Many complex networks have a small-world topology characterized by dense local clustering or cliquishness of connections between neighboring nodes yet a short path length between any (distant) pair of nodes due to the existence of relatively few long-range connections. This is an attractive model for the organization of brain anatomical and functional networks because a small-world topology can support both segregated/specialized and distributed/integrated information processing. Moreover, small-world networks are economical, tending to minimize wiring costs while supporting high dynamical complexity. The authors introduce some of the key mathematical concepts in graph theory required for small-world analysis and review how these methods have been applied to quantification of cortical connectivity matrices derived from anatomical tract-tracing studies in the macaque monkey and the cat. The evolution of small-world networks is discussed in terms of a selection pressure to deliver cost-effective information-processing systems. The authors illustrate how these techniques and concepts are increasingly being applied to the analysis of human brain functional networks derived from electroencephalography/magnetoencephalography and fMRI experiments. Finally, the authors consider the relevance of small-world models for understanding the emergence of complex behaviors and the resilience of brain systems to pathological attack by disease or aberrant development. They conclude that small-world models provide a powerful and versatile approach to understanding the structure and function of human brain systems.

Human learning is a complex phenomenon requiring flexibility to adapt existing brain function and precision in selecting new neurophysiological activities to drive desired behavior. These two attributes-flexibility and selection-must operate over multiple temporal scales as performance of a skill changes from being slow and challenging to being fast and automatic. Such selective adaptability is naturally provided by modular structure, which plays a critical role in evolution, development, and optimal network function. Using functional connectivity measurements of brain activity acquired from initial training through mastery of a simple motor skill, we investigate the role of modularity in human learning by identifying dynamic changes of modular organization spanning multiple temporal scales. Our results indicate that flexibility, which we measure by the allegiance of nodes to modules, in one experimental session predicts the relative amount of learning in a future session. We also develop a general statistical framework for the identification of modular architectures in evolving systems, which is broadly applicable to disciplines where network adaptability is crucial to the understanding of system performance.complex network | time-dependent network | fMRI | motor learning | community structure T he brain is a complex system, composed of many interacting parts, which dynamically adapts to a continually changing environment over multiple temporal scales. Over relatively short temporal scales, rapid adaptation and continuous evolution of those interactions or connections form the neurophysiological basis for behavioral adaptation or learning. At small spatial scales, stable neurophysiological signatures of learning have been best demonstrated in animal systems at the level of individual synapses between neurons (1-3). At a larger spatial scale, it is also well-known that specific regional changes in brain activity and effective connectivity accompany many forms of learning in humans-including the acquisition of motor skills (4, 5).Learning-associated adaptability is thought to stem from the principle of cortical modularity (6). Modular, or nearly decomposable (7), structures are aggregates of small subsystems (modules) that can perform specific functions without perturbing the remainder of the system. Such structure provides a combination of compartmentalization and redundancy, which reduces the interdependence of components, enhances robustness, and facilitates behavioral adaptation (8, 9). Modular organization also confers evolvability on a system by reducing constraints on change (8,(10)(11)(12). Indeed, a putative relationship between modularity and adaptability in the context of human neuroscience has recently been posited (13,14). To date, however, the existence of modularity in large-scale cortical connectivity during learning has not been tested directly.Based on the aforementioned theoretical and empirical grounds, we hypothesized that the principle of modularity would characterize the fundamental organiz...