^{1,2}, Jie Ren

^{3,4,6}, Ying-Cheng Lai

^{2,5}and Baowen Li

^{3,4}

### Abstract

Reverse engineering of complex dynamical networks is important for a variety of fields where uncovering the full topology of unknown networks and estimating parameters characterizing the network structure and dynamical processes are of interest. We consider complex oscillatornetworks with *time-delayed interactions* in a noisy environment, and develop an effective method to infer the full topology of the network and evaluate the amount of time delay based solely on noise-contaminated time series. In particular, we develop an analytic theory establishing that the dynamical correlation matrix, which can be constructed purely from time series, can be manipulated to yield both the network topology and the amount of time delay simultaneously. Extensive numerical support is provided to validate the method. While our method provides a viable solution to the networkinverse problem, significant difficulties, limitations, and challenges still remain, and these are discussed thoroughly.

Time-delayed interactions are common in complex systems arising from various fields of science and engineering. Consider, for example, a coupled oscillatornetwork in a physical environment where noise is present. Time delay can typically occur in the node-to-node interactions. Now suppose that no prior knowledge about the nodal dynamics and the network topology is available but only a set of noise-contaminated time series can be obtained through measurements. The question is whether it is possible to deduce the full topology of the network and to estimate the amount of average time delay using the time series only. This issue belongs to the recently emerged subfield of research in complex systems: reverse engineering of complex networks (or the inverse problem). While a number of methods address the networkinverse problem have appeared, to our knowledge, the issue of time-delayed interactions has not been considered. Here we present an effective method to infer the full network topology and, at the same time, to estimate the amount of average time delay in the network. In particular, we develop a physical theory to obtain a formula relating the network topology and time delay to the dynamical correlation matrix, which can be constructed purely from time series. We then show how information about the time delay encrypted in the dynamical correlation matrix can be separated from that of network topology, allowing both to be inferred in a computationally extremely efficient manner. We present numerical examples from both model and real-world complex networks to demonstrate the working of our method. Difficulties, limitations, and challenges are also discussed. Reverse-engineering of complex dynamical systems has potential applications in many disciplines, and our work represents a small step forward in this extremely challenging area.

J.R. thanks Dr. Gang Yan for useful discussions. W.X.W. and Y.C.L. are supported by AFOSR under Grant No. FA9550-10-1-0083. W.X.W. is supported by NSFC under Grant No. 11105011.

I. INTRODUCTION

II. THEORY AND PREDICTION METHOD

III. NUMERICAL RESULTS

IV. VARIANT OF COUPLED OSCILLATORNETWORK MODEL

V. CONCLUSION AND OUTLOOK

### Key Topics

- Network topology
- 31.0
- Time series analysis
- 20.0
- Networks
- 16.0
- Oscillators
- 16.0
- Inverse problems
- 14.0

## Figures

Diagonal elements of the dynamical correlation matrix as a function of node degree *k* for three dynamical processes with different values of the time delay on scale-free and random networks. Square, circle, triangle and reverse triangle denote , 0.05, 0.07, and 0.09, respectively. The curves are the theoretical prediction from Eq. (15). The sizes of model networks are 100 and the average degree is 10. The noise strength is 0.1 and the coupling strength *c* is 0.2.

Diagonal elements of the dynamical correlation matrix as a function of node degree *k* for three dynamical processes with different values of the time delay on scale-free and random networks. Square, circle, triangle and reverse triangle denote , 0.05, 0.07, and 0.09, respectively. The curves are the theoretical prediction from Eq. (15). The sizes of model networks are 100 and the average degree is 10. The noise strength is 0.1 and the coupling strength *c* is 0.2.

Example of the distribution of the values of elements of the generalized inverse of the dynamical correlation matrix **C** for consensus dynamics associated with a scale-free network, where . The bimodal behavior is present for Kuramoto model and Rössler dynamics as well.

Example of the distribution of the values of elements of the generalized inverse of the dynamical correlation matrix **C** for consensus dynamics associated with a scale-free network, where . The bimodal behavior is present for Kuramoto model and Rössler dynamics as well.

Success rate of prediction of existent links for (a) consensus dynamics, (b) Kuramoto oscillators, and (c) Rössler dynamics as a function of time delay for a number of model and real-world networks: scale-free networks (scale-free),^{27} random network (random),^{28} small-world network (small-world),^{29} dolphin social network (dolphins),^{30} network of American football games among colleges (football),^{31} friendship network of karate club (karate),^{32} and network of political book purchases (book).^{33} Other parameters are the same as in Fig. 1. The success rate of nonexistent links is higher than 0.99 for all considered cases and thus are not shown.

Success rate of prediction of existent links for (a) consensus dynamics, (b) Kuramoto oscillators, and (c) Rössler dynamics as a function of time delay for a number of model and real-world networks: scale-free networks (scale-free),^{27} random network (random),^{28} small-world network (small-world),^{29} dolphin social network (dolphins),^{30} network of American football games among colleges (football),^{31} friendship network of karate club (karate),^{32} and network of political book purchases (book).^{33} Other parameters are the same as in Fig. 1. The success rate of nonexistent links is higher than 0.99 for all considered cases and thus are not shown.

Predicted time delay from Eq. (19) versus the true (pre-assumed) values for the three dynamical processes on a number of model and real-world networks. The symbols denote the same networks as in Fig. 3. The lines are . Other parameters are the same as in Fig. 1.

Predicted time delay from Eq. (19) versus the true (pre-assumed) values for the three dynamical processes on a number of model and real-world networks. The symbols denote the same networks as in Fig. 3. The lines are . Other parameters are the same as in Fig. 1.

(a) Success rate of prediction of existent links as a function of the average time delay for different ranges of time delays for random consensus networks. (b) Predicted average time delay versus the original time delay for different ranges of time delay. The lines are . Other parameters are the same as in Fig. 1.

(a) Success rate of prediction of existent links as a function of the average time delay for different ranges of time delays for random consensus networks. (b) Predicted average time delay versus the original time delay for different ranges of time delay. The lines are . Other parameters are the same as in Fig. 1.

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