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Self-organized topology of recurrence-based complex networks
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10.1063/1.4829877
Hui Yang1,a) and Gang Liu1
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Affiliations:
1 Complex Systems Monitoring, Modeling and Analysis Laboratory, University of South Florida, Tampa, Florida 33620, USA
a) Author to whom correspondence should be addressed. Electronic mail: huiyang@usf.edu. Tel.: (813) 974-5579. Fax: (813) 974-5953.
Chaos 23, 043116 (2013)
/content/aip/journal/chaos/23/4/10.1063/1.4829877
http://aip.metastore.ingenta.com/content/aip/journal/chaos/23/4/10.1063/1.4829877

## Figures

FIG. 1.

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.

FIG. 2.

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

FIG. 3.

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.

FIG. 4.

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

FIG. 5.

(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.

FIG. 6.

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).

FIG. 7.

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.

FIG. 8.

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.

FIG. 9.

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.

FIG. 10.

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).

FIG. 11.

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.

FIG. 12.

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.

FIG. 13.

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

FIG. 14.

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

FIG. 15.

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

FIG. 16.

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

FIG. 17.

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

FIG. 18.

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

FIG. 19.

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

FIG. 20.

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

## Tables

Table I.

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

/content/aip/journal/chaos/23/4/10.1063/1.4829877
2013-11-08
2014-04-21

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