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      Complex Network Geometry and Frustrated Synchronization

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      1 , 1 , , 2
      Scientific Reports
      Nature Publishing Group UK

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          Abstract

          The dynamics of networks of neuronal cultures has been recently shown to be strongly dependent on the network geometry and in particular on their dimensionality. However, this phenomenon has been so far mostly unexplored from the theoretical point of view. Here we reveal the rich interplay between network geometry and synchronization of coupled oscillators in the context of a simplicial complex model of manifolds called Complex Network Manifold. The networks generated by this model combine small world properties (infinite Hausdorff dimension) and a high modular structure with finite and tunable spectral dimension. We show that the networks display frustrated synchronization for a wide range of the coupling strength of the oscillators, and that the synchronization properties are directly affected by the spectral dimension of the network.

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          Most cited references31

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          Emergence of scaling in random networks

          Systems as diverse as genetic networks or the world wide web are best described as networks with complex topology. A common property of many large networks is that the vertex connectivities follow a scale-free power-law distribution. This feature is found to be a consequence of the two generic mechanisms that networks expand continuously by the addition of new vertices, and new vertices attach preferentially to already well connected sites. A model based on these two ingredients reproduces the observed stationary scale-free distributions, indicating that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
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            Community detection in graphs

            The modern science of networks has brought significant advances to our understanding of complex systems. One of the most relevant features of graphs representing real systems is community structure, or clustering, i. e. the organization of vertices in clusters, with many edges joining vertices of the same cluster and comparatively few edges joining vertices of different clusters. Such clusters, or communities, can be considered as fairly independent compartments of a graph, playing a similar role like, e. g., the tissues or the organs in the human body. Detecting communities is of great importance in sociology, biology and computer science, disciplines where systems are often represented as graphs. This problem is very hard and not yet satisfactorily solved, despite the huge effort of a large interdisciplinary community of scientists working on it over the past few years. We will attempt a thorough exposition of the topic, from the definition of the main elements of the problem, to the presentation of most methods developed, with a special focus on techniques designed by statistical physicists, from the discussion of crucial issues like the significance of clustering and how methods should be tested and compared against each other, to the description of applications to real networks.
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              Fast unfolding of communities in large networks

              We propose a simple method to extract the community structure of large networks. Our method is a heuristic method that is based on modularity optimization. It is shown to outperform all other known community detection method in terms of computation time. Moreover, the quality of the communities detected is very good, as measured by the so-called modularity. This is shown first by identifying language communities in a Belgian mobile phone network of 2.6 million customers and by analyzing a web graph of 118 million nodes and more than one billion links. The accuracy of our algorithm is also verified on ad-hoc modular networks. .
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                Author and article information

                Contributors
                jtorres@onsager.ugr.es
                Journal
                Sci Rep
                Sci Rep
                Scientific Reports
                Nature Publishing Group UK (London )
                2045-2322
                2 July 2018
                2 July 2018
                2018
                : 8
                : 9910
                Affiliations
                [1 ]ISNI 0000000121678994, GRID grid.4489.1, Departamento de Electromagnetismo y Física de la Materia and Instituto Carlos I de Física Teórica y Computacional, , Universidad de Granada, ; 18071 Granada, Spain
                [2 ]ISNI 0000 0001 2171 1133, GRID grid.4868.2, School of Mathematical Sciences, , Queen Mary University of London, ; E1 4NS London, United Kingdom
                Author information
                http://orcid.org/0000-0001-6175-9676
                http://orcid.org/0000-0002-3380-887X
                Article
                28236
                10.1038/s41598-018-28236-w
                6028575
                29967410
                2f482c7f-93a2-485a-b4d7-f47903c58ea9
                © The Author(s) 2018

                Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/.

                History
                : 14 February 2018
                : 19 June 2018
                Funding
                Funded by: Spanish MINECO (under excelence project FIS2017-84256-P; FEDER funds)
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