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Complex Networks and Symmetry I: A Review
LEM, Sant’Anna School of Advanced Studies, P.zza Martiri della Libert`a 33, 56127 Pisa, Italy
Center for the Study of Complex Systems, University of Siena, Via Roma 56, 53100 Siena, Italy
Department of Chemistry, University of Siena, Via Aldo Moro 1, 53100 Siena, Italy
* Author to whom correspondence should be addressed.
Received: 25 August 2010; in revised form: 17 September 2010 / Accepted: 20 September 2010 / Published: 27 September 2010
Abstract: In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques.
Keywords: complexity; networks; symmetry
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Garlaschelli, D.; Ruzzenenti, F.; Basosi, R. Complex Networks and Symmetry I: A Review. Symmetry 2010, 2, 1683-1709.
Garlaschelli D, Ruzzenenti F, Basosi R. Complex Networks and Symmetry I: A Review. Symmetry. 2010; 2(3):1683-1709.
Garlaschelli, Diego; Ruzzenenti, Franco; Basosi, Riccardo. 2010. "Complex Networks and Symmetry I: A Review." Symmetry 2, no. 3: 1683-1709.