Special Issue "Symmetry Measures on Complex Networks"


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

Deadline for manuscript submissions: closed (30 September 2011)

Special Issue Editor

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
Phone: 34913987237
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

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


  • 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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Displaying article 1-6
p. 116-128
Symmetry 2012, 4(1), 116-128; doi:10.3390/sym4010116
Received: 1 November 2011 / Revised: 8 February 2012 / Accepted: 9 February 2012 / Published: 15 February 2012
Show/Hide Abstract | PDF Full-text (359 KB)
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
p. 767-779
Symmetry 2011, 3(4), 767-779; doi:10.3390/sym3040767
Received: 26 October 2011 / Revised: 11 November 2011 / Accepted: 16 November 2011 / Published: 29 November 2011
Show/Hide Abstract | Cited by 7 | PDF Full-text (105 KB)
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
p. 487-502
Symmetry 2011, 3(3), 487-502; doi:10.3390/sym3030487
Received: 27 April 2011 / Revised: 6 July 2011 / Accepted: 6 July 2011 / Published: 20 July 2011
Show/Hide Abstract | Cited by 3 | PDF Full-text (357 KB)
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
p. 472-486
by  and
Symmetry 2011, 3(3), 472-486; doi:10.3390/sym3030472
Received: 27 April 2011 / Revised: 20 June 2011 / Accepted: 22 June 2011 / Published: 15 July 2011
Show/Hide Abstract | Cited by 1 | PDF Full-text (275 KB)
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
p. 72-83
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 | Cited by 2 | PDF Full-text (232 KB)
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
p. 1-15
Symmetry 2011, 3(1), 1-15; doi:10.3390/sym3010001
Received: 16 November 2010 / Revised: 4 January 2011 / Accepted: 7 January 2011 / Published: 10 January 2011
Show/Hide Abstract | Cited by 2 | PDF Full-text (224 KB)
(This article belongs to the Special Issue Symmetry Measures on Complex Networks)
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Last update: 5 March 2014

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