Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems

Wenxu Wang; Ying-Cheng Lai; Celso Grebogi

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Data Based Identification and Prediction of Nonlinear and Complex Dynamical Systems

Wenxu Wang, Ying-Cheng Lai, Celso Grebogi

arXiv:1704.08764v1Apr 27, 2017physics.data-annlin.CDphysics.soc-ph

TL;DR

Reconstructing and controlling nonlinear, complex dynamical systems is difficult because network structure and nodal dynamics are often unknown while measured time series are limited. This review synthesizes data-based approaches including compressive sensing, reverse engineering, hidden-node detection, network reconstruction, and control, while identifying challenges such as strong noise, weak coupling, and representing nonlinear interactions.

Problem
Network structure and nodal dynamics are often unknown, while only limited measured time series are available for reconstructing complex systems and ultimately controlling their collective dynamics.
Method
The review synthesizes data-based reconstruction and control approaches, including compressive sensing, power-series approximation, sparse estimation, and attractor-network formulations.
Results
The reviewed methods support network reconstruction, hidden-node detection, and control across nonlinear and complex dynamical systems, with reported success rates reaching 100% in one coupled-oscillator reconstruction example.
Takeaways & Limitations
Data-based methods can infer network structure and dynamics from indirect or incomplete observations and provide frameworks for studying and controlling complex systems.
Takeaways & Limitations
Performance is constrained when strong noise entangles hidden-node effects with noise or when hidden-node coupling is weak, and gene-network applications require an appropriate representation of the Hill function.

Abstract

from arXiv · show

The problem of reconstructing nonlinear and complex dynamical systems from measured data or time series is central to many scientific disciplines including physical, biological, computer, and social sciences, as well as engineering and economics. In this paper, we review the recent advances in this forefront and rapidly evolving field, aiming to cover topics such as compressive sensing (a novel optimization paradigm for sparse-signal reconstruction), noised-induced dynamical mapping, perturbations, reverse engineering, synchronization, inner composition alignment, global silencing, Granger Causality and alternative optimization algorithms. Often, these rely on various concepts from statistical and nonlinear physics such as phase transitions, bifurcation, stabilities, and robustness. The methodologies have the potential to significantly improve our ability to understand a variety of complex dynamical systems ranging from gene regulatory systems to social networks towards the ultimate goal of controlling such systems. Despite recent progress, many challenges remain. A purpose of this Review is then to point out the specific difficulties as they arise from different contexts, so as to stimulate further efforts in this interdisciplinary field.

1 Introduction

Nonlinear and complex systems reconstruction infers governing equations, interactions, and hidden structure from limited time-series data. This review surveys data-driven approaches, especially compressive sensing, across dynamical systems and networks.

Motivation: The inverse problem seeks to reconstruct unknown nodal dynamics and network structure from limited measured time series.Successful reconstruction may support causal-relation identification and hidden-node detection.
Existing approaches: Delay-coordinate embedding provides a traditional foundation for reconstructing unknown dynamical systems from time series.It can support estimates of dimensionality, Lyapunov exponents, and unstable periodic orbits.
Scope: The reviewed approaches address equation and parameter-function prediction, coupled-oscillator reconstruction, evolutionary-game networks, and hidden-node detection.The review also covers alternative approaches including synchronization, Granger causality, and other optimization methods.
Applications: Data-driven reconstruction has applications spanning climate, biological, social, and other complex networks, including gene-regulatory reverse engineering.The review discusses both model-based and model-free approaches to recovering interactions and dynamics.
Compressive sensing: Compressive sensing estimates sparse coefficients in series expansions of the vector fields governing nonlinear network dynamics from limited observations.The approach exploits the sparsity of realistic complex networks and can reduce observation requirements relative to conventional signal reconstruction.

2 Compressive sensing based nonlinear dynamical systems identification

Compressive sensing reconstructs nonlinear dynamical equations from limited time-series measurements by exploiting sparse representations. The reviewed results extend this approach from chaotic-system identification to bifurcation prediction, tipping-point analysis, and forecasting future attractors in time-varying systems.

Compressive sensing framework: Compressive sensing recovers sparse dynamical equations from far fewer measurements than unknown terms by solving an l1-norm convex optimization problem.The measurements are linear projections, with M ≪ N, and recovery uses a convex optimization formulation.
Compressive sensing framework: Nonlinear-system identification remains difficult because network structure, nodal dynamics, and governing equations are often unknown while only limited time series are available.The inverse problem seeks both mathematical evolution equations and interaction topology from data.
Chaotic-system identification: For chaotic maps and oscillators, compressive sensing identifies governing coefficients accurately from extremely few data points, including the Hénon map, standard map, Lorenz system, and Rössler oscillator.Narrow coefficient distributions identify existent terms, while similar results were reported across these systems.
Chaotic-system identification: Predicted Lorenz and Rössler equations reproduce the original systems’ bifurcation diagrams, allowing critical bifurcation points to be predicted from time series alone.The predicted system equations agree well with those of the original systems.
Time-varying systems: Compressive sensing can model nonstationarity by incorporating time as an independent expansion variable, enabling prediction of future states and asymptotic attractors from measured time series.The approach targets systems whose equations or parameters vary adiabatically with time but are otherwise unknown.
Time-varying systems: For a time-varying Lorenz system, accurate prediction required only about 5% of data points in the sparsest tested expansion, despite the underlying system’s time dependence.This case used N = 357 possible terms, with the threshold defined by Enz = 10^-3.

3 Compressive sensing based reconstruction of complex networked systems

Compressive sensing has been applied to reconstruct several classes of complex dynamical networks, including coupled oscillators, evolutionary games, hidden-node systems, synchronization dynamics, spreading processes, and geospatial networks.

Compressive sensing has been introduced for reconstructing continuous-time coupled oscillator networks and evolutionary game dynamics on networks.
The approach has also been used to detect hidden nodes and predict or control synchronization dynamics.
Further applications include reconstructing spreading dynamics from binary data and estimating time delays in complex geospatial networks.

3.1 Reconstruction of coupled oscillator networks

The review presents compressive sensing as a framework for reconstructing sparse coupled-oscillator network dynamics, including both node equations and interaction topology from limited time-series data. Simulations show accurate, robust recovery when measurements exceed a critical amount, with performance supported by sparse connectivity.

Method: Compressive sensing estimates sparse coefficients in power-series expansions to reconstruct both oscillator dynamics and network topology.Sparse network structure and sparse dynamical couplings make the coefficient vector suitable for compressive-sensing reconstruction.
Method: Nonzero coupling terms identify links between nodes, while separating coupling terms from local terms recovers the nodal dynamics.A nonzero term associated with another node indicates a coupling; once the coefficients are determined, local dynamics and couplings are known.
Method: The formulation expands each component into power-series terms, constructs a measurement vector from sampled derivatives, and solves X = G ·a for the coefficient vector.For m dynamical variables and expansion order n, the coefficient vector contains N(n + 1)^m components.
Results: All existing couplings were accurately inferred in representative random Lorenz and scale-free Rössler networks, including heterogeneous interactions.For the Rössler example, the combined term −8z represents the local term −z plus coupling to seven neighboring nodes.
Results: Prediction errors become effectively zero after measurements exceed a critical value, and remain small for sparse networks across network sizes.The method requires far fewer data points than the number of power-series terms, but sufficiently low sampling frequency is needed to cover the phase space.
Results: The separated coefficient distributions make existing and nonexistent links distinguishable, producing 100% success rates in the reported reconstruction.Existing links have nonzero coupling strength whereas nonexistent links are effectively zero.

3.2 Reconstruction of complex networks with evolutionary-game dynamics

The review applies compressive sensing to infer evolutionary-game network topology from strategy and payoff time series. Simulations recover standard network types with very little data, while experiments recover social ties with a higher data requirement and reveal degree-dependent payoff patterns.

Method: Evolutionary-game reconstruction uses the relationship between agents’ payoffs and strategies to estimate sparse neighboring vectors and assemble the adjacency matrix.The payoff data determine the measurement vectors and matrices, while network sparsity enables compressive sensing.
Experimental setting: The method was tested on prisoner’s dilemma and snowdrift games implemented on random, small-world, and scale-free networks.Success rates for existent and nonexistent links quantify topology-reconstruction performance.
Simulation results: 100% topology-reconstruction success requires normalized data lengths of 0.3–0.4 for random and small-world networks and about 0.5 for scale-free networks.The larger scale-free requirement is attributed to hubs with denser connections, although neighboring vectors remain sparse.
Noise: Strategy-update noise can improve reconstruction, while compressive sensing provides immunity to measurement noise.The reported positive role of update noise is associated with increased exploration of the system’s state space.
Weighted networks: For randomly weighted scale-free networks, link-weight and nonexistent-link errors approach zero when relative data size exceeds about 0.4.The result indicates successful prediction of random link weights without failure or redundancy.
Experimental results: The experimental winner had only two neighbors, whereas high-degree players had approximately average normalized payoffs.Payoffs were highly non-uniform at smaller degrees and more similar at higher degrees.
Scope and limitation: Applying the framework to gene-regulatory networks requires representing Hill interactions with a basis that preserves the sparsity requirement.This is identified as a challenge for extending the method beyond evolutionary-game interactions.

3.3 Detection of hidden nodes in complex networks

Hidden-node detection infers inaccessible nodes by identifying anomalous reconstruction patterns in their accessible neighbors. Compressive sensing and cancellation-factor analysis help separate hidden-node effects from local noise, while detection remains constrained by weak coupling and strong noise.

Principle of hidden-node detection: Hidden nodes are inferred from observations by identifying accessible nodes whose reconstruction quantities exhibit characteristic anomalies.A hidden node is located by identifying its immediate neighbors, which are affected by incomplete measurement information.
Compressive-sensing formulation: Compressive sensing reconstructs weighted network dynamics and topology from measured time series, using sparse coefficient representations and basis-function expansions.The approach represents isolated nodal dynamics with sparse coefficients and determines coefficients and coupling weights from time series.
Limitations: Detection becomes harder when hidden-node coupling is weak or comparable to background noise, and intrinsic hubs can mimic reconstruction-induced dense patterns.Links with strength comparable to or below background noise may not be detected, while intrinsic network density can resemble hidden-node anomalies.
Noise mitigation: Cancellation-factor analysis distinguishes hidden-node effects from local noise by tracking cancellation ratios and coefficient variances as the data amount increases.For two nodes influenced by a hidden node, the cancellation ratio approaches unity and variance decreases toward zero; local noise produces different trends.
Extensions: The method extends to discrete-time dynamics and, under certain coupling conditions, to multiple entangled hidden nodes.For discrete-time systems, derivatives can be replaced by agent payoffs when calculating cancellation factors.
Examples without local noise: For the network with hidden node #20, anomalously dense predicted linkages and elevated variance identify nodes #3 and #7 as its immediate neighbors.Using different data segments produces much larger predicted-coefficient variances for nodes #3 and #7 than for other accessible nodes.
Examples without local noise: The variance gap between neighboring and non-neighboring nodes provides a quantitative measure of hidden-node detectability.A larger gap indicates more reliable distinction between the hidden node’s neighbors and other nodes.

3.4 Identifying chaotic elements in neuronal networks

Identifying intrinsically chaotic neurons is difficult because coupling can make regular and chaotic neurons appear similarly random. A compressive-sensing reconstruction of neuron equations, parameters, and topology enables isolated-neuron Lyapunov analysis to distinguish them.

Identification challenge: Traditional delay-coordinate embedding and whole-system Lyapunov analysis do not identify which individual neurons are chaotic in a high-dimensional coupled network.The largest Lyapunov exponent of the reconstructed whole system indicates only whether the entire coupled system is chaotic or nonchaotic.
Compressive-sensing approach: The compressive-sensing approach first estimates each neuron’s model parameters, coupling functions, and network topology from measured time series.The framework is applied to reconstruct the FHN parameters and network structure before analyzing each neuron in isolation.
Chaotic-neuron classification: Setting reconstructed coupling parameters to zero and calculating each neuron’s largest Lyapunov exponent distinguishes chaotic neurons from nonchaotic ones.The procedure evaluates the reconstructed isolated neuron dynamics rather than the coupled network as a whole.
Identification challenge: Coupling can make isolated chaotic and regular neurons produce qualitatively similar network time series, obscuring which neurons are intrinsically chaotic.Visual inspection of coupled trajectories provides little indication of each neuron’s isolated dynamical regime.
Reconstruction performance: The reconstruction works with sparse spiky neuronal time series, and estimated single-neuron coefficients converge toward their true values as data points increase.For the single FHN example, all estimated coefficients agree with their true values, with convergence reported after more than 10 data points.
Network reconstruction: The method can recover all network links despite small errors in predicted coupling weights, which mainly arise because system-equation coefficients are large while coupling weights are small.The reported link-identification result concerns the FHN network reconstruction.
Chaotic-neuron classification: In the example network, neuron #1 has a positive largest Lyapunov exponent while all other neurons have negative largest exponents.The exponents are calculated after extracting each neuron’s isolated velocity field from the reconstructed weighted adjacency matrix.

3.5 Data based reconstruction of complex geospatial networks and nodal positioning

Compressive sensing reconstructs complex geospatial networks from time series collected at a single location, recovering topology, coupling delays, and nodal positions despite unknown network details. The framework also supports identifying hidden nodes and estimating their locations.

Reconstruction of geospatial networks: Compressive sensing equations use time-series derivatives to recover nodal dynamics, link weights, and inhomogeneous time delays from a single collection location.The coefficient relations identify coupling weights as B_ij = w_ij and delay terms as C_ij = −w_ij × τ_ij.
Nodal positioning and reconstruction: The reconstructed time delays provide pairwise distances that can be converted into node positions using beacon nodes and triangular localization.At least three known positions are required in two dimensions and at least four in three dimensions; noise generally increases the required number of beacons.
Hidden-node detection: Hidden nodes can be detected through abnormally dense or unstable reconstructed neighborhoods, after which their physical locations can be estimated from neighboring accessible nodes.The approach is demonstrated for a 30-node network initially observed through only 29 normal-node time series.
Hidden-node detection: The method can identify multiple hidden nodes when they do not share common neighboring nodes.Neighboring nodes show the reconstruction irregularities used to ascertain hidden-node existence and support localization.

3.6 Reconstruction of complex spreading networks from binary data

Binary time series can support compressive-sensing reconstruction of spreading-network topology, heterogeneous infection rates, and hidden-source neighborhoods. The framework achieves high accuracy with small data amounts and remains robust to noise and missing observations.

3.6.1 Mathematical formulation: A binary-data compressive-sensing framework reconstructs spreading-network neighborhoods independently, enabling recovery of the full topology, including directed links.The neighboring vector is sparse, and each node’s reconstructed neighborhood can be combined with the others.
3.6.1 Mathematical formulation: The reconstruction requires binary time series and uses base strings whose averaging yields the vector X and matrix G for sparse neighborhood recovery.The strict evaluation criterion requires both existent-link and null-connection success rates to reach 100% for full reconstruction.
Reconstructing networks: Nearly perfect link reconstruction emerges once the normalized number of base strings reaches a relatively small value for SIS and CP dynamics on homogeneous and heterogeneous networks.At n̂_t = 0.1, link and null-connection values overlap; at n̂_t = 0.4, a clear threshold gap enables correct link identification.
Reconstructing networks: About 80% success rates remain achievable when 25% of nodal states flip, while high success rates remain mostly unchanged as unobservable nodes increase from zero to 25%.Even for missing-information fraction n_f = 0.3, a clear gap between actual links and null connections remains, indicating that full link recovery is achievable.
Infection and recovery rates: The framework estimates individual infection rates from binary states after topology reconstruction, with reproduced SIS and CP rates closely agreeing with their true values.The rates are inferred from infection probabilities approximated by infection frequencies and conditioned on the number of infected neighbors.
Hidden-source localization: Structural variance identifies the immediate neighbors of an externally hidden spreading source, allowing the source to be located topologically.In the example, the four source neighbors have much larger structural variance than other nodes.

4 Alternative methods for reconstructing complex, nonlinear dynamical networks

Alternative reconstruction methods infer network structure or dynamical equations using perturbation responses, synchronization, sparse optimization, noise, and automated reverse engineering.

4.1 Reconstructing complex networks from response dynamics: Response dynamics reconstructs oscillator-network topology from measurable phase and frequency differences, with the network matrix obtained by solving ˆJ = D ·θ−1.The method can also estimate interaction strengths, while generally requiring M = N experimental realizations for a unique solution.
4.1 Reconstructing complex networks from response dynamics: 17: Numerical tests found that substantially fewer than N experimental realizations can still yield reasonable reconstruction results.This result exploits sparsity and optimization constraints on the network matrix.
4.2 Reconstructing complex networks via system clone: Synchronization-based reconstruction uses a feedback-controlled clone system to recover connectivities and interactions without prior network-structure knowledge.Lipschitz constraints and sufficiently strong feedback ensure convergence of the clone to the original system with small errors.
4.4 Reconstruction of oscillator networks based on noise induced dynamical correlation: Noise-induced methods can approximate the network adjacency matrix from dynamical correlations without assuming nodal dynamics or imposing external perturbations.This approximation is argued under conditions where noise dominates the evolution of infinitesimal tangent vectors.
4.5 Reverse engineering of complex systems: Automated reverse engineering combines partitioning, probing, and snipping to decouple variables, test candidate models, and simplify the resulting equations.Minimal models retain essential underlying-mechanism features, and the reconstructed subspace is robust to parameter selection.

5 Inference approaches to reconstruction of biological networks

Biological-network reconstruction uses expression, knockout, and transcription-factor data with correlation, rank-based, and local-pattern measures to infer gene associations and interactions.

Biological network data and representations: Gene-network inference distinguishes co-expression networks from transcription-regulatory networks according to node and edge meanings.Common experimental inputs include gene co-expression data, gene knockout data, and transcriptional-factor data.
Value-based association measures: Pearson correlation measures linear association but assumes normally distributed data and is sensitive to outliers.Distance covariance instead provides a nonparametric test for statistical dependence, while Theil-Sen estimation is robust and less outlier-sensitive.
Rank-based methods: Rank-based measures such as Spearman and Kendall correlation are more robust and insensitive to outliers than value-based correlation measures.Kendall’s τ uses concordant and discordant pairs to characterize ranked relationships.
Rank-based methods: Inner Composition Alignment infers directed networks from short time series and, in its partial form, can eliminate indirect interactions.ICA extends Kendall’s τ by reordering one series according to another series’ ranks.
Local expression-rank methods: Local rank-pattern measures address interactions that vary with cellular state or appear only under specific conditions.W1 counts matching or reverse rank patterns in continuous subsequences, whereas W2 generalizes the comparison when temporal order is not meaningful.

5.2 Causality based measures

Causality-based reconstruction includes Wiener–Granger causality and convergent cross mapping, with differing assumptions about linearity, data requirements, and nonlinear dynamics.

Wiener-Granger causality: Wiener–Granger causality identifies Y as causing X when histories of both variables predict X_{t+1} better than X’s history alone.The method is fundamentally linear and assumes a multivariate stochastic-process description.
Limitations and alternatives: Measurement noise can reduce detected Granger causal influence monotonically as noise amplitude increases, producing spurious detection outcomes.Transfer entropy applies to linear and nonlinear systems but often requires prohibitively large data amounts.
Convergent cross mapping: Convergent cross mapping reconstructs delay-coordinate phase spaces and can infer causal influence for linear and nonlinear systems with small data sets.Its applications include neural, ecological, economic, and cerebral-autoregulation data.
Convergent cross mapping: In CCM, cross-map prediction accuracy is quantified by ρY|MX, while R = ρX|MY − ρY|MX measures the relative causal strength.A positive R indicates that x is the CCM cause of y.

5.3 Information-theoretic based methods

Information-theoretic reconstruction methods quantify dependence with mutual information and refine network inference through relevance, redundancy, indirect-edge filtering, and nonparametric association measures.

5.3.1 Mutual information: Mutual information quantifies pairwise mutual dependence, is nonnegative and symmetric, and increases with the variables’ statistical association.Estimating unbiased mutual information from continuous data remains difficult.
MI-based network methods: RELNET, CLR, MRNET, and ARACNE are four commonly used mutual-information-based network reconstruction methods.They respectively use thresholding, z-score significance, relevance–redundancy selection, and indirect-interaction filtering.
MI-based network methods: MRNET repeatedly applies maximum-relevance, minimum-redundancy feature selection to associate direct interactions with high-ranked variables.The greedy procedure selects variables highly informative about a target while reducing redundancy among selected variables.
MI-based network methods: ARACNE applies the Data Processing Inequality to remove indirect interactions, including the weakest edge in qualifying triplets.The procedure begins with edges whose mutual information exceeds a threshold τ and uses tolerance ε for filtering.
Maximal Information Coefficient: MIC searches grid partitions of paired data and maximizes mutual information across grids to uncover relationships in scatterplots.Claims in the literature have questioned whether MIC outperforms mutual information.
Neural network reconstruction: The Ising model fits effective coupling strengths so modeled single-cell and pairwise activity averages agree with numerical or experimental results.Spiking activity is represented by σ_i = ±1 within a maximum-entropy distribution.

5.4 Bayesian network

Bayesian networks represent probabilistic relationships as directed acyclic graphs, with nodes as random variables and links denoting conditional dependencies.

Bayesian networks are probabilistic graphical models represented as directed acyclic graphs.Nodes represent random variables, while directed links signify their conditional dependencies.

5.5 Regression and resampling

Regression-based network reconstruction treats gene regulation as feature selection, commonly assuming sparse links and extending Lasso with resampling or joint data sources.

Lasso infers regulatory networks by predicting a target gene’s expression from transcription factors under an L1 sparsity constraint.
Direct Lasso feature selection is unstable and does not provide confidence scores, motivating integrated stability selection.
Stability selection bootstraps sub-datasets, applies Lasso to each, and aggregates feature scores to select more confident links.
Group Lasso jointly uses steady-state and time-series data, constraining paired transcription-factor coefficients to be simultaneously zero or nonzero.
Nonlinear gene regulation can be modeled with polynomial regression or sigmoid functions.

5.6 Supervised and semi-supervised methods

Supervised and semi-supervised approaches formulate network reconstruction as classification, using labeled expression and prior-information data to infer regulatory links.

5.6 Supervised and semi-supervised methods: Supervised and semi-supervised methods treat network reconstruction as a classification problem requiring expression profiles and prior regulatory information.
5.6 Supervised and semi-supervised methods: SVM learns a decision boundary for regulatory-link classification, while GENIE uses tree ensembles for regression and feature selection.
5.6 Supervised and semi-supervised methods: Semi-supervised inference incorporates unlabeled data into the training set.
5.7 Transfer and joint entropies: Transfer entropy uses historical time series to reveal causal relationships between variables, unlike mutual information alone.
5.7 Transfer and joint entropies: For neuronal networks, spike counts within time windows can serve as transfer-entropy inputs, with historical lengths often set to l=1 and m=1.
5.7 Transfer and joint entropies: Joint entropy uses cross-inter-spike intervals, defined as cISI = t_y − t_x, to incorporate temporal spike patterns.
5.7 Transfer and joint entropies: TE and JE were reported to outperform other methods, while higher-order histories and multiple delays markedly improved TE.

5.8 Distinguishing between direct and indirect interactions

Network inference must distinguish direct from indirect interactions because correlations can misidentify indirect paths as direct links; deconvolution and global silencing address this problem.

5.8 Distinguishing between direct and indirect interactions: Motif analysis identifies fan-out, fan-in, and cascade errors as generic systematic errors in network inference.
5.8 Distinguishing between direct and indirect interactions: Cascade errors arise when indirect links are incorrectly interpreted as direct shortcuts.
5.8 Distinguishing between direct and indirect interactions: Deconvolution models observed network weights as sums of direct and indirect-path contributions to recover direct dependencies.
5.8 Distinguishing between direct and indirect interactions: A matrix similarity transformation separates observed and direct networks, using λ_dir = λ_obs/(1+λ_obs) to infer direct links.
5.8 Distinguishing between direct and indirect interactions: Modular response analysis defines local response coefficients as sensitivity of one module to another while other module states remain unchanged.
5.8 Distinguishing between direct and indirect interactions: Global response coefficients measure changes after an external parameter perturbation propagates through the system and reaches a new steady state.
5.8 Distinguishing between direct and indirect interactions: Global silencing obtains local responses from observed global responses, helping distinguish direct and indirect interactions.

6 Discussions and future perspectives

The Review surveys data-based paradigms for reconstructing complex-network structure and dynamics, emphasizing compressive sensing and related approaches. It also identifies persistent challenges in source localization, universal approximation, and controlling high-dimensional nonlinear networks.

Reconstruction paradigms: Compressive sensing reconstructs sparse signals from limited observations and supports inverse problems in complex networks.The Review applies it to topology, nodal dynamics, hidden-node detection, neuronal chaos, geospatial networks, and binary spreading data.
Reconstruction paradigms: Power-series approximations convert nonlinear system identification into estimation of coefficients governing nodal dynamics and interactions.Although high-order expansions can create many unknown coefficients, the Review describes compressive sensing as suited to this sparse reconstruction task.
Reconstruction paradigms: Alternative reconstruction methods use external driving, synchronization, phase-space linearization, noise-induced correlations, reverse engineering, correlation, causality, information theory, Bayesian inference, regression, and resampling.These approaches extend beyond the compressive-sensing paradigm and include methods reviewed for biological networks.
Open problems and applications: A compressive-sensing framework combined with minimum output analysis can quantify and efficiently locate diffusion sources in complex networks.The framework combines controllability and observability theories with compressive-sensing-based localization.
Universal structural estimator and dynamics approximator: An SDBM represents stochastic first-order Markovian dynamics with local interactions through node biases and undirected link weights.Its joint state distribution is defined from clique potentials, while conditional probabilities depend on current neighboring states and sometimes each node’s own state.
Universal structural estimator and dynamics approximator: Across 14 dynamical processes, a compressive-sensing and K-means framework recovered underlying network structures with almost zero error when data were modeled by equivalent SDBMs.The result supports simultaneous network reconstruction and dynamics approximation under the stated modeling assumption.
Controlling nonlinear and complex dynamical networks: Controlling nonlinear dynamical networks remains difficult because high dimensionality and diverse behaviors limit universal control frameworks.Existing chaos-control methods have mostly addressed low-dimensional systems with few unstable directions, whereas nonlinear complex networks are generally high dimensional.
Controlling nonlinear and complex dynamical networks: Attractor networks provide a nonlinear multistability control framework in which feasible temporary parameter changes define transitions between attractors.Each node denotes an attractor, and directed edges represent experimentally feasible finite transitions between states.