Reconstruction of dynamic networks with time-delayed interactions in the presence of fast-varying noises.

Most complex social, biological and technological systems can be described by dynamic networks. Reconstructing network structures from measurable data is a fundamental problem in almost all interdisciplinary fields. Network nodes interact with each other and those interactions often have diversely distributed time delays. Accurate reconstruction of any targeted interaction to a node requires measured data of all its neighboring nodes together with information on the time delays of interactions from these neighbors. When networks are large, these data are often not available and time-delay factors are deeply hidden. Here we show that fast-varying noise can be of great help in solving these challenging problems. By computing suitable correlations, we can infer the intensity and time delay of any targeted interaction with the data of two related nodes (driving and driven nodes) only while all other nodes in the network are hidden. This method is analytically derived and fully justified by extensive numerical simulations.

[1]  Diogo M. Camacho,et al.  Wisdom of crowds for robust gene network inference , 2012, Nature Methods.

[2]  Xiao Han,et al.  Robust Reconstruction of Complex Networks from Sparse Data , 2015, Physical review letters.

[3]  Yu Qian,et al.  Pattern formation in oscillatory complex networks consisting of excitable nodes. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Ying-Cheng Lai,et al.  Detection of time delays and directional interactions based on time series from complex dynamical systems. , 2017, Physical review. E.

[5]  M. Rosenblum,et al.  Estimation of delay in coupling from time series. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[6]  Jieping Ye,et al.  Network Reconstruction Based on Evolutionary-Game Data via Compressive Sensing , 2011, Physical Review X.

[7]  J Kurths,et al.  Inner composition alignment for inferring directed networks from short time series. , 2011, Physical review letters.

[8]  Si Wu,et al.  Solving the inverse problem of noise-driven dynamic networks. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[9]  A. Pikovsky,et al.  Reconstruction of a neural network from a time series of firing rates. , 2016, Physical review. E.

[10]  Debin Huang.  Synchronization-based estimation of all parameters of chaotic systems from time series. , 2004, Physical review. E, Statistical, nonlinear, and soft matter physics.

[11]  Francesco Sorrentino,et al.  Identification of delays and discontinuity points of unknown systems by using synchronization of chaos. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Chaoyang Zhang,et al.  Inference of targeted interactions of networks with data of driving and driven nodes only by applying fast-varying noise signals , 2017 .

[13]  Gordon Pipa,et al.  Transfer entropy—a model-free measure of effective connectivity for the neurosciences , 2010, Journal of Computational Neuroscience.

[14]  V. Latora,et al.  Complex networks: Structure and dynamics , 2006 .

[15]  S. Frenzel,et al.  Partial mutual information for coupling analysis of multivariate time series. , 2007, Physical review letters.

[16]  Wei Xing Zheng,et al.  Inferring topologies of complex networks with hidden variables. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[17]  Matthäus Staniek,et al.  Symbolic transfer entropy. , 2008, Physical review letters.

[18]  Michael Rosenblum,et al.  Reconstructing networks of pulse-coupled oscillators from spike trains. , 2017, Physical review. E.

[19]  Rainer Goebel,et al.  Mapping directed influence over the brain using Granger causality and fMRI , 2005, NeuroImage.

[20]  A. Seth,et al.  Granger causality and transfer entropy are equivalent for Gaussian variables. , 2009, Physical review letters.

[21]  Wen-Xu Wang,et al.  Noise bridges dynamical correlation and topology in coupled oscillator networks. , 2010, Physical review letters.

[22]  T. Sauer,et al.  Reconstructing the topology of sparsely connected dynamical networks. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[23]  Dongchuan Yu,et al.  Real-time estimation of interaction delays. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[24]  Marc Timme,et al.  Revealing network connectivity from response dynamics. , 2006, Physical review letters.

[25]  Y. Lai,et al.  Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems , 2016, 1704.08764.

[26]  Th. Meyer,et al.  Recovery of the time-evolution equation of time-delay systems from time series , 1997, chao-dyn/9907009.

[27]  M. Elowitz,et al.  A synthetic oscillatory network of transcriptional regulators , 2000, Nature.

[28]  V I Ponomarenko,et al.  Reconstruction of time-delay systems using small impulsive disturbances. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[29]  Yuanyuan Mi,et al.  Diverse self-sustained oscillatory patterns and their mechanisms in excitable small-world networks. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Wen-Xu Wang,et al.  Detecting hidden nodes in complex networks from time series. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[31]  Holger Kantz,et al.  Identifying and Modeling Delay Feedback Systems. , 1998, chao-dyn/9907019.

[32]  Pik-Yin Lai,et al.  Reconstructing weighted networks from dynamics. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[33]  S. Brunton,et al.  Discovering governing equations from data by sparse identification of nonlinear dynamical systems , 2015, Proceedings of the National Academy of Sciences.

[34]  I V Sysoev,et al.  Reconstruction of ensembles of coupled time-delay systems from time series. , 2014, Physical review. E, Statistical, nonlinear, and soft matter physics.

[35]  Gang Hu,et al.  Reconstruction of noise-driven nonlinear networks from node outputs by using high-order correlations , 2017, Scientific Reports.

[36]  Wen-Xu Wang,et al.  Predicting catastrophes in nonlinear dynamical systems by compressive sensing. , 2011, Physical review letters.

[37]  M Siefert,et al.  Practical criterion for delay estimation using random perturbations. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  Ying-Cheng Lai,et al.  Uncovering hidden nodes in complex networks in the presence of noise , 2014, Scientific Reports.

[39]  Schreiber,et al.  Measuring information transfer , 2000, Physical review letters.

[40]  Duncan J. Watts,et al.  Collective dynamics of ‘small-world’ networks , 1998, Nature.

[41]  Albert,et al.  Emergence of scaling in random networks , 1999, Science.

[42]  M D Prokhorov,et al.  Reconstruction of time-delay systems from chaotic time series. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[43]  Parlitz,et al.  Estimating model parameters from time series by autosynchronization. , 1996, Physical review letters.

[44]  Pik-Yin Lai,et al.  Extracting connectivity from dynamics of networks with uniform bidirectional coupling. , 2013, Physical review. E, Statistical, nonlinear, and soft matter physics.

[45]  Isao T Tokuda,et al.  Inferring phase equations from multivariate time series. , 2007, Physical review letters.

[46]  M D Prokhorov,et al.  Recovery of systems with a linear filter and nonlinear delay feedback in periodic regimes. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[47]  S. Strogatz.  Exploring complex networks , 2001, Nature.

[48]  J. P. Hamilton,et al.  Investigating neural primacy in Major Depressive Disorder: Multivariate granger causality analysis of resting-state fMRI time-series data , 2010, Molecular Psychiatry.

[49]  M. C. Soriano,et al.  Permutation-information-theory approach to unveil delay dynamics from time-series analysis. , 2010, Physical review. E, Statistical, nonlinear, and soft matter physics.

[50]  Marc Timme,et al.  Model-free inference of direct network interactions from nonlinear collective dynamics , 2017, Nature Communications.

[51]  I V Sysoev,et al.  Recovery of couplings and parameters of elements in networks of time-delay systems from time series. , 2016, Physical review. E.

[52]  Ljupco Kocarev,et al.  Estimating topology of networks. , 2006, Physical review letters.