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# Symmetry Measures on Complex Networks

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: closed (30 September 2011) | Viewed by 34899

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## Special Issue Editor

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Guest Editor
Department of Fundamental Mathematics, Faculty of Sciences, National University of Distance Learning (UNED), 28040 Madrid, Spain
Interests: mathematical analysis; measure theory; fuzzy measures, in particular symmetry and entropy; graph theory; discrete mathematics; automata theory; mathematical education; heuristics; artificial intelligence
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## Special Issue Information

Dear Colleagues,

As we know, Symmetry in a system means invariance of its elements under conditions of transformations. When we take network structures, its symmetry means invariance of adjacency of nodes under the permutations on node set. The graph isomorphism is an equivalence relation on the set of graphs. Therefore, it partitions the class of all graphs into equivalence classes. The underlying idea of isomorphism is that some objects have the same structure, if we omit the individual character of their components. A set of graphs isomorphic to each other is denominated 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 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. So, 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 either graph invariant or graph property, when it depends only on the abstract structure, not on graph representations, such as particular labelings or drawings of the graph. So, we may define a graph property as every property that is preserved under all its possible isomorphisms of the graph. Therefore, it will be 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 property can be interpreted as a class of graphs, composed by the graphs that have in common the accomplishment of some conditions.

We need to analyze here very interrelated concepts about 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 so, 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. Among its developers, we may remember Duncan Watts, with the Small-World Network; Réka Albert and Albert-László Barabasi, who developed the Scale-Free Network. In his work, Barabási found that the WWW, as a network, has very interesting mathematical properties. Network Theory is a quickly expanding area of Computer Science and Mathematics, and may be considered as an essential component of Graph Theory. Usually we may distinguish four structural models when we describe Complex Systems by Complex Networks, i.e. using Graph Theory. So, we can mention Regular Networks; Random Networks; Small-World Networks, and Scale-Free Networks. But also it is possible to introduce some new versions, according to the new measures of Symmetry/Asymmetry Level Measures. Complex Networks are everywhere. Many phenomena in nature can be modelled as a network. The topology of different networks may be very similar. They are rooted on the Power Law, with a scale free structure. How can very different systems have the same underlying topological features? Searching the hidden laws of these networks, modelling, and characterizing them are the 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 dichotomous classification shows a lack of necessary and realistic grades. So, it is convenient to introduce "shade regions", modulating their degrees. The parallel version of different mathematical fields adapted to degrees of truth is advancing. The basic idea according to which an element does not necessarily belong totally, or does not belong in absolute, to a set, but it can belong more or less, i.e. in some degree, signifies a change of paradigm, adapting mathematics to the features of the real world. So, we create new tools and fields, as Fuzzy Measure Theory, which generalizes the classical Measure Theory. We wish to dedicate this Special Issue to show such measures of symmetry, very related with the measures of information and entropy.

Contributions are invited on all aspects of symmetry measures as applied to every 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 measures
• Near Symmetry
• Fuzzy Symmetry
• Fuzzy Optimization
• Combinatorial Optimization
• Complex Networks
• Complex Systems
• Clustering
• Preferential attachment
• Graph Theory
• Combinatorial and Computational Group Theory
• Entropy Measures
• Information Theory
• Chirality
• Similarity
• Stability
• Complexity Theory
• Symmetry as a bridge between the sciences and humanities
Prof. Dr. Angel Garrido
Guest Editor

## Keywords

• measure theory
• fuzzy measure theory
• mathematical analysis
• graph theory
• discrete applied mathematics
• theoretical computer science
• complex networks
• complex systems
• symmetry measures
• entropy measures
• chirality
• similarity
• stability
• complexity theory
• combinatorial and computational group theory
• information theory
• combinatorial optimization
• fuzzy optimization

## Published Papers (6 papers)

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## Research

359 KiB
Article
Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems
Symmetry 2012, 4(1), 116-128; https://doi.org/10.3390/sym4010116 - 15 Feb 2012
Viewed by 4350
Abstract
The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity, and also many [...] Read more.
The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity, and also many examples. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
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105 KiB
Article
Information Theory of Networks
by Matthias Dehmer
Symmetry 2011, 3(4), 767-779; https://doi.org/10.3390/sym3040767 - 29 Nov 2011
Cited by 27 | Viewed by 5742
Abstract
The paper puts the emphasis on surveying information-theoretic network measures for analyzing the structure of networks. In order to apply the quantities interdisciplinarily, we also discuss some of their properties such as their structural interpretation and uniqueness. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
357 KiB
Article
Classifying Entropy Measures
by Angel Garrido
Symmetry 2011, 3(3), 487-502; https://doi.org/10.3390/sym3030487 - 20 Jul 2011
Cited by 12 | Viewed by 5606
Abstract
Our paper analyzes some aspects of Uncertainty Measures. We need to obtain new ways to model adequate conditions or restrictions, constructed from vague pieces of information. The classical entropy measure originates from scientific fields; more specifically, from Statistical Physics and Thermodynamics. With time [...] Read more.
Our paper analyzes some aspects of Uncertainty Measures. We need to obtain new ways to model adequate conditions or restrictions, constructed from vague pieces of information. The classical entropy measure originates from scientific fields; more specifically, from Statistical Physics and Thermodynamics. With time it was adapted by Claude Shannon, creating the current expanding Information Theory. However, the Hungarian mathematician, Alfred Rényi, proves that different and valid entropy measures exist in accordance with the purpose and/or need of application. Accordingly, it is essential to clarify the different types of measures and their mutual relationships. For these reasons, we attempt here to obtain an adequate revision of such fuzzy entropy measures from a mathematical point of view. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
275 KiB
Article
On Symmetry of Independence Polynomials
by Vadim E. Levit and Eugen Mandrescu
Symmetry 2011, 3(3), 472-486; https://doi.org/10.3390/sym3030472 - 15 Jul 2011
Cited by 5 | Viewed by 5182
Abstract
An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum [...] Read more.
An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. If sk is the number of independent sets of size k in G, then I(G; x) = s0 + s1x + s2x2 + ... + sαxα, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = sαj for all 0 ≤ j ≤ [α/2], then I(G; x) is called symmetric (or palindromic). It is known that the graph G ° 2K1, obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ μ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G ° 2K1; x) = (1 + x)k · I (H; x). Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
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232 KiB
Article
Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian
by Remi Leandre
Symmetry 2011, 3(1), 72-83; https://doi.org/10.3390/sym3010072 - 21 Mar 2011
Cited by 6 | Viewed by 4834
Abstract
We show that in long-time the heat semi-group on a path group associated to a Bilaplacian on the group tends to the Haar distribution on a path group. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
224 KiB
Article
Symmetry in Complex Networks
by Angel Garrido
Symmetry 2011, 3(1), 1-15; https://doi.org/10.3390/sym3010001 - 10 Jan 2011
Cited by 23 | Viewed by 8211
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 [...] Read more.
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. Full article
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
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