Recent advances have demonstrated the effectiveness of a machine-learning approach known as “reservoir computing” for model-free prediction of chaotic systems. We find that a well-trained reservoir computer can synchronize with its learned chaotic systems by linking them with a common signal. A necessary condition for achieving this synchronization is the negative values of the sub-Lyapunov exponents. Remarkably, we show that by sending just a scalar signal, one can achieve synchronism in trained reservoir computers and a cascading synchronization among chaotic systems and their fitted reservoir computers. Moreover, we demonstrate that this synchronization is maintained even in the presence of a parameter mismatch. Our findings possibly provide a path for accurate production of all expected signals in unknown chaotic systems using just one observational measure.

We investigate, for the first time, navigation on networks with a Lévy walk strategy such that the step probability scales as p_ij ~ d_ij^–α, where d_ij is the Manhattan distance between nodes i and j and α is the transport exponent. We find that the optimal transport exponent α^opt of such a diffusion process is determined by the fractal dimension d_f of the underlying network. Specially, we theoretically derive the relation α^opt = d_f + 2 for synthetic networks and we demonstrate that this holds for a number of real-world networks. Interestingly, the relationship we derive is different from previous results for Kleinberg navigation without or with a cost constraint, where the optimal conditions are α = d_f and α = d_f + 1, respectively. Our results uncover another general mechanism for how network dimension can precisely govern the efficient diffusion behavior on diverse networks.

Significant advances have recently been made in modeling chaotic systems with the reservoir computing approach, especially for prediction. We find that although state prediction of the trained reservoir computer will gradually deviate from the actual trajectory of the original system, the associated geometric features remain invariant. Specifically, we show that the typical geometric metrics including the correlation dimension, the multiscale entropy, and the memory effect are nearly identical between the trained reservoir computer and its learned chaotic systems. We further demonstrate this fact on a broad range of chaotic systems ranging from discrete and continuous chaotic systems to hyperchaotic systems. Our findings suggest that the successfully reservoir computer may be topologically conjugate to an observed dynamical system.

We construct temporal networks from time series via unfolding the temporal information into an additional topological dimension of the networks. Thus, we are able to introduce memory entropy analysis to unravel the memory effect within the considered signal. We find distinct patterns in the entropy growth rate of the aggregate network at different memory scales for time series with different dynamics ranging from white noise, 1/f noise, autoregressive process, periodic to chaotic dynamics. Interestingly, for a chaotic time series, an exponential scaling emerges in the memory entropy analysis. We demonstrate that the memory exponent can successfully characterize bifurcation phenomenon, and differentiate the human cardiac system in healthy and pathological states. Moreover, we show that the betweenness preference analysis of these temporal networks can further characterize dynamical systems and separate distinct electrocardiogram recordings. Our work explores the memory effect and betweenness preference in temporal networks constructed from time series data, providing a new perspective to understand the underlying dynamical systems.

We investigate multiple random walks traversing independently and concurrently on complex networks and introduce the concept of mean first parallel passage time (MFPPT) to quantify their search efficiency. The mean first parallel passage time represents the expected time required to find a given target by one or some of the multiple walkers. We develop a general theory that allows us to calculate the MFPPT analytically. Interestingly, we find that the global MFPPT follows a harmonic law with respect to the global mean first passage times of the associated walkers. Remarkably, when the properties of multiple walkers are identical, the global MFPPT decays in a power law manner with an exponent of unity, irrespective of network structure. These findings are confirmed by numerical and theoretical results on various synthetic and real networks. The harmonic law reveals a universal principle governing multiple random walks on networks that uncovers the contribution and role of the combined walkers in a target search. Our paradigm is also applicable to a broad range of random search processes.

Anomalous random walks having long-range jumps are a critical branch of dynamical processes on networks, which can model a number of search and transport processes. However, traditional measurements based on mean first passage time are not useful as they fail to characterize the cost associated with each jump. Here we introduce a new concept of mean first traverse distance (MFTD) to characterize anomalous random walks that represents the expected traverse distance taken by walkers searching from source node to target node, and we provide a procedure for calculating the MFTD between two nodes. We use Lévy walks on networks as an example, and demonstrate that the proposed approach can unravel the interplay between diffusion dynamics of Lévy walks and the underlying network structure. Moreover, applying our framework to the famous PageRank search, we show how to inform the optimality of the PageRank search. The framework for analyzing anomalous random walks on complex networks offers a useful new paradigm to understand the dynamics of anomalous diffusion processes, and provides a unified scheme to characterize search and transport processes on networks.