### Abstract

With the rapid technological advancement, network is almost everywhere in our daily life. Network theory leads to a new way to investigate the dynamics of complex systems. As a result, many methods are proposed to construct a network from nonlinear time series, including the partition of state space, visibility graph, nearest neighbors, and recurrence approaches. However, most previous works focus on deriving the adjacency matrix to represent the complex network and extract new network-theoretic measures. Although the adjacency matrix provides connectivity information of nodes and edges, the network geometry can take variable forms. The research objective of this article is to develop a self-organizing approach to derive the steady geometric structure of a network from the adjacency matrix. We simulate the recurrence network as a physical system by treating the edges as springs and the nodes as electrically charged particles. Then, force-directed algorithms are developed to automatically organize the network geometry by minimizing the system energy. Further, a set of experiments were designed to investigate important factors (i.e., dynamical systems, network construction methods, force-model parameter, nonhomogeneous distribution) affecting this self-organizing process. Interestingly, experimental results show that the self-organized geometry recovers the attractor of a dynamical system that produced the adjacency matrix. This research addresses a question, i.e., “what is the self-organizing geometry of a recurrence network?” and provides a new way to reproduce the attractor or time series from the recurrence plot. As a result, novel network-theoretic measures (e.g., average path length and proximity ratio) can be achieved based on actual node-to-node distances in the self-organized network topology. The paper brings the physical models into the recurrence analysis and discloses the spatial geometry of recurrence networks.

The theory of recurrence network derives network measures (or graph theoretical properties) from the recurrence-based adjacency matrix to describe dynamical properties of complex systems. However, the adjacency matrix only defines the connectivity of network nodes but not the geometrical structure of complex networks. A question remains in the literature, i.e., “what is the self-organizing geometry of a recurrence network?” In this present paper, we simulate the recurrence network as a physical system with the use of spring-electrical models. The edges are treated as springs and the nodes are electrically charged particles. As a result, the attractive and repulsive forces direct the network to self-organize into a stable geometry that minimizes the system energy. Experimental results show that the self-organized geometry of a recurrence network recovers the attractor of a nonlinear dynamical system that produced the recurrence adjacency matrix. This finding demonstrates that the recurrence matrix retains sufficient information of nonlinear dynamical systems. In addition, it provides a new way to reproduce the attractor or time series from the recurrence plot. This present work lays out solid foundations for the theory of recurrence network. The paper discloses the spatial geometry of recurrence networks and brings the physical model into recurrence analysis.

The authors would like to thank the National Science Foundation (CMMI-1266331 and IOS-1146882) for the support of this research.

I. INTRODUCTION

II. RECURRENCE-BASED NETWORKS

III. FORCE-DIRECTED RECURRENCE NETWORKS

IV. EXPERIMENTAL DESIGN

V. RESULTS

A. Effects of network construction methods ( = 1 and unequally spaced attractor)

B. Effects of force-model parameter (recurrence network and unequally spaced attractor)

C. Effects of nonhomogeneous distribution ( = 2 and recurrence network)

VI. DISCUSSION

VII. CONCLUSIONS

### Key Topics

- Networks
- 66.0
- Attractors
- 63.0
- Network topology
- 44.0
- Self assembly
- 22.0
- Topology
- 22.0

## Figures

Graphical illustration of the state space (a) and its recurrence plot (b). The recurrence plot characterizes the proximity of state vectors, i.e., whether or not the state-space distance between two “state” and is below a certain recurrence threshold . It is mathematically defined as , where Θ is the Heaviside function and is a distance measure.

Graphical illustration of the state space (a) and its recurrence plot (b). The recurrence plot characterizes the proximity of state vectors, i.e., whether or not the state-space distance between two “state” and is below a certain recurrence threshold . It is mathematically defined as , where Θ is the Heaviside function and is a distance measure.

The example of an adjacency matrix in a complex recurrence network.

The example of an adjacency matrix in a complex recurrence network.

Illustrations of the self-organizing process of complex network based on the nodes and edges in the recurrence matrix. (a)–(f) Iterative self-organization of 292 nodes in a network at 300, 600, 900, 1200, 1800, and 1900 iterations. The optimal two-wing layout is achieved at 1900 iterations with a minimal energy and a steady topological structure.

Illustrations of the self-organizing process of complex network based on the nodes and edges in the recurrence matrix. (a)–(f) Iterative self-organization of 292 nodes in a network at 300, 600, 900, 1200, 1800, and 1900 iterations. The optimal two-wing layout is achieved at 1900 iterations with a minimal energy and a steady topological structure.

Cause and effect diagram for performance evaluation of self-organizing network algorithms.

Cause and effect diagram for performance evaluation of self-organizing network algorithms.

(a) Lorenz attractor (i.e., unequally spaced) generated from the equations: with the step , (b)equally spaced Lorenz attractor, (c) original Rossler attractor (i.e., unequally spaced) generated from the equations: with the step , and (d) equally spaced Rossler attractor.

(a) Lorenz attractor (i.e., unequally spaced) generated from the equations: with the step , (b)equally spaced Lorenz attractor, (c) original Rossler attractor (i.e., unequally spaced) generated from the equations: with the step , and (d) equally spaced Rossler attractor.

The adjacency matrices of original Lorenz attractor that are derived with the use of (a) recurrence network (i.e., a fixed size of the neighborhood) and (b) k-nearest neighbors network (i.e., a fixed number of neighbors).

The adjacency matrices of original Lorenz attractor that are derived with the use of (a) recurrence network (i.e., a fixed size of the neighborhood) and (b) k-nearest neighbors network (i.e., a fixed number of neighbors).

The self-organizing process of complex network based on the nodes and edges in the recurrence-based adjacency matrix of Lorenz system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of complex network based on the nodes and edges in the recurrence-based adjacency matrix of Lorenz system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of complex network based on the nodes and edges in the KNN-based adjacency matrix of Lorenz system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of complex network based on the nodes and edges in the KNN-based adjacency matrix of Lorenz system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The iterative variations of the system energy for the self-organizing process in the complex network using the (a) recurrence-based adjacency matrix and (b) KNN-based adjacency matrix.

The iterative variations of the system energy for the self-organizing process in the complex network using the (a) recurrence-based adjacency matrix and (b) KNN-based adjacency matrix.

The adjacency matrices of original Rossler attractor that are derived with the use of (a) recurrence network (i.e., a fixed size of the neighborhood) and (b) k-nearest neighbors network (i.e., a fixed number of neighbors).

The adjacency matrices of original Rossler attractor that are derived with the use of (a) recurrence network (i.e., a fixed size of the neighborhood) and (b) k-nearest neighbors network (i.e., a fixed number of neighbors).

The self-organizing process of complex networks based on the nodes and edges in the recurrence-based adjacency matrix of Rossler system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of complex networks based on the nodes and edges in the recurrence-based adjacency matrix of Rossler system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of complex networks based on the nodes and edges in the KNN-based adjacency matrix of Rossler system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of complex networks based on the nodes and edges in the KNN-based adjacency matrix of Rossler system. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Lorenz system with the force-model parameter p equal to 2. (a-h): Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Lorenz system with the force-model parameter p equal to 2. (a-h): Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Lorenz system with the force-model parameter p equal to 3. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Lorenz system with the force-model parameter p equal to 3. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Rossler system with the force-model parameter p equal to 2. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Rossler system with the force-model parameter p equal to 2. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Rossler system with the force-model parameter p equal to 3. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network of Rossler system with the force-model parameter p equal to 3. (a)-(h) Iterative self-organization for the random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The recurrence-based adjacency matrices of equally spaced (a) Lorenz attractor and (b) Rossler attractor.

The recurrence-based adjacency matrices of equally spaced (a) Lorenz attractor and (b) Rossler attractor.

The self-organizing process of recurrence-based complex network for the equally spaced Lorenz attractor with the force-model parameter p equal to 2. (a)-(h) The random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network for the equally spaced Lorenz attractor with the force-model parameter p equal to 2. (a)-(h) The random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network for the equally spaced Rossler attractor with the force-model parameter p equal to 2. (a)-(h) The random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The self-organizing process of recurrence-based complex network for the equally spaced Rossler attractor with the force-model parameter p equal to 2. (a)-(h) The random layout, 1000, 2000, 3000, 4000, 5000, 6000, and 7000 iterations.

The flow chart of both geometrical and statistical investigations of recurrence networks.

The flow chart of both geometrical and statistical investigations of recurrence networks.

## Tables

Design of experiments to evaluate the geometry of self-organizing network.

Design of experiments to evaluate the geometry of self-organizing network.

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