Special Issue "Symmetry in Complex Networks II"
A special issue of Symmetry (ISSN 2073-8994).
Deadline for manuscript submissions: closed (30 November 2016)
Prof. Dr. Angel Garrido
Department of Fundamental Mathematics, Faculty of Sciences, UNED, Paseo Senda del Rey No. 9, 28040 Madrid, Spain
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Interests: mathematical analysis; measure theory; fuzzy measures, in particular symmetry and entropy; graph theory; discrete mathematics; automata theory; mathematical education; heuristics; automata theory; artificial intelligence
In 2011 we published a Special Issue on “Complex Networks” and, now, I have been asked to be the Guest Editor of a collection of new contributions on the subject. I gladly accepted this new challenge, and offer to colleagues a new ocassion for research on these exciting topics.
As we know, symmetry in a system means the invariance of its elements under conditions of transformation. When we take network structures, their symmetry means an invariance of the adjacency of nodes under the permutations of the node set. Graph isomorphism is an equivalence relation on the set of graphs. Therefore, we have partitioned the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structures, if we omit the individual characteristics of their components. A set of graphs isopmorphic to each other is usually known as an isomorphism class of graphs. The automorphism of a graph will be an isomorphism from G onto itself. The family of all automorphisms of a graph G is a permutation group. The inner operation of such a group will be the composition of permutations. It is called the Automorphism Group of G, and is denoted by Aut (G). Conversely, all groups may be represented as the automorphism group of a connected graph. The automorphism group is an algebraic invariant of a graph. Thus, we can say that the automorphism of a graph is a form of symmetry, in which the graph is mapped onto itself while preserving the edge-node connectivity. We will say that we have either a graph invariant or a graph property, when it only depends on the abastract stucture and not on graph representations, such as particular labeling or drawings on the graph. Thus, we may define a graph property as an property that is preserved under all its possible isomorphism of the graph. Because there exist a property of the graph itself, not depending on the representation of the graph. The semantic difference also consists in its character: a qualitative or quantitative one. From a strictly mathematical viewpoint, a graph’s property can be interpreted as a class of graphs, composed by the graphs that have the accomplishment of having some conditions in common.
Here, we need to analyze very interrelated concepts regarding graphs, such as their Symmetry/Asymmetry levels, or degrees, their entropies, etc. It may be applied when we study the different types of Systems; particularly, analyzing 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, thus, we can focus on those aspects that interact more closely than others. Network Science is a new scientific field that analyzes the interconnection among diverse networks; for instance, among Physics, Engineering, Biology, Semantics, and so on. We may distinguish four structural models when we describe Complex Systems by Complex Networks, i.e., using Graph Theory. Thus, we can mention Regular Networks, Random Networks, Small-World Networks, and Scale-Free Networks. However, it is also possible to introduce some new versions, according to new measures.
Complex Networks are everywhere. Many phenomena in nature can be modeled as a network. The topology of different networks may be very similar. They are rooted in the Power Law, with a scale-free structure. How can very different systems have the same underlying topological features? Searching for the hidden laws of these networks, modelling, and characterizing them are current lines of research.
Symmetry and Asymmetry may be considered (on graphs and networks in general) as two sides of the same coin, but such a dichotomous classification shows a lack of necessary and realistic grades. Thus, it is convenient to introduce "shade regions", modulating their degrees. The parallel versions of different mathematical fields adapted to degrees of truth are advancing. The basic idea according to which an element does not necessarily totally belong, or does not belong in absolute, to a set, but it can belong more or less, i.e., to some degree, which signifies a change of paradigm, adapting mathematics to features of the real world. Thus, we create new tools and fields, such as Fuzzy Measure Theory, which generalizes the classical Measure Theory. We wish to dedicate this Special Issue to show measures of symmetry, very related to the measures of information and entropy.
Contributions are invited on all aspects of symmetry measures, as applied to all complex networks and systems. Pure mathematical treatments that are applicable to such concepts are welcome. Possible themes include, but are not limited to:
Symmetry and Asymmetry
Symmetry as a new and very important bridge between sciences and humanities
Prof. Dr. Angel Garrido
Manuscript Submission Information
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