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Exploring complex networks via topological embedding on surfaces

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  • Tomaso Aste
  • Ruggero Gramatica
  • T. Di Matteo

Abstract

We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive because any network can be embedded on a surface with sufficiently high genus. The local properties of the network are affected by the surface genus which, for example, produces significant changes in the degree distribution and in the clustering coefficient. The global properties of the graph are also strongly affected by the surface genus which is constraining the degree of interwoveness, changing the scaling properties from large-world-kind (small genus) to small- and ultra-small-world-kind (large genus). Two elementary moves allow the exploration of all networks embeddable on a given surface and naturally introduce a tool to develop a statistical mechanics description. Within such a framework, we study the properties of topologically-embedded graphs at high and low `temperatures' observing the formation of increasingly regular structures by cooling the system. We show that the cooling dynamics is strongly affected by the surface genus with the manifestation of a glassy-like freezing transitions occurring when the amount of topological disorder is low.

Suggested Citation

  • Tomaso Aste & Ruggero Gramatica & T. Di Matteo, 2011. "Exploring complex networks via topological embedding on surfaces," Papers 1107.3456, arXiv.org, revised Aug 2012.
    • Handle: RePEc:arx:papers:1107.3456
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    References listed on IDEAS

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    1. Aste, T. & Di Matteo, T. & Hyde, S.T., 2005. "Complex networks on hyperbolic surfaces," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 346(1), pages 20-26.
    2. H.M. Ohlenbusch & T. Aste & B. Dubertret & N. Rivier, 1998. "The topological structure of 2D disordered cellular systems," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 2(2), pages 211-220, March.
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    Cited by:

    1. Le, Anbo & Gao, Fei & Xi, Lifeng & Yin, Shuhua, 2015. "Complex networks modeled on the Sierpinski gasket," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 436(C), pages 646-657.
    2. Nicolò Musmeci & Vincenzo Nicosia & Tomaso Aste & Tiziana Di Matteo & Vito Latora, 2017. "The Multiplex Dependency Structure of Financial Markets," Complexity, Hindawi, vol. 2017, pages 1-13, September.
    3. M. Raddant & T. Di Matteo, 2023. "A look at financial dependencies by means of econophysics and financial economics," Journal of Economic Interaction and Coordination, Springer;Society for Economic Science with Heterogeneous Interacting Agents, vol. 18(4), pages 701-734, October.
    4. Nicol'o Musmeci & Tomaso Aste & Tiziana Di Matteo, 2014. "Risk diversification: a study of persistence with a filtered correlation-network approach," Papers 1410.5621, arXiv.org.
    5. Musmeci, Nicoló & Nicosia, Vincenzo & Aste, Tomaso & Di Matteo, Tiziana & Latora, Vito, 2017. "The multiplex dependency structure of financial markets," LSE Research Online Documents on Economics 85337, London School of Economics and Political Science, LSE Library.

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