Special Issue "Symmetry Measures on Complex Networks"

Guest Editor
Prof. Dr. Angel Garrido
Department of Fundamental Mathematics, Faculty of Sciences, UNED, Paseo Senda del Rey No. 9, 28040 Madrid, Spain
Website: http://www.telefonica.net/web2/angabu
E-Mail: agarrido@mat.uned.es
Phone: +34 91 6103797
Fax: +34 91 3987237
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

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

Submission

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are refereed through a peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed Open Access quarterly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 500 CHF (Swiss Francs). English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.

Angel Garrido Article: Symmetry in Complex Networks Symmetry 2011, 3(1), 1-15; doi:10.3390/sym3010001 Received: 16 November 2010; in revised form: 4 January 2011 / Accepted: 7 January 2011 / Published: 10 January 2011 Show/Hide Abstract | Download PDF Full-text (224 KB) 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.

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Remi Leandre Article: Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian Symmetry 2011, 3(1), 72-83; doi:10.3390/sym3010072 Received: 16 February 2011 / Accepted: 7 March 2011 / Published: 21 March 2011 Show/Hide Abstract | Download PDF Full-text (232 KB) 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.

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Vadim E. Levit and Eugen Mandrescu Article: On Symmetry of Independence Polynomials Symmetry 2011, 3(3), 472-486; doi:10.3390/sym3030472 Received: 27 April 2011; in revised form: 20 June 2011 / Accepted: 22 June 2011 / Published: 15 July 2011 Show/Hide Abstract | Download PDF Full-text (275 KB) 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 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).

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Angel Garrido Article: Classifying Entropy Measures Symmetry 2011, 3(3), 487-502; doi:10.3390/sym3030487 Received: 27 April 2011; in revised form: 6 July 2011 / Accepted: 6 July 2011 / Published: 20 July 2011 Show/Hide Abstract | Download PDF Full-text (357 KB) 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 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.

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Matthias Dehmer Article: Information Theory of Networks Symmetry 2011, 3(4), 767-779; doi:10.3390/sym3040767 Received: 26 October 2011; in revised form: 11 November 2011 / Accepted: 16 November 2011 / Published: 29 November 2011 Show/Hide Abstract | Download PDF Full-text (105 KB) 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.

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Serban E. Vlad Article: Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems Symmetry 2012, 4(1), 116-128; doi:10.3390/sym4010116 Received: 1 November 2011; in revised form: 8 February 2012 / Accepted: 9 February 2012 / Published: 15 February 2012 Show/Hide Abstract | Download PDF Full-text (359 KB) 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 examples.

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