Review TRENDS in Cognitive Sciences Vol.8 No.9 September 2004 41 Box 1.Complex networks:small-world and scale-free architectures of nnodes and kcor a RANDOM GRAPH (s ility.Fot m dom al and tec Iments 17 mirst rev ery large and B-like connec s with a sr ths,pr d by orks 11 This h hitecture g e of highly nal istributions of most 060808 toothetodesedlh dynamics of neurons and neuronal populations result in I depende (functional conn conn Box 2).Humar ion is associated with rapidly on structueat 12-15.Two major organizational principles of the erebral corte regat ion and fur Structural organiza tion of cortical networks ntegr 6],ena have Which structural and functional prineinles of complex tion patterns of the cerebral cortex of rat [201,cat [21,22] s pr functional segreg or,in general, range an revealed several in this review we examine re networks ent insights gained about patter om the application of novel either cor nd t interco neuron networks have revealed their complex mornhology used either graph the retica dynamics of neurons and neuronal populations result in patterns of statistical dependencies (functional connec￾tivity) and causal interactions (effective connectivity), defining three major modalities of complex brain networks (Box 2). Human cognition is associated with rapidly changing and widely distributed neural activation pat￾terns, which involve numerous cortical and sub-cortical regions activated in different combinations and contexts [12–15]. Two major organizational principles of the cerebral cortex are functional segregation and functional integration [16–18], enabling the rapid extraction of information and the generation of coherent brain states. Which structural and functional principles of complex networks promote functional segregation and functional integration, or, in general, support the broad range and flexibility of cognitive processes? In this review we examine recent insights gained about patterns of brain connectivity from the application of novel quantitative computational tools and theoretical models to empirical datasets. Whereas many studies of single neuron networks have revealed their complex morphology and wiring [19], our focus is on the large-scale and intermediate-scale networks of the cerebral cortex, allow￾ing us to examine links between neural organization and cognition arising at the ‘systems’ level. We divide this review into three parts, devoted in turn to the organiz￾ation (structure), development (growth) and function (dynamics) of brain networks. Structural organization of cortical networks Most structural analyses of brain networks have been carried out on datasets describing the large-scale connec￾tion patterns of the cerebral cortex of rat [20], cat [21,22], and monkey [23] – structural connection data for the human brain is largely missing [24]. These analyses have revealed several organizational principles expressed within structural brain networks. All studies confirmed that cerebral cortical areas in mammalian brains are neither completely connected with each other nor ran￾domly linked; instead, their interconnections show a specific and intricate organization. Methodologically, investigations have used either graph theoretical Box 1. Complex networks: small-world and scale-free architectures For any number of n nodes and k connections, a RANDOM GRAPH (see Glossary) can be constructed by assigning connections between pairs of nodes with uniform probability. For many years, random graphs (Figure Ia) have served as a major class of models for describing the topology of natural and technological networks. However, although random graphs have yielded numerous and often surprising math￾ematical insights [5], they are probably only poor approximations of the connectivity structure of most complex systems. Classical experiments [71] first revealed the existence of a small world in large social networks. Small worlds are characterized by the prevalence of surprisingly short PATHS linking pairs of nodes within very large networks. In a seminal paper [10], Watts and Strogatz demonstrated the emergence of small world connectivity in networks that combined ordered lattice-like connections with a small admixture of random links (Figure Ib). Combining elements of order and randomness, such networks were characterized by high degrees of local clustering as well as short path lengths, properties shared by genetic, metabolic, ecological and information networks [1–3]. The nodes in random graphs have approximately the same DEGREE (number of connections). This homogeneous architecture generates a normal (or Poisson) degree distribution. However, the degree distributions of most natural and technological networks follow a power law [11], with very many nodes that have few connections and a few nodes (hubs) that have very many connections (Figure Ic). This inhomogeneous architecture lacks an intrinsic scale and is thus called SCALE-FREE. Scale-free networks are surprisingly robust with respect to random deletion of nodes, but are vulnerable to targeted attack on heavily connected hubs [45], which often results in disintegration of the network. A corollary of this finding is that the connection topology of scale-free networks cannot be efficiently captured by random sampling, as most nodes have few connections and hubs will tend to be under-represented. Sampling is thus a crucial issue for determining if brain networks have scale-free topology. Systematic investigations of large-scale [32–35] and intermediate￾scale [35,36] structural cortical networks have revealed small-world attributes, with path lengths that are close to those of equivalent random networks but with significantly higher values for the clustering coefficient. At the structural level, cortical networks either do not appear to be scale-free [35] or exhibit scale-free architectures with low maximum degrees [44], owing to saturation effects in the number of synaptic connections, which prevent the emergence of highly connected hubs. Instead, functional brain networks exhibit power law degree distributions as well as small￾world attributes [52,62]. L=1.73 (0.06) C=0.52 (0.05) L=1.79 (0.04) C=0.52 (0.04) L=1.68 (0.01) C=0.35 (0.03) (a) (b) (c) Figure I. Structure of random, small-world and scale-free networks. All networks have 24 nodes and 86 connections with nodes arranged on a circle. The characteristic path length L and the clustering coefficient C are shown (mean and standard deviation for 100 examples in each case; only one example network is drawn). (a) Random network. (b) Small-world network. Most connections are among neighboring nodes on the circle (dark blue), but some connections (light blue) go to distant nodes, creating short-cuts across the network. (c) Scale-free network. Most of the 24 nodes have few connections to other nodes (red), but some nodes (black connections) are linked to more than 12 other nodes. For comparison, an ideal lattice with 24 nodes and 86 connections has LZ1.96 and CZ0.64. Review TRENDS in Cognitive Sciences Vol.8 No.9 September 2004 419 www.sciencedirect.com