黑 Review TRENDS in Cognitive Sciences Vol.8 No.9 September 2004 d- Organization,development and function of complex brain networks Olaf Sporns1,Dante R.Chialvo2,Marcus Kaiser3 and Claus C.Hilgetag3 many Recent research has revealed general principles in the coupled neighborhoods.but maintain very short DISTANCEs structural and functional organization of complex net- among nodes across the entire network,giving rise to a works which are shared by various natural,social and small world within the network [10].The degree to which logical syste his revi individual nodes are connect d forms a distribution that. spe for many but not amine the structu cale anatom existence of highly connected nodes (hubs)[111. ical and functional brain networks and discuss how they What about the brain?Nervous systems are complex networks par excellence,capable of generating and reove the integrating informatior from mu iple external and fu uctura patterns that underlie human cognition.We suggest trate that network analysis offers new fundamental insights Glossary:Graph theory and networks d coherent cognitive states ionohe Complex networks.in a range of disciplines from biolog to physics,social sciences and informatics,have received significant attention in recent years [1-3).What can an stigationo I netw namics contrib- addr estion by hig series of recent studies of complex brain networkss and by attempting to identify promising areas and questions for Netw ally sets by connections athcma as GRAPHS (14-6];see Glossary their sto oecu ad their interations 00e10 18),or web pages and hyperlinks (91,often numbering in the thousands or mil lions.What makes such networks ce:Th c8petdeg9rceneejndaargetnodeisequalo only t ze but also the inte raction pond to the which gives rise to global states and 'emergent behaviors Recent work across a broad spectrum of complex networks has revealed common organizational principles(Box 1).In R0eeot tworks,the nor to th thetwor n net In many networks,clusters of nodes seg doe9epdoange2avahaanenpmng9 Corresponding author:Olaf Sporns (ospornsindiana.edu) d.dot10.101e5.6cs.2004.07.00Organization, development and function of complex brain networks Olaf Sporns1 , Dante R. Chialvo2 , Marcus Kaiser3 and Claus C. Hilgetag3 1 Department of Psychology and Programs in Cognitive and Neural Science, Indiana University, Bloomington, IN 47405, USA 2 Department of Physiology, Northwestern University Medical School, Chicago, IL 60611, USA; and Instituto Mediterraneo de Estudios Avanzados, IMEDEA (CSIC-UIB), E07122 Palma de Mallorca, Spain 3 International University Bremen, School of Engineering and Science, Campus Ring 6, 28759 Bremen, Germany Recent research has revealed general principles in the structural and functional organization of complex net￾works which are shared by various natural, social and technological systems. This review examines these principles as applied to the organization, development and function of complex brain networks. Specifically, we examine the structural properties of large-scale anatom￾ical and functional brain networks and discuss how they might arise in the course of network growth and rewiring. Moreover, we examine the relationship between the structural substrate of neuroanatomy and more dynamic functional and effective connectivity patterns that underlie human cognition. We suggest that network analysis offers new fundamental insights into global and integrative aspects of brain function, including the origin of flexible and coherent cognitive states within the neural architecture. Complex networks, in a range of disciplines from biology to physics, social sciences and informatics, have received significant attention in recent years [1–3]. What can an investigation of network structure and dynamics contrib￾ute to our understanding of brain and cognitive function? In our review, we address this question by highlighting a series of recent studies of complex brain networks and by attempting to identify promising areas and questions for future experimental and theoretical inquiry. Networks are sets of nodes linked by connections, mathematically described as GRAPHS ([4–6]; see Glossary). The nodes and connections may represent persons and their social relations [7], molecules and their interactions [8], or web pages and hyperlinks [9], often numbering in the thousands or millions. What makes such networks complex is not only their size but also the interaction of architecture (the network’s connection topology) and dynamics (the behavior of the individual network nodes), which gives rise to global states and ‘emergent’ behaviors. Recent work across a broad spectrum of complex networks has revealed common organizational principles (Box 1). In many complex networks, the non-linear dynamics of individual network components unfolds within network topologies that are strikingly irregular, yet non-random. In many networks, clusters of nodes segregate into tightly coupled neighborhoods, but maintain very short DISTANCES among nodes across the entire network, giving rise to a small world within the network [10]. The degree to which individual nodes are connected forms a distribution that, for many but not all networks, decays as a power law, producing a SCALE-FREE architecture characterized by the existence of highly connected nodes (hubs) [11]. What about the brain? Nervous systems are complex networks par excellence, capable of generating and integrating information from multiple external and internal sources in real time. Within the neuroanatomical substrate (structural connectivity), the non-linear Glossary: Graph theory and networks For the following definitions of graph theory terms used in this review we essentially follow the nomenclature of ref. 4 (see also [27] for additional definitions and more detail). A Matlab toolbox allowing the calculation of these and other graph theory measures is available at http://www.indiana.edu/ (cortex/connectivity.html. Adjacency (connection) matrix: The adjacency matrix of a graph is a n!n matrix with entries aijZ1 if node j connects to node i, and aijZ0 is there is no connection from node j to node i. Characteristic path length: The characteristic path length L (also called ‘path length’ or ‘average shortest path’) is given by the global mean of the finite entries of the distance matrix. In some cases, the median or the harmonic mean can provide better estimates.[10] Clustering coefficient: The clustering coefficient Ci of a node i is calculated as the number of existing connections between the node’s neighbors divided by all their possible connections. The clustering coefficient ranges between 0 and 1 and is typically averaged over all nodes of a graph to yield the graph’s clustering coefficient C.[10] Connectedness: A connected graph has only one component, that is a set of nodes, for which every pair of nodes is joined by at least one path. A disconnected graph has at least two components. Cycle: A cycle is a path that links a node to itself. Degree: The degree of a node is the sum of its incoming (afferent) and outgoing (efferent) connections. The number of afferent and efferent connections is also called the ‘in-degree’ and ‘out-degree’, respectively. Distance: The distance between a source node j and a target node i is equal to the length of the shortest path. Distance matrix: The entries dij of the distance matrix correspond to the distance between node j and i. If no path exists, dijZinfinity Graph: Graphs are a set of n nodes (vertices, points, units) and k edges (connections, arcs). Graphs may be undirected (all connections are symmetri￾cal) or directed. Because of the polarized nature of most neural connections, we focus on directed graphs, also called digraphs. Path: A path is an ordered sequence of distinct connections and nodes, linking a source node j to a target node i. No connection or node is visited twice in a given path. The length of a path is equal to the number of distinct connections. Random graph: A graph with uniform connection probabilities and a binomial degree distribution. All nodes have roughly the same degree (‘single-scale’). Scale-free graph: Graph with a power-law degree distribution. ‘Scale-free’ means that degrees are not grouped around one characteristic average degree (scale), but can spread over a very wide range of values, often spanning several orders of magnitude. Corresponding author: Olaf Sporns (osporns@indiana.edu). www.sciencedirect.com 1364-6613/$ - see front matter Q 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.tics.2004.07.008 Review TRENDS in Cognitive Sciences Vol.8 No.9 September 2004