Reconstruction of noise-driven nonlinear dynamic networks with some hidden nodes

The problem of network reconstruction, particularly exploring unknown network structures by analyzing measurable output data from networks, has attracted significant interest in many interdisciplinary fields in recent times. In practice, networks may be very large, and data can often be measured for only some of the nodes in a network while data for other variables are hidden. It is thus crucial to be able to infer networks from partial data. In this article, we study the problem of noise-driven nonlinear networks with some hidden nodes. Various difficulties appear jointly: nonlinearity of network dynamics, the impact of strong noise, the complexity of interaction structures between network nodes, and missing data from certain hidden nodes. We propose using high-order correlation to treat nonlinearity and structural complexity, two-time correlation to decorrelate noise, and higherorder derivatives to overcome the difficulties of hidden nodes. A closed form of network reconstruction is derived, and numerical simulations confirm the theoretical predictions.

[1]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[2]  M. Timme,et al.  Revealing networks from dynamics: an introduction , 2014, 1408.2963.

[3]  Emily S C Ching,et al.  Reconstructing links in directed networks from noisy dynamics. , 2017, Physical review. E.

[4]  Ying-Cheng Lai,et al.  Scaling of noisy fluctuations in complex networks and applications to network prediction. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Gang Hu,et al.  Depicting network structures from variable data produced by unknown colored-noise driven dynamics , 2016 .

[6]  R. FitzHugh.  Impulses and Physiological States in Theoretical Models of Nerve Membrane. , 1961, Biophysical journal.

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

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

[9]  Si Wu,et al.  Long-period rhythmic synchronous firing in a scale-free network , 2013, Proceedings of the National Academy of Sciences.

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

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

[12]  Alexander S Mikhailov,et al.  Evolutionary reconstruction of networks. , 2002, Physical review. E, Statistical, nonlinear, and soft matter physics.

[13]  Michael J. Berry,et al.  Weak pairwise correlations imply strongly correlated network states in a neural population , 2005, Nature.

[14]  Marc Timme,et al.  Inferring network topology from complex dynamics , 2010, 1007.1640.

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

[16]  Arkady Pikovsky,et al.  Network reconstruction from random phase resetting. , 2010, Physical review letters.

[17]  Muhammad Sahimi,et al.  Approaching complexity by stochastic methods: From biological systems to turbulence , 2011 .

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

[19]  Aneta Stefanovska,et al.  Inference of time-evolving coupled dynamical systems in the presence of noise. , 2012, Physical review letters.

[20]  Arkady Pikovsky,et al.  Untangling complex dynamical systems via derivative-variable correlations , 2014, Scientific Reports.

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

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

[23]  Zoran Levnajić,et al.  Derivative-variable correlation reveals the structure of dynamical networks , 2012, 1210.7446.