Many complex networks are built up from empirical data prone to experimental error. Thus, the determination of the specific weights of the links is a noisy measure. Noise propagates to those macroscopic variables researchers are interested in, such as the critical threshold for synchronization of coupled oscillators or for the spreading of a disease. Here, we apply error propagation to estimate the macroscopic uncertainty in the critical threshold for some dynamical processes in networks with noisy links. We obtain closed form expressions for the mean and standard deviation of the critical threshold depending on the properties of the noise and the moments of the degree distribution of the network. The analysis provides confidence intervals for critical predictions when dealing with uncertain measurements or intrinsic fluctuations in empirical networked systems. Furthermore, our results unveil a nonmonotonous behavior of the uncertainty of the critical threshold that depends on the specific network structure.
REFERENCES
1.
S. N.
Dorogovtsev
, A. V.
Goltsev
, and J. F. F.
Mendes
, “Critical phenomena in complex networks
,” Rev. Mod. Phys.
80
, 1275
–1335
(2008
). 2.
M.
Newman
, Networks: An Introduction
(Oxford University Press, Inc.
, New York, NY
, 2010
).3.
R.
Pastor-Satorras
, C.
Castellano
, P.
Van Mieghem
, and A.
Vespignani
, “Epidemic processes in complex networks
,” Rev. Mod. Phys.
87
, 925
–979
(2015
). 4.
A.
Arenas
, A.
Díaz-Guilera
, J.
Kurths
, Y.
Moreno
, and C.
Zhou
, “Synchronization in complex networks
,” Phys. Rep.
469
, 93
–153
(2008
). 5.
S.
Boccaletti
, V.
Latora
, Y.
Moreno
, M.
Chavez
, and D. U.
Hwang
, “Complex networks: Structure and dynamics
,” Phys. Rep.
424
, 175
–308
(2006
). 6.
J.
Matamalas
, A.
Arenas
, and S.
Gómez
, “Effective approach to epidemic containment using link equations in complex networks
,” Sci. Adv.
4
, eaau4212
(2018
). 7.
I.
Dobson
, B.
Carreras
, V.
Lynch
, and D.
Newman
, “Complex systems analysis of series of blackouts: Cascading failure, critical points, and self-organization
,” Chaos
17
, 026103
(2007
). 8.
P. S.
Skardal
and A.
Arenas
, “Control of coupled oscillator networks with application to microgrid technologies
,” Sci. Adv.
1
, e1500339
(2015
). 9.
A.
Clauset
, C.
Moore
, and M. E.
Newman
, “Hierarchical structure and the prediction of missing links in networks
,” Nature
453
, 98
(2008
). 10.
R.
Guimerà
and M.
Sales-Pardo
, “Missing and spurious interactions and the reconstruction of complex networks
,” Proc. Natl. Acad. Sci.
106
, 22073
–22078
(2009
). 11.
D.
Hric
, T. P.
Peixoto
, and S.
Fortunato
, “Network structure, metadata, and the prediction of missing nodes and annotations
,” Phys. Rev. X
6
, 031038
(2016
). 12.
T.
Hoffmann
, L.
Peel
, R.
Lambiotte
, and N. S.
Jones
, “Community detection in networks without observing edges,” Sci. Adv.
6
(4
), eaav1478
(2020
). 13.
M. E. J.
Newman
, “Analysis of weighted networks
,” Phys. Rev. E
70
, 056131
(2004
). 14.
S.
Gómez
, A.
Arenas
, J.
Borge-Holthoefer
, S.
Meloni
, and Y.
Moreno
, “Discrete-time Markov chain approach to contact-based disease spreading in complex networks
,” Eurphys. Lett.
89
, 38009
(2010
). 15.
E.
Wigner
, “Characteristic vectors of bordered matrices with infinite dimensions
,” Ann. Math.
62
, 548
–564
(1955
). 16.
17.
D. A.
Spielman
, “Spectral graph theory and its applications,” in Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science (IEEE, 2007), pp. 29–38.18.
19.
F.
Chung
, L.
Lu
, and V.
Vu
, “Spectra of random graphs with given expected degrees
,” Proc. Natl. Acad. Sci.
100
, 6313
–6318
(2003
). 20.
J. G.
Restrepo
, E.
Ott
, and B. R.
Hunt
, “Onset of synchronization in large networks of coupled oscillators
,” Phys. Rev. E
71
, 036151
(2005
). 21.
J. G.
Restrepo
, E.
Ott
, and B. R.
Hunt
, “Approximating the largest eigenvalue of network adjacency matrices
,” Phys. Rev. E
76
, 056119
(2007
). 22.
C.
Castellano
and R.
Pastor-Satorras
, “Thresholds for epidemic spreading in networks
,” Phys. Rev. Lett.
105
, 218701
(2010
). 23.
H. H.
Ku
, “Notes on the use of propagation of error formulas
,” J. Res. Nat. Bureau Stand. C
70C
(4
), 263
–273
(1966
).24.
G.
Mana
and F.
Pennecchi
, “Uncertainty propagation in non-linear measurement equations
,” Metrologia
44
, 246
(2007
). 25.
J. P.
Gleeson
, S.
Melnik
, J. A.
Ward
, M. A.
Porter
, and P. J.
Mucha
, “Accuracy of mean-field theory for dynamics on real-world networks
,” Phys. Rev. E
85
, 026106
(2012
). 26.
A.-L.
Barabasi
and R.
Albert
, “Emergence of scaling in random networks
,” Science
286
, 509
–512
(1999
). 27.
S. N.
Dorogovtsev
, J. F. F.
Mendes
, and A. N.
Samukhin
, “Structure of growing networks with preferential linking
,” Phys. Rev. Lett.
85
, 4633
–4636
(2000
). 28.
S.
Valverde
, R.
Ferrer-i Cancho
, and R. V.
Sole
, “Scale-free networks from optimal design
,” Eurphys. Lett.
60
, 512
–517
(2002
). 29.
D. B.
Larremore
, W. L.
Shew
, and J. G.
Restrepo
, “Predicting criticality and dynamic range in complex networks: Effects of topology
,” Phys. Rev. Lett.
106
, 058101
(2011
). 30.
B.
Corominas-Murtra
, J.
Goñi
, R. V.
Solé
, and C.
Rodríguez-Caso
, “On the origins of hierarchy in complex networks
,” Proc. Natl. Acad. Sci.
110
, 13316
–13321
(2013
). 31.
O.
Kinouchi
and M.
Copelli
, “Optimal dynamical range of excitable networks at criticality
,” Nat. Phys.
2
, 348
–351
(2006
). 32.
D. R.
Chialvo
, “Emergent complex neural dynamics
,” Nat. Phys.
6
, 744
–750
(2010
). 33.
L.
Arola-Fernández
, A.
Díaz-Guilera
, and A.
Arenas
, “Synchronization invariance under network structural transformations
,” Phys. Rev. E
97
, 060301
(2018
). 34.
S.
Jalan
and J.
Bandyopdhyay
, “Random matrix analysis of complex networks
,” Phys. Rev. E
76
, 046107
(2007
). 35.
C.
Sarkar
and S.
Jalan
, “Spectral properties of complex networks
,” Chaos
28
, 102101
(2018
). © 2020 Author(s).
2020
Author(s)
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