A small world of weak ties provides optimal global

The human brain is organized in functional modules. Such an organization presents a basic conundrum: Modules ought to be sufficiently independent to guarantee functional specialization and sufficiently connected to bind multiple

processors for efficient information transfer. It is commonly accepted that small-world architecture of short paths and large local clustering may solve this problem. However, there is intrinsic tension between shortcuts generatingsmallworldsandthepersistenceofmodularity,aglobal property unrelated to local clustering. Here, we present a possible solution to this puzzle. We first show that a modified percolation theory can define a set of hierarchically organized modules made of strong links in functional brain networks. These modules are “large-world” self-similar structures and, therefore, are far from being small-world. However, incorporating weaker ties to the net- work converts it into a small world preserving an underlying back- bone of well-defined modules. Remarkably, weak ties are precisely organized as predicted by theory maximizing information transfer with minimal wiring cost. This trade-off architecture is reminiscent of the “strength of weak ties” crucial concept of social networks. Such a design suggests a natural solution to the paradox of effi- cient information flowin the highlymodular structure of the brain. O ne of the main findings in neuroscience is the modular or- ganization of the brain, which in turn implies the parallel nature of brain computations (1–3). For example, in the visual modality, more than 30 visual areas analyze simultaneously dis- tinct features of the visual scene: motion, color, orientation, space, form, luminance, and contrast, among others (4). These features, as well asinformation from different sensory modalities, have to be integrated, as one of the main aspects of perception is its unitary nature (1, 5). This leads to a basic conundrum of brain networks: Modular processors have to be sufficiently isolated to achieve independent computations, but also globally connected to be integrated in coherent functions (1, 2, 6). A current view is that small-world networks provide a solution to this puzzle because they combine high local clustering and short path length (7–9). This view has been fueled by the systematic finding of small-world topology in a wide range ofhuman brainnetworks derived fromstructural (10), functional (11–13), and diffusion tensor MRI (14). Small-world topology has also been identified at the cellular-network scale in functional cortical neuronal circuits in mammals (15, 16) and even in the nervous system of the nematode Caenorhabditis ele- gans (8).Moreover, small-world property seemsto berelevant for brainfunction becauseitisaffected bydisease(17),normalaging, and by pharmacological blockade of dopamine neurotransmis- sion (13). Although brain networks show small-world properties, several experimental studies have also shown that they are hierarchical, fractal and highly modular (2, 3, 18)....

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