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Symmetry 2011, 3(1), 1-15; doi:10.3390/sym3010001
Symmetry in Complex Networks
Received: 16 November 2010; in revised form: 4 January 2011 / Accepted: 7 January 2011 / Published: 10 January 2011
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract: In this paper, we analyze a few interrelated concepts about graphs, such as their degree, entropy, or their symmetry/asymmetry levels. These concepts prove useful in the study of different types of Systems, and particularly, in the analysis of Complex Networks. A System can be defined as any set of components functioning together as a whole. A systemic point of view allows us to isolate a part of the world, and so, we can focus on those aspects that interact more closely than others. Network Science analyzes the interconnections among diverse networks from different domains: physics, engineering, biology, semantics, and so on. Current developments in the quantitative analysis of Complex Networks, based on graph theory, have been rapidly translated to studies of brain network organization. The brain's systems have complex network features—such as the small-world topology, highly connected hubs and modularity. These networks are not random. The topology of many different networks shows striking similarities, such as the scale-free structure, with the degree distribution following a Power Law. How can very different systems have the same underlying topological features? Modeling and characterizing these networks, looking for their governing laws, are the current lines of research. So, we will dedicate this Special Issue paper to show measures of symmetry in Complex Networks, and highlight their close relation with measures of information and entropy.
Keywords: graph theory; applications of graph theory; group theory fuzzy sets; fuzzy logic; logic of vagueness; fuzzy topology; Fuzzy Measure theory; fuzzy real analysis; Small World graphs; Complex Networks; artificial intelligence