This paper is concerned with the study of the global emerging behavior in complex networks where each node can be modeled as a cyber-physical system. We recast the problem of characterizing the behavior of such systems as a stability problem and give two technical results to assess this property. We then illustrate the effectiveness of our approach by considering two testbed examples arising in applications where networks, arising from Internet of Things applications, need to be designed so as to fulfill a given task.

1
E.
Lee
, “
The past, present and future of cyber-physical systems: A focus on models
,”
Sensors
15
,
4837
(
2015
).
2
A. R.
Al-Ali
,
R.
Gupta
, and
A. A.
Nabulsi
, “
Cyber physical systems role in manufacturing technologies
,”
AIP Conf. Proc.
1957
,
050007
(
2018
).
3
V.
Muliukha
,
A.
Ilyashenko
,
V.
Zaborovsky
, and
A.
Lukashin
, “
Cyber-physical approach to the network-centric robotics control task
,”
AIP Conf. Proc.
1776
,
090056
(
2016
).
4
S. H.
Strogatz
, “
Exploring complex networks
,”
Nature
410
,
268
(
2001
).
5
R.
Albert
and
A.-L.
Barabási
, “
Statistical mechanics of complex networks
,”
Rev. Mod. Phys.
74
,
47
97
(
2002
).
6
S.
Boccaletti
,
R.
Criado
,
M.
Romance
, and
J. J.
Torres
, “
Introduction to focus issue: Complex dynamics in networks, multilayered structures and systems
,”
Chaos
26
,
065101
(
2016
).
7
L. M.
Pecora
and
T. L.
Carroll
, “
Synchronization of chaotic systems
,”
Chaos
25
,
097611
(
2015
).
8
R.
Olfati-Saber
and
R.
Murray
, “
Consensus problems in networks of agents with switching topology and time-delays
,”
IEEE Trans. Automat. Contr.
49
,
1520
1533
(
2004
).
9
C.
Qian
,
J.
Cao
,
J.
Lu
, and
J.
Kurths
, “
Adaptive bridge control strategy for opinion evolution on social networks
,”
Chaos
21
,
025116
(
2011
).
10
G.
Russo
and
J. J. E.
Slotine
, “
Global convergence of quorum-sensing networks
,”
Phys. Rev. E
82
,
041919
(
2010
).
11
J. M. V.
Grzybowski
,
E. E. N.
Macau
, and
T.
Yoneyama
, “
On synchronization in power-grids modelled as networks of second-order Kuramoto oscillators
,”
Chaos
26
,
113113
(
2016
).
12
K.
Xi
,
J. L. A.
Dubbeldam
, and
H. X.
Lin
, “
Synchronization of cyclic power grids: Equilibria and stability of the synchronous state
,”
Chaos
27
,
013109
(
2017
).
13
Y.
Tang
,
F.
Qian
,
H.
Gao
, and
J.
Kurths
, “
Synchronization in complex networks and its application—A survey of recent advances and challenges
,”
Annu. Rev. Control
38
,
184
198
(
2014
).
14
W.
Lu
,
B.
Liu
, and
T.
Chen
, “
Cluster synchronization in networks of coupled nonidentical dynamical systems
,”
Chaos
20
,
013120
(
2010
).
15
G.
Russo
, “
How to desynchronize quorum-sensing networks
,”
Phys. Rev. E
95
,
042312
(
2017
).
16
G.
Russo
, “
Loss of coordination in complex directed networks: An incremental approach based on matrix measures
,”
Int. J. Robust Nonlinear Control
28
,
120
131
(
2018
).
17
G.
Fortino
,
R.
Gravina
,
W.
Russo
, and
C.
Savaglio
, “
Modeling and simulating internet-of-things systems: A hybrid agent-oriented approach
,”
Comput. Sci. Eng.
19
,
68
76
(
2017
).
18
D.
Zhang
,
Z.
Xu
,
H. R.
Karimi
,
Q.
Wang
, and
L.
Yu
, “
Distributed h output-feedback control for consensus of heterogeneous linear multiagent systems with aperiodic sampled-data communications
,”
IEEE Trans. Ind. Electron.
65
,
4145
4155
(
2018
).
19
D.
Zhang
,
L.
Liu
, and
G.
Feng
, “
Consensus of heterogeneous linear multiagent systems subject to aperiodic sampled-data and DoS attack
,”
IEEE Trans. Cybern.
49
,
1501
1511
(
2019
).
20
J.
Sztipanovits
,
X.
Koutsoukos
,
G.
Karsai
,
N.
Kottenstette
,
P.
Antsaklis
,
V.
Gupta
,
B.
Goodwine
,
J.
Baras
, and
S.
Wang
, “
Toward a science of cyber-physical system integration
,”
Proc. IEEE
1000
,
29
44
(
2012
).
21
C.
Nowzari
and
J.
Cortes
, “
Team-triggered coordination for real-time control of networked cyber-physical systems
,”
IEEE Trans. Automat. Contr.
61
,
34
47
(
2016
).
22
E.
Lee
and
S.
Seshia
,
Introduction to Embedded Systems: A Cyber-Physical Systems Approach
, 2nd ed. (
MIT Press
,
2017
).
23
T. H.
Cormen
,
Introduction to Algorithms
(
MIT Press
,
2009
).
24
J.
Cortes
, “
Discontinuous dynamical systems
,”
IEEE Control Syst. Mag.
44
,
36
73
(
2005
).
25
W.
Lohmiller
and
J. J. E.
Slotine
, “
On contraction analysis for non-linear systems
,”
Automatica
34
,
683
696
(
1998
).
26
G.
Russo
,
M.
di Bernardo
, and
J.
Slotine
, “
A graphical algorithm to prove contraction of nonlinear circuits and systems
,”
IEEE Trans. Circuits Syst. I
58
,
336
348
(
2011
).
27
Q. C.
Pham
and
J. J. E.
Slotine
, “
Stable concurrent synchronization in dynamic system networks
,”
Neural Netw.
20
,
62
77
(
2007
).
28
M.
Mobilia
, “
Does a single zealot affect an infinite group of voters?
,”
Phys. Rev. Lett.
91
,
028701
(
2003
).
29
D. D.
Chinellato
,
I. R.
Epstein
,
D.
Braha
,
Y.
Bar-Yam
, and
M. A. M.
de Aguiar
, “
Dynamical response of networks under external perturbations: Exact results
,”
J. Stat. Phys.
159
,
221
230
(
2015
).
30
H.
Chen
,
G.
He
,
F.
Huang
,
C.
Shen
, and
Z.
Hou
, “
Explosive synchronization transitions in complex neural networks
,”
Chaos
23
,
033124
(
2013
).
31
X.
Li
,
J.
Zhang
, and
M.
Small
, “
Self-organization of a neural network with heterogeneous neurons enhances coherence and stochastic resonance
,”
Chaos
19
,
013126
(
2009
).
32
M. A. D.
Aguiar
,
A. P. S.
Dias
, and
F.
Ferreira
, “
Patterns of synchrony for feed-forward and auto-regulation feed-forward neural networks
,”
Chaos
27
,
013103
(
2017
).
33
P.
Beim Graben
and
R.
Potthast
, “
Inverse problems in dynamic cognitive modeling
,”
Chaos
19
,
015103
(
2009
).
34
S.
Arianos
,
E.
Bompard
,
A.
Carbone
, and
F.
Xue
, “
Power grid vulnerability: A complex network approach
,”
Chaos
19
,
013119
(
2009
).
35
A.
Yazdani
and
P.
Jeffrey
, “
Complex network analysis of water distribution systems
,”
Chaos
21
,
016111
(
2011
).
36
M.
Vidyasagar
,
Nonlinear Systems Analysis
, 2nd ed. (
Prentice-Hall
,
Englewood Cliffs, NJ
,
1993
).
37
M.
di Bernardo
,
D.
Liuzza
, and
G.
Russo
, “
Contraction analysis for a class of nondifferentiable systems with applications to stability and network synchronization
,”
SIAM J. Control Optim.
52
,
3203
3227
(
2014
).
38
D.
Fiore
,
G.
Russo
, and
M.
di Bernardo
, “
Exploiting nodes symmetries to control synchronization and consensus patterns in multiagent systems
,”
IEEE Control Syst. Lett.
1
,
364
369
(
2017
).
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