Betweenness Centrality in Large Complex Networks
TL;DR: In this article, the authors analyzed the betweenness centrality of nodes in large complex networks and showed that for trees or networks with a small loop density, a larger density of loops leads to the same result.
Abstract: We analyze the betweenness centrality (BC) of nodes in large complex networks. In general, the BC is increasing with connectivity as a power law with an exponent $\eta$. We find that for trees or networks with a small loop density $\eta=2$ while a larger density of loops leads to $\eta<2$. For scale-free networks characterized by an exponent $\gamma$ which describes the connectivity distribution decay, the BC is also distributed according to a power law with a non universal exponent $\delta$. We show that this exponent $\delta$ must satisfy the exact bound $\delta\geq (\gamma+1)/2$. If the scale free network is a tree, then we have the equality $\delta=(\gamma+1)/2$.
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TL;DR: This article reviews studies investigating complex brain networks in diverse experimental modalities and provides an accessible introduction to the basic principles of graph theory and highlights the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
Abstract: Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
9,700 citations
TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
9,441 citations
Book•
23 Aug 2010TL;DR: This chapter discusses random network models, which are based on the Erdos-Renyi models, and their application in the context of complex networks, where distances in scale-free networks are small and distances in complex networks are large.
Abstract: 1. Introduction Part I. Random Network Models: 2. The Erdos-Renyi models 3. Observations in real-world networks 4. Models for complex networks 5. Growing network models Part II. Structure and Robustness of Complex Networks: 6. Distances in scale-free networks - the ultra small world 7. Self-similarity in complex networks 8. Distances in geographically embedded networks 9. The network's structure - the generating function method 10. Percolation on complex networks 11. Structure of random directed networks - the bow tie 12. Introducing weights - bandwidth allocation and multimedia broadcasting Part III. Network Function - Dynamics and Applications: 13. Optimization of the network structure 14. Epidemiological models 15. Immunization 16. Thermodynamic models on networks 17. Spectral properties, transport, diffusion and dynamics 18. Searching in networks 19. Biological networks and network motifs Part IV. Appendices References Index.
891 citations
TL;DR: This report reviews the developmental arc of theoretical epidemiology with emphasis on vaccination, as it led from classical models assuming homogeneously mixing populations and ignoring human behavior, to recent models that account for behavioral feedback and/or population spatial/social structure.
789 citations
TL;DR: In this paper, the authors present OSMnx, a new tool to make the collection of data and creation and analysis of street networks simple, consistent, automatable and sound from the perspectives of graph theory, transportation, and urban design.
Abstract: Urban scholars have studied street networks in various ways, but there are data availability and consistency limitations to the current urban planning/street network analysis literature. To address these challenges, this article presents OSMnx, a new tool to make the collection of data and creation and analysis of street networks simple, consistent, automatable and sound from the perspectives of graph theory, transportation, and urban design. OSMnx contributes five significant capabilities for researchers and practitioners: first, the automated downloading of political boundaries and building footprints; second, the tailored and automated downloading and constructing of street network data from OpenStreetMap; third, the algorithmic correction of network topology; fourth, the ability to save street networks to disk as shapefiles, GraphML, or SVG files; and fifth, the ability to analyze street networks, including calculating routes, projecting and visualizing networks, and calculating metric and topological measures. These measures include those common in urban design and transportation studies, as well as advanced measures of the structure and topology of the network. Finally, this article presents a simple case study using OSMnx to construct and analyze street networks in Portland, Oregon.
738 citations
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TL;DR: Simple models of networks that can be tuned through this middle ground: regular networks ‘rewired’ to introduce increasing amounts of disorder are explored, finding that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs.
Abstract: Networks of coupled dynamical systems have been used to model biological oscillators, Josephson junction arrays, excitable media, neural networks, spatial games, genetic control networks and many other self-organizing systems. Ordinarily, the connection topology is assumed to be either completely regular or completely random. But many biological, technological and social networks lie somewhere between these two extremes. Here we explore simple models of networks that can be tuned through this middle ground: regular networks 'rewired' to introduce increasing amounts of disorder. We find that these systems can be highly clustered, like regular lattices, yet have small characteristic path lengths, like random graphs. We call them 'small-world' networks, by analogy with the small-world phenomenon (popularly known as six degrees of separation. The neural network of the worm Caenorhabditis elegans, the power grid of the western United States, and the collaboration graph of film actors are shown to be small-world networks. Models of dynamical systems with small-world coupling display enhanced signal-propagation speed, computational power, and synchronizability. In particular, infectious diseases spread more easily in small-world networks than in regular lattices.
39,297 citations
TL;DR: A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
Abstract: 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 was found to be a consequence of two generic mechanisms: (i) networks expand continuously by the addition of new vertices, and (ii) new vertices attach preferentially to sites that are already well connected. A model based on these two ingredients reproduces the observed stationary scale-free distributions, which indicates that the development of large networks is governed by robust self-organizing phenomena that go beyond the particulars of the individual systems.
33,771 citations
TL;DR: This article reviews studies investigating complex brain networks in diverse experimental modalities and provides an accessible introduction to the basic principles of graph theory and highlights the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
Abstract: Recent developments in the quantitative analysis of complex networks, based largely on graph theory, have been rapidly translated to studies of brain network organization. The brain's structural and functional systems have features of complex networks--such as small-world topology, highly connected hubs and modularity--both at the whole-brain scale of human neuroimaging and at a cellular scale in non-human animals. In this article, we review studies investigating complex brain networks in diverse experimental modalities (including structural and functional MRI, diffusion tensor imaging, magnetoencephalography and electroencephalography in humans) and provide an accessible introduction to the basic principles of graph theory. We also highlight some of the technical challenges and key questions to be addressed by future developments in this rapidly moving field.
9,700 citations
TL;DR: The major concepts and results recently achieved in the study of the structure and dynamics of complex networks are reviewed, and the relevant applications of these ideas in many different disciplines are summarized, ranging from nonlinear science to biology, from statistical mechanics to medicine and engineering.
9,441 citations
01 Mar 1977
TL;DR: A family of new measures of point and graph centrality based on early intuitions of Bavelas (1948) is introduced in this paper, which define centrality in terms of the degree to which a point falls on the shortest path between others and there fore has a potential for control of communication.
Abstract: A family of new measures of point and graph centrality based on early intuitions of Bavelas (1948) is introduced. These measures define centrality in terms of the degree to which a point falls on the shortest path between others and there fore has a potential for control of communication. They may be used to index centrality in any large or small network of symmetrical relations, whether connected or unconnected.
8,026 citations