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In this article, we try to merge two completely different (at first glance) areas of research: complex networks and chaotic systems visualization.
Large-scale networks, exhibiting complex patterns of interaction amongst vertices exist in both nature and in man-made systems (i.e., communication networks, genetic pathways, ecological or economical networks, social networks, networks of various scientific collaboration, Internet, World Wide Web, power grid, etc.). The structure of complex networks thus can be observed in many systems.
The term “complex networks” (Dorogovtsev & Mendes, 2002; Boccaletti, Latora, Moreno, Chavez, & Hwang, 2006) comes from the fact that they exhibit substantial and non-trivial topological features, with patterns of connection between vertices that are neither purely regular nor purely random. Such features include a heavy tail in the degree distribution, a high clustering coefficient, and hierarchical structure, amongst other features. In the case of directed networks, these features also include reciprocity, triad significance profile and other features.
Amongst many studies, two well-known and much studied classes of complex networks are the scale-free networks and small-world networks (see examples in Figure 1, Figure 2, and Figure 3, with edges that are self-loops) (positive feedback), vertices having more outgoing edges than incoming, vertices have the same outgoing and incoming edges, and the most profitable vertices – more incoming than outgoing edges), whose discovery and definition are vitally important in the scope of this research. Specific structural features can be observed in both classes i.e., so called power-law degree distributions for the scale-free networks and short path lengths with high clustering for the small-world networks. Research in the field of complex networks has joined together researchers from many areas, which were outside of this interdisciplinary research in the past like mathematics, physics, biology, chemistry computer science, epidemiology, etc. As an example of another exhibition of the existence of complex networks are evolutionary algorithms, whose dynamics can be visualized like complex network structure and dynamics.
Figure 1. Example 1: Complex network
Figure 2. Example 2: Complex network with more vertices
Figure 3. Example 3: Complex network, note isolated vertices that are not connected to the main network
Evolutionary computation is a sub-discipline of computer science belonging to the “bio-inspired” computing area. Since the end of the Second World War, the main ideas of evolutionary computation has been published (Turing, 1969) and widely introduced to the scientific community (Holland, 1975). Hence, the “golden era” of evolutionary techniques began, when Genetic Algorithms (GA) by Holland (1975), Evolutionary Strategies (ES), by Schwefel (1974), Rechenberg (1973), and Evolutionary Programming (EP) by Fogel (1998) had been introduced. All these designs were favored by the forthcoming of more powerful and more easily programmable computers, so that for the first time interesting problems could be tackled and evolutionary computation started to compete with and became a serious alternative to other optimization methods.