Introduction: Why Complex Systems?
In our daily lives, we have to run many routines for regular errands, such as putting on clothes in the morning, brushing teeth before breakfast, and tying shoe laces prior to leaving the house. If you think much about these routines, you should realize that they are not really the same steps or daily procedures. However, we give them the same labels and treat them the same, although these routines are constantly changing "routines." To survive, we human beings have the tendency to turn a dynamic world into named objects, and assume these terms have been fixed. Otherwise, we cannot move forward at all if we have to wonder from moment to moment which foot has to be extended. Even scientists have to simplify real-world phenomena to construct models, theories or laws. More than one hundred years ago, physicists began to realize the flaws of reductionism and started the topic of statistical physics. Even so, that theory is still based on an analytic method in forms of formulae. A half century ago, the technological progress of computers brought the ability to handle non-linear differential equations in a better way. Researchers began to study complex systems by paying attention to the small or negligible factors which have explosive results due to the non-linear effects. Starting in the late eighties, with a growing discontentment with scientific analytical methods in science and the growing power of computers, researchers began to study complex systems such as living organisms, evolution of genes, biological systems, brain neural networks, epidemics, ecology, economy, social networks, etc. In the early nineties, the research gradually spread into the language field. Simply put, complex system theories are beginning to reflect the real world in a more realistic way.
Topology of Complex Systems
A complex system is made of nodes and links. Nodes are the elements or members of the system, and links are the connections between nodes. Links can be directed or undirected, weighted or unweighted. Based on the structure of nodes and links, one can characterize the system by calculating the average node degree, average path length, and clustering coefficient of the system. The calculation of average node degree can be illustrated by the following diagram:
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Similarly, the average path length can be figured out as the following:
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The local clustering coefficient of a node i is given by the ratio between the number of links among the neighbors of node i and the maximum possible number of links among these neighbors. This can be expressed as [c.sub.i] = [e.sub.i]/[[k.sub.i]([k.sub.i] - 1)]/2 in which [c.sub.i] is the clustering coefficient, [e.sub.i] is the number of links, and [k.sub.i] is the number of nodes (3). The difference between a strong and weak local clustering system can be shown in the following:
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When networks have a short average path length and strong local clustering, they are called small-world systems, in which the neighbors of a given node are more likely to be connected to one another than would be expected through chance alone. Research has also discovered that many real-world systems demonstrate the scale-free characteristic that the system starts with a simple structure and duplicates or iterates the structure to construct a complex system. This can be illustrated by the following artificially constructed depiction:
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The scale-free network indicates that the majority of the nodes have a small amount of links, but a few nodes called hubs, can link to most of nodes in the system. The frequency distribution of the system as shown below has a power law relationship:
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In summary, a complex system has the following characteristics: (a) its behavior emerges from the interactions of its components and the interaction sometimes is leading to self-organization and the emergence of new behavior; (b) the components change and adapt in response to feedback; (c) it is nonlinear, unpredictable and disproportional to its causal factors; and (d) the emergence is temporary but coherent, coming into existence of new forms through an ongoing process intrinsic to the system. …