Identification of complex networks from limited and noise contaminated data is an important yet challenging task, which has attracted researchers from different disciplines recently. In this paper, the underlying feature of a complex network identification problem was analyzed and translated into a sparse linear programming problem. Then, a general framework based on the Bayesian model with independent Laplace prior was proposed to guarantee the sparseness and accuracy of identification results after analyzing influences of different prior distributions. At the same time, a three-stage hierarchical method was designed to resolve the puzzle that the Laplace distribution is not conjugated to the normal distribution. Last, the variational Bayesian was introduced to improve the efficiency of the network reconstruction task. The high accuracy and robust properties of the proposed method were verified by conducting both general synthetic network and real network identification tasks based on the evolutionary game dynamic. Compared with other five classical algorithms, the numerical experiments indicate that the proposed model can outperform these methods in both accuracy and robustness.
REFERENCES
1.
Z.
Wang
, L.
Wang
, A.
Szolnoki
, and M.
Perc
, “Evolutionary games on multilayer networks: A colloquium
,” Eur. Phys. J. B
88
, 124
(2015
). 2.
L.
Wang
and X.
Li
, “Spatial epidemiology of networked metapopulation: An overview
,” Chin. Sci. Bull.
59
, 3511
–3522
(2014
). 3.
Z.
Wang
, C.-Y.
Xia
, S.
Meloni
, C.-S.
Zhou
, and Y.
Moreno
, “Impact of social punishment on cooperative behavior in complex networks
,” Sci. Rep.
3
, 3055
(2013
). 4.
M.
Perc
, J.
Gomez-Gardenes
, A.
Szolnoki
, L. M.
Floría
, and Y.
Moreno
, “Evolutionary dynamics of group interactions on structured populations: A review
,” J. R. Soc. Interface
10
, 20120997
(2013
). 5.
C.-y.
Xia
, Z.
Wang
, J.
Sanz
, S.
Meloni
, and Y.
Moreno
, “Effects of delayed recovery and nonuniform transmission on the spreading of diseases in complex networks
,” Physica A
392
, 1577
–1585
(2013
). 6.
S.
Boccaletti
, V.
Latora
, Y.
Moreno
, M.
Chavez
, and D.-U.
Hwang
, “Complex networks: Structure and dynamics
,” Phys. Rep.
424
, 175
–308
(2006
). 7.
R.
Albert
and A.-L.
Barabási
, “Statistical mechanics of complex networks
,” Rev. Mod. Phys.
74
, 47
(2002
). 8.
H.
Zhang
, X.-Y.
Wang
, and X.-H.
Lin
, “Topology identification and module–phase synchronization of neural network with time delay
,” IEEE Trans. Syst. Man Cybern. Syst.
47
, 885
–892
(2016
). 9.
Y.-Y.
Liu
, J.-J.
Slotine
, and A.-L.
Barabási
, “Controllability of complex networks
,” Nature
473
, 167
(2011
). 10.
M.
Rohden
, A.
Sorge
, D.
Witthaut
, and M.
Timme
, “Impact of network topology on synchrony of oscillatory power grids
,” Chaos
24
, 013123
(2014
). 11.
J.
Yan
, H.
He
, and Y.
Sun
, “Integrated security analysis on cascading failure in complex networks
,” IEEE Trans. Inf. Forensics Secur.
9
, 451
–463
(2014
). 12.
H.
Dong
, N.
Hou
, Z.
Wang
, and W.
Ren
, “Variance-constrained state estimation for complex networks with randomly varying topologies
,” IEEE Trans. Neural Netw. Learn. Syst.
29
, 2757
–2768
(2018
). 13.
Z.
Zhang
, C.
Xia
, S.
Chen
, T.
Yang
, and Z.
Chen
, “Reachability analysis of networked finite state machine with communication losses: A switched perspective
,” IEEE J. Sel. Areas Commun.
38
, 845
–853
(2020
). 14.
H.-J.
Li
, Z.
Bu
, A.
Li
, Z.
Liu
, and Y.
Shi
, “Fast and accurate mining the community structure: Integrating center locating and membership optimization
,” IEEE Trans. Knowl. Data Eng.
28
, 2349
–2362
(2016
). 15.
C.
Xia
, C.
Gracia-Lázaro
, and Y.
Moreno
, “Effect of memory, intolerance, and second-order reputation on cooperation
,” Chaos
30
, 063122
(2020
). 16.
Z.
Wang
, C.
Xia
, Z.
Chen
, and G.
Chen
, “Epidemic propagation with positive and negative preventive information in multiplex networks
,” IEEE Trans. Cybern.
(to be published
).17.
S. G.
Shandilya
and M.
Timme
, “Inferring network topology from complex dynamics
,” New J. Phys.
13
, 013004
(2011
). 18.
K.
You
and L.
Xie
, “Network topology and communication data rate for consensusability of discrete-time multi-agent systems
,” IEEE Trans. Automat. Contr.
56
, 2262
–2275
(2011
). 19.
Z.
Meng
, W.
Ren
, Y.
Cao
, and Z.
You
, “Leaderless and leader-following consensus with communication and input delays under a directed network topology
,” IEEE Trans. Syst. Man Cybern. B Cybern.
41
, 75
–88
(2011
). 20.
F.
Pasqualetti
, F.
Dörfler
, and F.
Bullo
, “Attack detection and identification in cyber-physical systems
,” IEEE Trans. Automat. Contr.
58
, 2715
–2729
(2013
). 21.
S.
Hempel
, A.
Koseska
, J.
Kurths
, and Z.
Nikoloski
, “Inner composition alignment for inferring directed networks from short time series
,” Phys. Rev. Lett.
107
, 054101
(2011
). 22.
H.
Liu
, J.-A.
Lu
, J.
Lü
, and D. J.
Hill
, “Structure identification of uncertain general complex dynamical networks with time delay
,” Automatica
45
, 1799
–1807
(2009
). 23.
S.
Ganguli
and H.
Sompolinsky
, “Compressed sensing, sparsity, and dimensionality in neuronal information processing and data analysis
,” Annu. Rev. Neurosci.
35
, 485
–508
(2012
). 24.
K.
Klitenik
, J.
Sweeney
, F. C.
Wheatley
, and M. A.
Held
, “Systems, apparatuses and methods for supply chain network identification and optimization,” U.S. patent application 16/695,736 (May 5 2020).25.
G.
Pio
, M.
Ceci
, F.
Prisciandaro
, and D.
Malerba
, “Exploiting causality in gene network reconstruction based on graph embedding
,” Mach. Learn.
109
, 1231
–1279
(2020
). 26.
B.
Prasse
and P.
Van Mieghem
, “Network reconstruction and prediction of epidemic outbreaks for general group-based compartmental epidemic models
,” IEEE Trans. Netw. Sci. Eng.
(to be published).27.
X.
Han
, Z.
Shen
, W.-X.
Wang
, and Z.
Di
, “Robust reconstruction of complex networks from sparse data
,” Phys. Rev. Lett.
114
, 028701
(2015
). 28.
Y.
Zhang
, C.
Yang
, K.
Huang
, M.
Jusup
, Z.
Wang
, and X.
Li
, “Reconstructing heterogeneous networks via compressive sensing and clustering
,” IEEE Trans. Emerg. Topics Comput. Intell.
(published online, 2020). 29.
Y. C.
Pati
, R.
Rezaiifar
, and P. S.
Krishnaprasad
, “Orthogonal matching pursuit: Recursive function approximation with applications to wavelet decomposition,” in Proceedings of 27th Asilomar Conference on Signals, Systems and Computers (IEEE, 1993), pp. 40–44.30.
S. G.
Mallat
and Z.
Zhang
, “Matching pursuits with time-frequency dictionaries
,” IEEE Trans. Signal Process.
41
, 3397
–3415
(1993
). 31.
R.
Tibshirani
, “Regression shrinkage and selection via the lasso
,” J. R. Stat. Soc. Series B Methodol.
58
, 267
–288
(1996
). 32.
S.
Ji
, Y.
Xue
, L.
Carin et al.
, “Bayesian compressive sensing
,” IEEE Trans. Signal Process.
56
, 2346
(2008
). 33.
L.
Yu
, C.
Wei
, J.
Jia
, and H.
Sun
, “Compressive sensing for cluster structured sparse signals: Variational Bayes approach
,” IET Signal Process.
10
, 770
–779
(2016
). 34.
C. K.
Williams
and D.
Barber
, “Bayesian classification with Gaussian processes
,” IEEE Trans. Pattern Anal. Mach. Intell.
20
, 1342
–1351
(1998
). 35.
C. K.
Williams
, “Prediction with Gaussian processes: From linear regression to linear prediction and beyond,” in Learning in Graphical Models (Springer, 1998), pp. 599–621.36.
K.
Bae
and B. K.
Mallick
, “Gene selection using a two-level hierarchical Bayesian model
,” Bioinformatics
20
, 3423
–3430
(2004
). 37.
M. A.
Nowak
and R. M.
May
, “Evolutionary games and spatial chaos
,” Nature
359
, 826
(1992
). 38.
W.-X.
Wang
, Y.-C.
Lai
, C.
Grebogi
, and J.
Ye
, “Network reconstruction based on evolutionary-game data via compressive sensing
,” Phys. Rev. X
1
, 021021
(2011
). 39.
Z.
Shen
, W.-X.
Wang
, Y.
Fan
, Z.
Di
, and Y.-C.
Lai
, “Reconstructing propagation networks with natural diversity and identifying hidden sources
,” Nat. Commun.
5
, 4323
(2014
). 40.
K.
Huang
, Z.
Wang
, and M.
Jusup
, “Incorporating latent constraints to enhance inference of network structure
,” IEEE Trans. Netw. Sci. Eng.
7
, 466
–475
(2020
). 41.
K.
Huang
, X.
Zheng
, Z.
Li
, and Y.
Yang
, “Understanding cooperative behavior based on the coevolution of game strategy and link weight
,” Sci. Rep.
5
, 14783
(2015
). 42.
A. C.
Faul
, “Analysis of sparse Bayesian learning,” in International Conference on Neural Information Processing Systems: Natural & Synthetic (The MIT Press, 2001).43.
M. E.
Tipping
, “Sparse Bayesian learning and the relevance vector machine
,” J. Mach. Learn. Res.
1
, 211
–244
(2001
).44.
M. W.
Seeger
and H.
Nickisch
, “Compressed sensing and Bayesian experimental design,” in Proceedings of the 25th International Conference on Machine Learning (ACM, 2008), pp. 912–919.45.
D.
Wipf
, J.
Palmer
, and B.
Rao
, “Perspectives on sparse Bayesian learning,” in International Conference on Neural Information Processing Systems (The MIT Press, 2003).46.
D.
Wipf
, J.
Palmer
, B.
Rao
, and K.
Kreutz-Delgado
, “Performance Evaluation of Latent Variable Models with Sparse Priors
,” 2007 IEEE International Conference on Acoustics, Speech and Signal Processing – ICASSP '07
, (IEEE
, Honolulu, HI
, 2007
), pp. II-453
–II-456
. 47.
A.
Antoniadis
and J.
Fan
, “Regularization of wavelet approximations
,” J. Am. Stat. Assoc.
96
, 939
–967
(2001
). 48.
H.
Zou
, “The adaptive lasso and its oracle properties
,” J. Am. Stat. Assoc.
101
, 1418
–1429
(2006
). 49.
M. A.
Figueiredo
, “Adaptive sparseness for supervised learning
,” IEEE Trans. Pattern Anal. Mach. Intell.
25
, 1150
–1159
(2003
). 50.
D. J.
Watts
and S. H.
Strogatz
, “Collective dynamics of ‘small-world’ networks
,” Nature
393
, 440
(1998
). 51.
A.-L.
Barabási
and R.
Albert
, “Emergence of scaling in random networks
,” Science
286
, 509
–512
(1999
). 52.
D.
Lusseau
, K.
Schneider
, O. J.
Boisseau
, P.
Haase
, E.
Slooten
, and S. M.
Dawson
, “The bottlenose dolphin community of doubtful sound features a large proportion of long-lasting associations
,” Behav. Ecol. Sociobiol.
54
, 396
–405
(2003
). 53.
M. E.
Newman
, “Finding community structure in networks using the eigenvectors of matrices
,” Phys. Rev. E
74
, 036104
(2006
). 54.
M.
Girvan
and M. E.
Newman
, “Community structure in social and biological networks
,” Proc. Natl. Acad. Sci. U.S.A.
99
, 7821
–7826
(2002
). 55.
D. L.
Donoho
, I.
Drori
, Y.
Tsaig
, and J.-L.
Starck
, “Sparse solution of underdetermined linear equations by stagewise orthogonal matching pursuit” (Department of Statistics, Stanford University, 2006).56.
Y.
Yang
, T.
Luo
, Z.
Li
, X.
Zhang
, and S. Y.
Philip
, “A robust method for inferring network structures
,” Sci. Rep.
7
, 5221
(2017
). 57.
M. J.
Beal
et al., “Variational algorithms for approximate Bayesian inference,” Ph.D. thesis (University of London, London, 2003).© 2021 Author(s).
2021
Author(s)
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