Response of degree-correlated scale-free networks to stimuli
Response of degree-correlated scale-free networks to stimuli
Sheng-Jun Wang,1 An-Cai Wu,1 Zhi-Xi Wu,1 Xin-Jian Xu,2 and Ying-Hai Wang1∗
1Institute of Theoretical Physics, Lanzhou University, Lanzhou Gansu 730000, China
2Departamento de Fı́sica da Universidade de Aveiro, 3810-193 Aveiro, Portugal
(Dated: August 24, 2021)
The response of degree-correlated scale-free attractor networks to stimuli is studied. We show that degree-
correlated scale-free networks are robust to random stimuli as well as the uncorrelated scale-free networks, while
assortative (disassortative) scale-free networks are more (less) sensitive to directed stimuli than uncorrelated
networks. We find that the degree-correlation of scale-free networks makes the dynamics of attractor systems
different from uncorrelated ones. The dynamics of correlated scale-free attractor networks result in the effects
of degree correlation on the response to stimuli.
PACS numbers: 89.75.Hc, 87.18.Sn, 05.50.+q, 05.40.-a
Many complex systems have the ability to react to low lev-
els of special stimuli, whereas, they can maintain their state
when exposed to high levels of other irrelevant stimuli [1]. If
we take the units of response as nodes and the interactions be-
tween responding units as edges, the structure of some these
systems can be described as complex networks. In neural net-
works or social networks, for example, the nodes are individ-
ual neurons or persons. It is an interesting problem that how
one system have both the sensitivity to the right stimuli and
robustness in the face of the wrong one. And the problem is
also important for designing large artificial complex systems.
The source of the ability of networked complex systems to in-
corporate the two complementary attributes have been investi-
gated using network models. It was shown that the power-law
shape degree distributions of networks give rise to the sensi-
tivity and robustness in a system [1].
The topology of real networks is also characterized by de-
gree correlation [2, 3, 4]. In a network with degree correla-
tion, there exist certain relationships between network nodes.
The degree correlations are often named respectively as “as-
sortative mixing”, i.e. a preference for high-degree nodes
to attach to other high-degree nodes, while “ disassortative
mixing” — high-degree nodes attach to low-degree ones [4].
It has been pointed out that the existence of degree correla-
tions among nodes is an important property of real networks
[5, 6, 7, 8, 9, 10, 11, 12, 13, 14]. The percolation [4] and
disease spreading [15] on correlated networks have been stud-
ied. And more effects of degree correlation on network struc-
ture and function have attracted attention [16, 17, 18]. There-
fore, the extension of previous results for uncorrelated net-
work model about responding to stimuli is necessary.
In this paper, we study the response of degree-correlated
scale-free networks to stimuli following the work contributed
by Bar-Yam and Epstein [1]. Numerical investigation reveals
that the dynamical process of the evolution of attractor sys-
tems on correlated scale-free networks is different from un-
correlated networked systems. The special dynamics of cor-
related attractor systems result in the different responding
behavior from uncorrelated systems. The degree-correlated
∗Electronic address: yhwang@lzu.edu.cn
scale-free network is robust in the face of wrong stimuli as
uncorrelated networks. In assortative networks, the sensitiv-
ity to right stimuli is enhanced. While in the disassortative
networks the sensitivity to right stimuli is weaker than uncor-
related networks. And, the relation between the sensitivity to
stimuli and the degree of correlation is not monotonic.
We consider the method for modelling the response of com-
plex systems proposed in [1]. We use a model of attrac-
tor networks [19, 20], where the node states si = ±1, i ∈
{1, · · · , N} are binary. The state of the system is the set of
node states {si}. The dynamical equations of the attractor
system are
si(t+ 1) = sign(
Jijsj(t)), (1)
with symmetric influence matrix Jij . Using the Hebbian im-
printing rule
Jij =
j , (2)
we can set the states {sαi }α=1,··· ,n as the stable states of the
network dynamics (attractor). cij is the entry of the symmet-
ric adjacent matrix which is equal to 1 when node i connects
to node j, and zero otherwise. An attractor is stable to pertur-
bation and thus can represent a functional state of systems. In
simulations, we randomly choose two functional states of the
system {sαi }α=1,2, and the influence is Jij =
α=1 cijs
External stimuli are modelled by changing the signs of a spec-
ified set of nodes. When the states of some nodes are flipped,
the system either evolves back to its initial state or switches
to other stable system states. The response of networked sys-
tems is described as a process of switching between attractors.
The size of the basin of attraction, the number of nodes whose
states can be changed before the dynamics of the network fails
to bring the system back to its original state, indicates the de-
gree of stability of the system. We calculate the size of the
basin of attraction in different cases of stimuli to reveal the
sensitivity and robustness of the network model.
Generally, degree-correlated networks can be generated
from uncorrelated ones by means of reshuffling method pro-
posed in [5]. Starting from a given network, at each step two
http://arxiv.org/abs/0704.1849v1
edges of the network are chosen at random. The four nodes
attached to the two edges are ordered with respect to their de-
grees. Then with probability p, the edges are rewired in such
a way that one edge connects the two nodes with the smaller
degrees and the other connects the two nodes with the larger
degrees; otherwise, the edges are randomly rewired. In the
case when one or both of these new edges already existed in
the network, the step is discarded and a pair of other edges
is selected. A repeated application of the rewiring step leads
to an assortative networks. For producing disassortative net-
works, we modify the way for building new edges used in
above reshuffling method as that the node of the largest degree
connects to the nodes of the smallest degree and two other
nodes are connected. It is worth noting that the algorithm does
not change the degree distribution in the given network [5].
Before investigating the effect of the degree correlation on
the response, we review the results on uncorrelated attrac-
tor networks [1], where the system was characterized by the
scale-free networks which have the power-law shape degree
distribution P (k) ∼ k−γ . The size of the basin of attrac-
tion for two kinds of stimuli, namely, the random stimuli
(randomly chosen nodes are flipped) and the directed stim-
uli (means flipping sequentially the nodes of greatest degree)
were studied on scale-free attractor network systems. The re-
lation between the size of the basin of attraction for random
stimuli br and directed stimuli bm, which are all normalized
by network size N , are derived:
bm = b
(γ−1)/(γ−2)
r . (3)
The derivation was based on a assumption that the response
of attractor networks occurs if the sum of edges coming from
stimulated nodes exceeds a threshold which is the same for
both random and directed stimuli. For Barabási-Albert (BA)
scale-free networks [21], the distribution exponent γ = 3 and
thus bm = b
r. So the scale-free networks are robust to random
stimuli and sensitive to directed stimuli.
Let us first calculate the average size of the basin of attrac-
tion for random stimuli br and directed stimuli bm on degree-
correlated BA networks. According to [1], we use the network
size N = 1000 and average degree 〈k〉 = 20 in all simula-
tions. Figure 1 shows the average size of attractor basin versus
the degree of correlation which is quantified by the Pearson
correlation coefficient r [4]. To compare with uncorrelated
case, in Fig. 1 we also plot the predicted size of the attrac-
tor basin for directed stimuli b′m which is calculated using the
size of the attractor basin for random stimuli br following Eq.
(3). Restricted by the reshuffling method, we can not gener-
ate networks with strong degree correlation |r| → 1 [5]. In
simulations, the region of the Pearson correlation coefficient
r is about from −0.3 to 0.3. Although the region is small, it
nearly covers all the values of the Pearson correlation coeffi-
cient r of realistic complex networks shown in [4]. Therefore,
we interest in systems with the Pearson correlation coefficient
belonging to the region about from -0.3 to 0.3.
In Fig. 1 we can see the effects of the degree correla-
tion of scale-free networks on the size of the basion of at-
traction. Comparing the size of attractor basin b
m predicted
using Eq.(3) (the curve with triangles) with the size obtained
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
FIG. 1: (Color online) The size of attractor basin of scale-free net-
works as a function of Pearson correlation coefficient r in the case of
directed (square) and random stimuli (circle). All networks have the
same network size N = 1000 and average degree 〈k〉 = 20. Each
curve is an average of 1000 realizations. The predicted curve of b′m
calculated using Eq.(3) is shown as the curve with triangles.
by computer simulations (the curve with squares), one can see
that the relation between the size of attractor basin for random
stimuli br and directed stimuli bm derived in uncorrelated case
is not satisfied in correlated scale-free networks. When r ≈ 0
the numerical result of the attractor basin for directed stimuli
bm is identical with the prediction of uncorrelated networks
b′m.[22] For assortative case r > 0, the basin of attraction
for directed stimuli is less than the value of uncorrelated net-
work. This means that the assortative scale-free network is
more sensitive to directed stimuli than uncorrelated scale-free
networks. For disassortative case, the size of attractor basin
undergoes a non-monotonic process with the variance of Pear-
son correlation coefficient. The sensitivity of disassortative
scale-free networks is weaker than uncorrelated systems. The
size of the basin of attraction for random stimuli br decreases
monotonically with the increase of r. And the slope is small.
The robustness of scale-free networks to random stimuli re-
tains when these networks are degree correlated.
To understand the underlying mechanism of the effect of
degree correlation on response, we analyze the dynamics of
attractor networks. We assume that there are n functional
states in an attractor system. Substitute of Eq. (2) into Eq.
(1) gives
si(t+ 1) = sign(
j sj(t))
= sign(
sαj sj(t)), (4)
where Gi is the set of nodes adjacent to node i (the neighbors
of node i). We use the functional state {s1i } as the original
system state, and the stimulated system state is denoted as
i }. Thus the first step of the evolution is like
si(1) = sign(s
sαj s
j ). (5)
The functional states {sαi }α=2,··· ,n are uncorrelated with the
stimulated state {sβi }, since the functional states are chosen at
random. Thus the second term in the bracket at the right side
of Eq. (5) is approximately equal to 0, and this term can be
taken as noise [19]. For an arbitrary node i, if much less than
half nodes in Gi are flipped by the stimulus, then si(1) = s
if much more than half nodes in Gi are flipped, si(1) = −s
In general, the fraction of flipped nodes in Gi increases as
stimuli are enhanced. Because of the influence of noise, when
the fraction of flipped nodes in Gi is near but less than 0.5, the
node i choose a state s1i or −s
i at random.
In the case of uncorrelated networks, for both random and
directed stimuli, the fraction of flipped nodes in neighbors of
each node is equal to the fraction f of edges coming from
flipped nodes in a network. This property determines a criti-
cal condition for uncorrelated systems responding to stimuli:
near half edges in a network come from the stimulated nodes.
We obtained the critical value of f on the system with two
functional states by numerical simulation, which is fc = 0.46
for both random and directed stimuli. When stimuli are large
enough to satisfy the critical condition, all nodes in uncorre-
lated networks choose their states at random with the help of
noise term. Then, the system state {si(1)} becomes a random
state, and evolves to one of attractors randomly. The analysis
of the above property gives an insight of the dynamics of un-
correlated networks that the uncorrelated networks responds
to both kinds of stimuli as a whole.
Figure 2 shows numerical results of the critical fraction of
edges attached to stimulated nodes versus the Pearson correla-
tion coefficient of reshuffling scale-free networks. When net-
works are degree-correlated, the difference between the crit-
ical fraction fc for random stimuli and directed stimuli is re-
markable. The result shows that the mentioned assumption
used for deriving Eq.(3) in [1] is not appropriate for degree-
correlated scale-free networks. In Fig. 2, one can note that the
critical fraction fc for random stimuli varies slightly. Under
random stimuli, for correlated scale-free networks, the frac-
tion of flipped nodes in the neighbor of each node is approx-
imately equal to the fraction f of edges coming from flipped
nodes in a network. The dynamics of degree-correlated scale-
free networks under random stimuli have the same character-
istic as uncorrelated networks: the attractor systems respond
to random stimuli as a whole. Under directed stimuli, the vari-
ation of fc versus the Pearson correlation coefficient indicates
that the dynamics of directed stimulated attractor networks are
affected seriously by degree-correlation.
Next we numerically investigate the dynamical process of
the evolution of the attractor system in the case of directed
stimuli, and reveal the underlying mechanism of the effect
of degree correlation. To do this, we give a directed stimuli
with size equal to 235 to a realization of the uncorrelated net-
work. The stimulus is larger than the average attractor basin
-0.3 -0.2 -0.1 0.0 0.1 0.2 0.3
f c  
FIG. 2: (Color online) The critical value of the number of edges
attached to flipped nodes as a function of Pearson correlation coef-
ficient in the case of directed (square) and random (circle) stimuli.
Each curve is an average of 1000 realizations.
0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5 6 7 8 9
FIG. 3: The number of flipped nodes in the process of the evolution
of the uncorrelated system. Inset: the number of nodes whose state
si(t) is the same as s
for uncorrelated scale-free attractor systems given in Fig. 1
which is equal to 215(±12). In Fig. 3 the dynamical process
of the evolution of the system is represented by the number of
flipped nodes (Nf ). At the first step of the evolution, the num-
ber of the flipped nodes is 488, which is near half of the net-
work size. And then the system evolves to another imprinted
functional state, as shown in the inset of Fig. 3. The evolu-
tion shows that the uncorrelated scale-free networks response
to directed stimuli as a whole, as the above analysis.
For assortative networks, we give a directed stimulus with
the size 170 to attractor systems. Although the size of the
0 10 20 30 40 50 60
0 10 20 30 40 50
FIG. 4: (Color online) The number of flipped nodes in the process of
the evolution of two assortative systems with r = 0.13 (square) and
r = 0.15 (circle). Inset: the number of nodes whose state si(t) is
the same as s2i .
stimuli is smaller than the mentioned average attractor basin
of uncorrelated networks, the system responds to the stim-
ulus with the process of the change of the system state, as
shown in Fig. 4. We note that the number of flipped nodes
increases gradually. In contrast with uncorrelated scale-free
networks, the evolution shows that the assortative scale-free
network system does not make response as a whole. In as-
sortative scale-free networks, a group of nodes of large de-
gree preferentially connect to the nodes of greatest degrees,
i.e. stimulated nodes, and thus they are easier to get the con-
dition for changing their states. So the set of flipped nodes can
be extended by the assortative mixing. The assortative scale-
free network system evolves as a hierarchical cascade [23]
that progresses from higher to lower degree classes. Therefore
the basin of attraction of assortative network system decreases
and the system is more sensitive to directed stimuli.
With the increase of Pearson correlation coefficient, the
cluster coefficient of assortative networks are increased by the
degree based reshuffling steps [5]. The cluster property also
effects the dynamics of assortative scale-free networks. In Fig.
4 we show two numerical simulations with different types of
dynamics. For one kind of dynamics (square), the stable sys-
tem states are the functional states imprinted by Hebbian rule,
as the uncorrelated networks. The upper curve (square) in the
inset of Fig. 4 shows that a system evolves into the second
functional state. For another kind of dynamics (circle), the
stable system state at the end of evolution is not the imprinted
functional state. The lower curve (circle) in the inset of Fig. 4
shows the discrepancy. In this kind of systems, cluster forms
between stimulated nodes which have a high density of edges
within them, with a lower density of edges between other
groups of nodes. So these stimulated nodes hold their states on
−s1i . Additionally, the state of some low-degree nodes which
connect tightly to the cluster is also held. These nodes held by
the cluster structure result in the difference between the sys-
0 5 10 15 20
0 5 10 15 20
FIG. 5: The number of flipped nodes in the process of disassortative
system, r = −0.16. Inset: system with r = −0.30. The size of
stimuli is 245.
tem state and the imprinted functional state. There is a critical
value rc, for the networks used in simulations rc = 0.32, be-
low which two types of dynamics are possible (and larger the
value of r is, more frequently the second type of dynamics oc-
cur), while above which systems only responde to stimuli by
the second type of dynamics. Because of the cluster property
of assortative networks, too larger assortative mixing is not
expected for response of networks. In the limit of r → 1, net-
works disintegrate into isolated clusters, each of them consists
of nodes with certain degree k. Directed stimuli cannot in-
duce these systems to change their functional states, but only
change few clusters and leave the other nodes on their initial
states.
For the disassortative system, we choose a reshuffling scale-
free network realization with Pearson correlation coefficient
r = −0.16 which has the lowest sensitivity to directed stim-
uli as shown in Fig. 1. We give the disassortative network a
directed stimulus with size 245 which is larger than the av-
erage attractor basin of the uncorrelated scale-free networks.
Fig. 5 shows the dynamical process of the evolution of the
system. Although more than half of nodes flip their states
at the first step, the system state is attracted into the origi-
nal functional state. In disassortative networks, nodes with
large degrees preferentially connect to the nodes with small
ones. Under directed stimuli, the fraction of stimulated nodes
in the neighbors of the nodes in middle degree class is less
than the fraction of the edge coming from stimulated nodes.
Thus, more nodes need to be stimulated than uncorrelated sys-
tems for inducing the system into random state, and the basin
of attraction of disassortative system extends.
For larger disassortative mixing systems, the second im-
printed functional state cannot be reached. The inset of Fig. 5
shows the dynamical process of evolution of a network real-
ization with r = −0.30. The system is induced into stable os-
cillation state, which is established by the interaction between
large and small nodes. The system with large disassortative
mixing property is easier to responde the directed stimuli by
evolving into stable oscillation states. This structural prop-
erty leads to the non-monotonic behavior of sensitivity versus
Pearson correlation coefficient shown in Fig. 1. Additionally,
it is notable that the too large disassortative degree correlation
also destroys the ability of systems to responde directed stim-
uli with imprinted functional states, as the too large assortative
degree correlation.
In summary, we have studied the effect of the degree corre-
lation on the response of scale-free networks to stimuli. Cor-
related scale-free networks retain the robustness to random
stimuli. In the region of Pearson correlation coefficient in
which we interest, assortative scale-free networks are more
sensitive to directed stimuli than uncorrelated ones; and the
sensitivity of scale-free networks are weaken when networks
are disassortative. We found that the effects of degree corre-
lation result from the properties of the dynamics of degree-
correlated network systems. Uncorrelated networks responde
to stimuli as a whole. While the degree correlation of a net-
work destroys the identical critical condition of all nodes for
the response to directed stimuli. Assortative scale-free net-
works reduce the need on the size of directed stimuli to be
responded via a cascade that progresses from higher to lower
degree classes. The disassortative correlation extends the size
of the basin of attraction by the nodes in middle degree class
which has less stimulated neighbors and stay on initial state.
But the response of too large assortative and disassortative
scale-free networks is destroyed by the structure property, and
imprinted functional states cannot be reached. Since many
realistic complex networks have both scale-free and degree-
correlated properties, the intuitive description of the dynamics
might contribute to understanding of the attributes of realistic
networks.
This work was supported by the Fundamental Research
Fund for Physics and Mathematics of Lanzhou Univer-
sity under Grant No. Lzu05008. X.-J. Xu acknowl-
edges financial support from FCT (Portugal), Grant No.
SFRH/BPD/30425/2006.
[1] Y. Bar-Yam and I. R. Epstein, Proc. Natl. Acad. Sci. USA 101,
4341 (2004).
[2] A. Vázquez and Y. Moreno, Phys. Rev. E 67, 015101(R) (2003).
[3] R. Pastor-Satorras, A. Vázquez and A. Vespignani, Phys. Rev.
Lett. 87, 258701 (2001).
[4] M. E. J. Newman, Phys. Rev. Lett. 89, 208701 (2002).
[5] R. Xulvi-Brunet and I. M. Sokolov, Phys. Rev. E 70, 066102
(2004).
[6] M. E. J. Newman, Phys. Rev. E 67, 026126 (2003).
[7] A. Vázquez, M. Boguñá, Y. Moreno, R. Pastor-Satorras and A.
Vespignani, Phys. Rev. E 67, 046111 (2003).
[8] A. Capocci, G. Caldarelli, and P. De Los Rios, Phys. Rev. E 68,
047101 (2003).
[9] M. E. J. Newman and J. Park, Phys. Rev. E 68, 036122 (2003).
[10] J. Berg and M. Lässig, Phys. Rev. Lett. 89, 228701 (2002).
[11] K.-I. Goh, E. Oh, B. Kahng, and D. Kim, Phys. Rev. E 67,
017101 (2003).
[12] S. Maslov and K. Sneppen, Science 296, 910 (2002).
[13] P. L. Krapivsky and S. Redner, Phys. Rev. E 63, 066123 (2001).
[14] S. N. Dorogovtsev, Phys. Rev. E 69, 027104 (2004).
[15] M. Boguñá and R. Pastor-Satorras, Phys. Rev. E 66, 047104
(2002).
[16] L. K. Gallos and P. Argyrakis, Phys. Rev. E 72, 017101 (2005).
[17] G. Bianconi and M. Marsili, Phys. Rev. E 73, 066127 (2006).
[18] A. Fronczak and P. Fronczak, Phys. Rev. E 74, 026121 (2006).
[19] J. J. Hopfield, Proc. Natl. Acad. Sci. USA 79, 2554 (1982).
[20] D. J. Amit, H. Gutfreund and H. Sompolinsky, Phys. Rev. Lett.
55, 1530 (1985).
[21] A.-L. Barabási and R. Albert, Science 286, 509 (1999).
[22] For r ≈ 0, we have checked that the size of the attractor basin
does not change when using neutral shuffled networks instead
of BA networks.
[23] M. Barthélemy, A. Barrat, R. Pastor-Satorras and A. Vespig-
nani, Phy. Rev. Lett. 92, 178701 (2004).
