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      Modular and Hierarchically Modular Organization of Brain Networks

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          Abstract

          Brain networks are increasingly understood as one of a large class of information processing systems that share important organizational principles in common, including the property of a modular community structure. A module is topologically defined as a subset of highly inter-connected nodes which are relatively sparsely connected to nodes in other modules. In brain networks, topological modules are often made up of anatomically neighboring and/or functionally related cortical regions, and inter-modular connections tend to be relatively long distance. Moreover, brain networks and many other complex systems demonstrate the property of hierarchical modularity, or modularity on several topological scales: within each module there will be a set of sub-modules, and within each sub-module a set of sub-sub-modules, etc. There are several general advantages to modular and hierarchically modular network organization, including greater robustness, adaptivity, and evolvability of network function. In this context, we review some of the mathematical concepts available for quantitative analysis of (hierarchical) modularity in brain networks and we summarize some of the recent work investigating modularity of structural and functional brain networks derived from analysis of human neuroimaging data.

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          The structure and function of complex networks

          M. Newman (2003)
          Inspired by empirical studies of networked systems such as the Internet, social networks, and biological networks, researchers have in recent years developed a variety of techniques and models to help us understand or predict the behavior of these systems. Here we review developments in this field, including such concepts as the small-world effect, degree distributions, clustering, network correlations, random graph models, models of network growth and preferential attachment, and dynamical processes taking place on networks.
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            Modularity and community structure in networks

            M. Newman (2006)
            Many networks of interest in the sciences, including a variety of social and biological networks, are found to divide naturally into communities or modules. The problem of detecting and characterizing this community structure has attracted considerable recent attention. One of the most sensitive detection methods is optimization of the quality function known as "modularity" over the possible divisions of a network, but direct application of this method using, for instance, simulated annealing is computationally costly. Here we show that the modularity can be reformulated in terms of the eigenvectors of a new characteristic matrix for the network, which we call the modularity matrix, and that this reformulation leads to a spectral algorithm for community detection that returns results of better quality than competing methods in noticeably shorter running times. We demonstrate the algorithm with applications to several network data sets.
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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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                Author and article information

                Journal
                Front Neurosci
                Front. Neurosci.
                Frontiers in Neuroscience
                Frontiers Research Foundation
                1662-4548
                1662-453X
                17 October 2010
                08 December 2010
                2010
                : 4
                : 200
                Affiliations
                [1] 1simpleCentre for Speech, Language and the Brain, Department of Experimental Psychology, University of Cambridge Cambridge, UK
                [2] 2simpleInstitute for Mathematical Sciences, Imperial College London, UK
                [3] 3simpleBehavioural and Clinical Neuroscience Institute, Department of Psychiatry, University of Cambridge Cambridge, UK
                Author notes

                Edited by: Marcus Kaiser, Seoul National University, South Korea

                Reviewed by: Claus Hilgetag, Jacobs University Bremen, Germany; Olaf Sporns, Indiana University, USA; Changsong Zhou, Hong Kong Baptist University, China

                *Correspondence: etb23@ 123456cam.ac.uk

                David Meunier and Renaud Lambiotte have contributed equally.

                Article
                10.3389/fnins.2010.00200
                3000003
                21151783
                ed1119ff-17b0-446b-a7a9-8f5b3f94aa75
                Copyright © 2010 Meunier, Lambiotte and Bullmore.

                This is an open-access article subject to an exclusive license agreement between the authors and the Frontiers Research Foundation, which permits unrestricted use, distribution, and reproduction in any medium, provided the original authors and source are credited.

                History
                : 03 September 2010
                : 17 November 2010
                Page count
                Figures: 4, Tables: 0, Equations: 0, References: 79, Pages: 11, Words: 7855
                Categories
                Neuroscience
                Focused Review

                Neurosciences
                near-decomposability,partition,fractal,cortex,graph
                Neurosciences
                near-decomposability, partition, fractal, cortex, graph

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