
croscopic dynamics.
DISCUSSION
The question of the dynamical evolution of networks
is crucial in the study of many dynamical processes and
complex systems. If the time scales governing the dy-
namics of the network and of the process taking place on
it are comparable, one can indeed expect a highly non
trivial behavior, which in principle could be very differ-
ent from the static network case. In this article, we have
used as a case study the US airline network, and we have

7100 101 102 103
s
max /smin
10-4
100
P(s
m
a
x
/s m
in
)
100 101 102 103τ
10-6
10-4
10-2
100
P(
τ)
10-2
10-1
100
f d
fd
A
B
FIG. 7: Model simulation with the same parameters as in Fig.
6. (A) Lifetime distribution P (τ ) displaying a broad behav-
ior. A power law fit gives an exponent of order −1.1 ± 0.1.
(B) Fraction of disappearing links fd versus smax/smin (cir-
cles). We also represent the logarithmically binned reference
distribution of smax/smin (line above shaded ares).
shown that it exhibits stationary distributions despite the
incessant creation and deletion of connections on broadly
distributed timescales. We have introduced a set of mea-
sures in a systematic way in order to characterize this
dynamics. Finally, we have proposed a model based on
simple assumptions which reproduces the main empirical
features, both for stationary and local dynamical prop-
erties.
The coexistence of stationary distributions and strong
microscopic activity taking place at very different
timescales occurs in many different systems and our
model can provide a framework that can easily be ex-
tended and serve as a basis for further and more detailed
modeling. For instance, we have observed that a bimodal
distribution of the thresholds ss(i) for the deletion of a
link results in the following picture: nodes with small
ss have typically a large degree, but are connected to
weak links, while nodes with large ss reach a smaller
number of stronger connections. This behavior does not
correspond to infrastructure networks such as the USAN
but could describe social behavior where individuals with
many connections do not have intense (i.e. with large
weight) relations. In these perspectives, the present work
should stimulate further studies on the coexistence of dy-
namics at different scales and on the impact of network
dynamics on different processes.
Acknowledgements: We thank V. Colizza and J.J. Ra-
masco for a careful reading of the manuscript and inter-
esting suggestions.
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