Nafise Naseri, Fatemeh Parastesh, Farnaz Ghassemi, Sajad Jafari,
Eckehard Schöll, Jürgen Kurths
Converting high-dimensional complex networks to
lower-dimensional ones preserving
synchronization features
Open Access via institutional repository of Technische Universität Berlin
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Citation details
Naseri, N., Parastesh, F., Ghassemi, F., Jafari, S., Schöll, E., & Kurths, J. (2022). Converting high-dimensional
complex networks to lower-dimensional ones preserving synchronization features. In Europhysics Letters (Vol.
140, Issue 2, p. 21001). IOP Publishing. https://doi.org/10.1209/0295-5075/ac98de.
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epl draft
Converting high dimensional complex networks to lower dimen-
sional ones preserving synchronization features
Nafise Naseri1, Fatemeh Parastesh1, Farnaz Ghassemi1, Sajad Jafari*1,2, Eckehard Sch¨
oll3,4,5,
J¨
urgen Kurths5,6
1Department of Biomedical Engineering, Amirkabir University of Technology (Tehran polytechnic), Iran
2Health Technology Research Institute, Amirkabir University of Technology (Tehran polytechnic), Iran
3Institut f¨ur Theoretische Physik, Technische Universit¨at Berlin, Hardenbergstrasse 36, 10623 Berlin, Germany
4Bernstein Center for Computational Neuroscience Berlin, Humboldt-Universit¨at, 10115 Berlin, Germany
5Potsdam Institute for Climate Impact Research, Telegrafenberg A 31, 14473 Potsdam, Germany
6Humboldt University Berlin, Department of Physics, Berlin, 12489, Germany
PACS 89.75.-k – Complex systems
PACS 05.45.Xt – Synchronization
Abstract –Studying the stability of synchronization of coupled oscillators is one of the prominent
topics in network science. However, in most cases, the computational cost of complex network
analysis is challenging because they consist of a large number of nodes. This study includes
overcoming this obstacle by presenting a method for reducing the dimension of a large-scale
network, while keeping the complete region of stable synchronization unchanged. To this aim,
the first and last non-zero eigenvalues of the Laplacian matrix of a large network are preserved
using the eigen-decomposition method and, Gram-Schmidt orthogonalization. The method is
only applicable to undirected networks and the result is a weighted undirected network with
smaller size. The reduction method is studied in a large-scale a small-world network of Sprott-B
oscillators. The results show that the trend of the synchronization error is well maintained after
node reduction for different coupling schemes.
Introduction. – The study of structural and dynami-1
cal features of real-world networks is facilitated using com-2
plex networks in different fields such as biology [1,2], neu-3
roscience [3,4], ecology [5,6], and social science [7,8]. Syn-4
chronization is an important topic in complex networks5
[9–11]. Different types of synchronization have been found6
there, including complete [12], phase [13], cluster [14–16],7
explosive [17], and lag synchronization [18]. These syn-8
chronized states can emerge as the effect of the static or9
time-varying interactions in either attractive or repulsive10
couplings [19]. Moreover, enormous effort has been de-11
voted to the controllability and observability of synchro-12
nized complex networks [20], improving the synchroniz-13
ability [21,22] and robustness of synchronization [23].14
Most real-world systems can be better modeled by com-15
plex networks or even wih considering higher-order inter-16
actions [24]. These models may contain many nodes, mak-17
ing their analysis difficult and costly. Therefore, any effi-18
cient reduction of the size is of interest. One of the basic19
methods to decrease network size is graph partitioning.20
Various criteria exist whose persistence has been consid- 21
ered in reducing the network nodes. For instance, authors 22
in [25] seek to keep some physical properties of the net- 23
work after node reduction. Graph partitioning methods 24
are mostly considered as non-deterministic polynomial- 25
time problems [26], which cannot be solved in polynomial 26
time. Therefore, researchers have tried to find other meth- 27
ods to reduce the network size. For example, Bona et al. 28
[27] proposed a reduced model for the public transporta- 29
tion complex network with a long sequence of 2-degree 30
nodes and some hubs. Despite removing 2-degree nodes, 31
the reduced network has the same topological characteris- 32
tics and skeleton as the original one. Besides, it was shown 33
that this reduction increases the network cluster coefficient 34
and the average degree while decreasing the path length. 35
Recently, different methods such as Spectral Coarse- 36
Graining [28] and a Search Algorithm to Dimension Re- 37
duction [29] have been proposed. These algorithms de- 38
crease the dimension of the Laplacian matrix of the graph, 39
while preserving some specific features of the parent net- 40
p-1
N.Naseri et al.
work to keep synchronization. Whereas Spectral Coarse-41
Graining [28] iteratively reduces the dimension by merging42
the nodes, the Search Algorithm [29] can effectively reduce43
the number of nodes through a fast search. Another sys-44
tematic approach for size reduction has been taken into45
account recently. In 2020, Thibeault et al. developed46
the Dynamics Approximate Reduction Technique to sim-47
plify a complex network [30]. Their method, which was48
based on spectral graph theory, enabled the prediction of49
the synchronization regimes of phase oscillators in large-50
scale networks by using dominant eigenvectors features.51
In this method, the reduced network size is not arbitrary52
and depends on the number of the network’s communi-53
ties. In [31], the authors have reduced the dimension of a54
non-locally coupled network by projecting the network dy-55
namics onto the subspace that corresponds to the unstable56
eigenvalues of the linear part of the network.57
In this paper, we introduce a novel approach to reduce58
the size of a complex undirected network with preserv-59
ing its synchronization pattern. The key point for main-60
taining the synchronization stability of a network is to61
keep the eigenvalues of the Laplacian matrix that affect62
the synchronization within the master stability function63
approach. To this end, the eigen-decomposition and the64
Gram-Schmidt methods are utilized, and a smaller adja-65
cency matrix which is weighted, is obtained.66
The paper is organized as follows: First, the dimen-67
sion reduction method is described in Section 2 in detail.68
Then, a large-scale network of chaotic Sprott-B systems is69
analyzed, and the preservation of synchronization pattern70
after reduction is checked. The results are presented in71
Section 3. Finally, the conclusions of the paper are given72
in Section 4.73
Dimension reduction method. – This section de-74
scribes the method used to reduce the dimension of a large75
undirected network to a smaller one. The aim is to pre-76
serve the synchronization pattern of the large-scale net-77
work after dimension reduction. It has been shown that78
the stability of synchronization in networks relies on the79
coupling topology [32]. According to the master stability80
function method [33], the region of stable synchronization81
depends on the eigenvalues of the connectivity matrix of82
the graph. Here, the reduction method is based on obtain-83
ing a reduced connectivity matrix with desired eigenvalues84
which are those involved in determining the synchroniza-85
tion stability region. The eigen-decomposition factoriza-86
tion is used for finding this reduced connectivity matrix.87
Master stability function. The master stability func-
tion (MSF) [33] is a method for finding the local stability
of synchronization. The description of this approach is
given in the following.
It is supposed that Nidentical oscillators with the indi-
vidual dynamics of F(.) are linearly coupled by the overall
coupling strength dthrough a Laplacian connection ma-
trix G. For the oscillator i, one can write
˙
Xi=F(Xi)−d
N
X
j=1
GijH(Xj), i = 1,2, . . . , N (1)
where Hindicates the coupling function. When all os-
cillators lie in the synchronization manifold, i.e., X1=
X2=. . . =XN=Xs, the linearization of Eq. (1) around
the synchronized solution Xsis defined as the variational
equation and can be written as
˙ηl= [DF (Xs)−αlDH (Xs)] ηl, l = 1,2, . . . , N (2)
in which αl=dλl, where λlis the lth eigenvalue of the ma- 88
trix G. Also, DH and DF are the Jacobian matrixes of 89
Hand F, respectively. The variational equation (Eq. (2)) 90
determines the stability of synchronization, which can be 91
found by calculating its maximum Lyapunov exponent. 92
The maximum Lyapunov exponent (Λ) of Eq. (2) as a 93
function of α=dλ is known as the master stability func- 94
tion (MSF). Considering a connected and undirected net- 95
work, the first eigenvalue of Gis zero (λ1= 0), which is 96
along the synchronization manifold. The other eigenval- 97
ues are sorted assendingly λ2≤λ3≤. . . ≤λN. When 98
Λ<0 for all eigenvalues λi,i= 2, . . . , N of the Laplacian 99
matrix, all the nodes of the network oscillate in complete 100
synchrony. 101
Huang et al. [34] proposed a general scheme for catego-
rizing the MSFs and introduced four classes. The classi-
fication is based on the number of zero-crossing points of
the master stability function curve versus α, such that Γk
represents a class in which Λ (α) crosses the zero ktimes.
In case the synchronization cannot be reached for any α
value, the master stability function has no zero-crossing
point and is classified as Γ0(Fig. 1a). The master sta-
bility function with only one zero-crossing point, αmin, is
known as class Γ1, which is shown in Fig. 1b. Sorting
the eigenvalues of the Laplacian matrix (G) in ascending
order (i.e., λ1= 0), the synchronization manifold of this
class is stable if
αmin < dλ2≤dλ3≤. . . ≤dλN(3)
holds. Hence, choosing the coupling strength as d > αmin
λ2
ensures the stability of the synchronization manifold. In
other words, the synchronization region, which is un-
bounded depends only on λ2. In class Γ2(Fig. 1c), the
master stability function versus αhas two zero-crossing
points, αmin and αmax, where the region αmin < α < αmax
is the stability region (Λ <0). Therefore, an upper bound
of the eigenvalues is also required for the stability region.
In this case, the synchronization is stable if
αmin < dλ2≤dλ3≤. . . ≤dλN< αmax (4)
Consequently, synchronization can be achieved for αmin
λ2<102
d < αmax
λN. By taking R≡λN
λ2as an eigenratio, the syn- 103
chronization can occur if R < αmax
αmin . Thus, in this class, 104
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Converting complex networks to lower dimensional ones to preserve synchronization
Fig. 1: Different classes of master stability function. a) Class
Γ0with no zero-crossing point, b) class Γ1with only one zero-
crossing point, c) class Γ2with two zero-crossing points, and
d) class Γ3with three zero-crossing.
the stability region of synchronization depends only on105
the value of R. Barahona and Pecora [35] investigated106
the stability of synchronization in small-world networks107
by using the concept of the first-non-zero and maximum108
eigenvalues of the Laplacian matrix.109
Finally, the fourth class belongs to the master stability110
function with more than two zero-crossing points; as an111
example, class Γ3with three zero-crossing is illustrated in112
Fig. 1d. For these systems, the synchronization can be113
achieved if all dλireside in the Λ <0 regions. Since this114
class is more complex and case-dependent, we ignore it in115
this study.116
According to the above definitions of the master stabil-117
ity function classifications, the synchronization region is118
only affected by λ2and λN. In fact, two networks have119
the same synchronization region if they have the same120
λ2and λN. Based on this concept, a reduced network121
can have the same synchronization pattern as the original122
network by choosing its λ2and λmax the same as the123
original network. To find the connectivity matrix with de-124
fined eigenvalues, the eigen-decomposition approach can125
be used which is explained in the next subsection.126
Eigen-decomposition and Gram-Schmidt orthogonaliza-
tion of Laplacian matrix. Consider λi, i = 1,2, . . . , N
and λ′i, i = 1,2, . . . , n as the ith eigenvalue of the original
and reduced Laplacian matrix, respectively, and Rand R′
as their eigenratio as well. To have the same synchroniza-
tion pattern, we must keep λ2∼
=λ′2and also R∼
=R′,
leading to λN∼
=λ′n. To determine Laplacian matrix of
the reduced network with desired eigenvalues, the eigen-
decomposition factorization can be utilized. According to
this factorization, any positive semidefinite matrix, e.g.,
A, can be factorized as
A=QDQ−1(5)
in which Dis a diagonal matrix whose diagonal elements
are the eigenvalues of A, and the corresponding eigenvec-
tors lie in the columns of Q. Therefore, by considering D
as the matrix of eigenvalues of the reduced matrix (n×n)
and finding an appropriate eigenvector matrix (Q), the
Laplacian matrix An×ncan be computed using Eq. (5).
Since the matrix Ais assumed symmetric, we can write,
A=AT=QDQ−1T=Q−1TDQT(6)
leading to Q−1=QT, where Tdenotes the transposed 127
matrix. Therefore, Qmust be an orthogonal matrix. To 128
form an orthogonal basis, the Gram-Schmidt process can 129
be used [see Appendix for more details]. Since the first 130
eigenvalue of Ais zero, its corresponding eigenvector must 131
be chosen as v1= [1,1,...,1]1×n
Tfor the Gram-Schmidt 132
process. Selecting the other independent basis vectors is 133
arbitrary. Then, using the orthogonal basis vectors, Qn×n134
can be obtained. 135
In order to determine Dn×n,neigenvalues in the as- 136
cending order are needed, where three of them are known: 137
λ′1= 0, λ′2=λ2, and λ′n=λN. The rest of the needed 138
eigenvalues (n−3 eigenvalues) are found by partitioning 139
the N−3 eigenvalues of the Laplacian matrix of the orig- 140
inal network. Here, we use the k-means clustering algo- 141
rithm. K-means is the most popular clustering method 142
due to its simplicity (for more detail, see [36]). After ob- 143
taining an orthogonal matrix Q, and a diagonal matrix 144
D, a Laplacian matrix Awith the desired dimension and 145
eigenvalues can be found by using Eq. (5). It should be 146
noted that the obtained matrix is weighted. The described 147
method for obtaining the reduced connectivity matrix A148
is presented in Fig. 2. 149
Simulation results. – In this section, we apply the
proposed method to reduce a high-dimensional Watts-
Strogatz small-world network with N= 500 nodes and
105links. It is assumed that the individual dynamics of
the node obey the chaotic Sprott-B equations [37]:
˙x=yz
˙y=x−y
˙z= 1 −xy
(7)
The size of the reduced network is assumed as n= 100 150
here. We consider different coupling functions to inves- 151
tigate different synchronization patterns. For the orig- 152
inal network, we have λ2= 339.47 and λN= 449.80. 153
Thus, we keep these eigenvalues and obtain the other 154
n−3 eigenvalues by classifying N−3 eigenvalues of the 155
original network. So, the matrix Dis found. Next, the 156
eigen-decomposition factorization and Gram-Schmidt or- 157
thogonalization are employed, and an orthogonal matrix 158
of eigenvectors is obtained (Q). Finally, a zero-row sum, 159
symmetry Laplacian matrix of size n= 100 with desired 160
eigenvalues is found using Eq. (5). The values of the 161
two most essential eigenvalues and eigenratio used in this 162
example are represented in Table 1. It can be seen that 163
p-3
N.Naseri et al.
Fig. 2: The schematic of the proposed method to reduce an N-dimensional network to n-dimensional one (N > n) using
eigen-decomposition factorization and Gram-Schmidt orthogonalization.
Table 1: Two eigenvalues and eigenratios of the reduced net-
work and its parent.
Original network Reduced network
λ2339.47 339.50
λmax 449.80 449.80
R1.32 1.32
the eigenvalues of the reduced and original networks are164
approximately equal.165
For more investigation, three couplings with different166
MSF classes are considered. In Fig. 3, the master stability167
functions versus αare plotted. Three different couplings168
y→x,x→y, and x→zare considered. The notation,169
e.g., x→z, means that the coupling which is defined on x170
state variables is added to zstate variables. According to171
the eigenvalues presented in Table 1, the stability regions172
in y→xcoupling are d > 0.003069 and d > 0.003081 for173
the original and reduced networks, respectively. For x→y174
coupling, the stability regions of the original and reduced175
networks are 0.002957 <d<0.0031030 and 0.002962 <176
d < 0.0031023, respectively.177
Next, the networks are solved numerically, and the syn-
chronization error is calculated using Eq. 8.
Err =1
T(N−1) limT→∞ RT
0
PN
k=2 q(x1−xk)2+ (y1−yk)2+ (z1−zk)2dt(8)
The synchronization errors for both networks and each178
coupling scheme are illustrated in Fig. 4. The upper and179
lower panels represent the errors of the parent and reduced180
networks, respectively. It can be observed that the syn-181
chronization regions, i.e., the region of coupling strength182
(d) with zero error, are the same for both networks. More-183
over, the synchronization errors have similar trends in the 184
original and reduced networks. 185
To better compare the synchronization behavior of both 186
networks, time series, spatiotemporal patterns, and time 187
snapshots are presented in Figs. 5-8 for synchronous and 188
asynchronous states for master stability function of class 189
Γ1and class Γ2. Figure 5 illustrates the patterns of both 190
networks for d= 3.7×10−3which is in the synchroniza- 191
tion regime under y→xcoupling. Also, the results 192
for d= 2.7×10−3in which the oscillators of networks 193
under y→xcoupling are asynchronous, are shown in 194
Fig. 6. Moreover, the networks have the same behavior 195
for class Γ2(x→ycoupling). In Fig. 7 and Fig. 8, 196
the synchronous and asynchronous behavior of both net- 197
works is represented by considering d= 3.0×10−3and 198
d= 3.2×10−3, respectively. It can be observed that the 199
networks have similar synchronous and asynchronous pat- 200
terns. 201
Conclusion. – Large-scale complex networks are im- 202
portant models for describing various real-world networks. 203
However, their high dimensionality often gives rise to 204
high computational costs for analysis, leading to be time- 205
consuming. Hence, reducing the dimension of these net- 206
works is essential. On the other hand, synchronization is a 207
significant phenomenon in complex networks. Therefore, 208
it is desired not to disturb the synchronization pattern 209
during dimension reduction. This study addressed this 210
issue by decreasing the size of the Laplacian matrix of a 211
large-scale network using the eigen-decomposition method 212
and the Gram-Schmidt orthogonalization process. The 213
original network is considered to be undirected; there- 214
fore, the eigenvalues of the Laplacian matrix are real. To 215
construct a network with eigen-decomposition approach, 216
firstly, the eigenvalues of the reduced Laplacian matrix 217
p-4
Converting complex networks to lower dimensional ones to preserve synchronization
Fig. 3: The master stability function versus αfor Sprott-B chaotic system (Eq. (7)) under three different couplings: a) y→x
(class Γ1), b) x→y(class Γ2), and c) x→z(class Γ0). Coupled Sprott-B systems represent different synchronization patterns
according to the coupling scheme.
Fig. 4: The synchronization errors of coupled Sprott-B systems for the original (upper plots) and reduced (lower plots) networks
as a function of coupling strength d. The coupling is on a) class Γ1(y→x), b) class Γ2(x→y), and c) class Γ0(x→z). The
synchronization region and the trend of error are similar for both networks in each class.
p-5
N.Naseri et al.
Fig. 5: a) Time series, b) spatiotemporal pattern, and c) time
snapshot at t= 4000 for y→xcoupling which is class Γ1.
The left and right panels are the results of the original network
(N= 500) and the reduced one (n= 100), respectively. The
coupling strength is d= 3.7×10−3, in which all oscillators lie
in the synchronous manifold. The oscillations of both original
and reduced networks are synchronous in this case.
must be defined. According to the master stability func-218
tion, the region of stable synchronization depends on the219
minimum and maximum non-zero eigenvalues. Thus, we220
kept them the same as the original network and selected221
the other eigenvalues by classifying the original eigenval-222
ues. Then, the matrix of eigenvectors was obtained by223
the Gram-Schmidt orthogonalization process. Finally, us-224
ing the eigenvalues and eigenvectors, a weighted reduced225
Laplacian matrix was obtained. The method was applied226
on a 500-node small-world network of Sprott-B systems.227
The results were validated via synchronization error, time228
series, spatiotemporal patterns, and snapshots of both net-229
works for different coupling functions in the synchronous230
and asynchronous states. Our findings indicate that the231
number of nodes of any complex network can be decreased232
regardless of network topology and node dynamics with233
preserving the synchronization stability region.234
Conflict of interest. – The authors declare that they235
have no conflict of interest.236
APPENDIX: the Gram-Schmidt process. –237
Suppose the arbitrary set {−→
v1,−→
v2,...,−→
vk}as the ba-238
sis for a given set V, whose vectors are linearly inde-239
pendent. The Gram-Schmidt process can generate an or-240
thogonal basis for V. The vectors {−→
u1,−→
u2,...,−→
uk}are241
said to be orthogonal if and only if the inner product of242
any two different vectors of them is equal to zero, i.e.,243
⟨−→
ui,−→
uj⟩= 0 ∀i=j. This set of new vectors can be244
constructed as follows:245
Fig. 6: a) Time series, b) spatiotemporal pattern, and c) time
snapshot at t= 4000 for y→xcoupling which is class Γ1. The
coupling strength is d= 2.7×10−3, that leads to asynchronous
oscillations in the original (left panel) and the reduced networks
(right panel). This case exhibits asynchronous oscillations in
both networks.
−→
u1=−→
v1
−→
u2=−→
v2−⟨−→
v2,−→
u1⟩
⟨−→
u1,−→
u1⟩
−→
u1
−→
u3=−→
v3−⟨−→
v3,−→
u1⟩
⟨−→
u1,−→
u1⟩
−→
u1−⟨−→
v3,−→
u2⟩
⟨−→
u2,−→
u2⟩
−→
u2
.
.
.
−→
uk=−→
vk−Pk−1
p=1
⟨−→
vk,−→
up⟩
⟨−→
up,−→
up⟩
−→
up
where ⟨.⟩denotes the inner product. 246
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