While any network that has a small-world coefficient larger than one is formally a small-world network, for many researchers, the term has become associated with the specific Watts and Strogatz model that is based on the partial random rewiring of a regular network (Fig. 2006), which is defined as the clustering coefficient of a given network (normalized by the clustering coefficient of a same-size random network) divided by the network’s normalized average shortest pathlength. These features can be mathematically summarized by the small-world coefficient (Humphries et al. Such an organization results in short paths across the whole network-almost as small as in random networks-combined with local ‘cliquishness’ (or clustering) of neighboring nodes, due to dense local interconnections. The concept was formalized by Watts and Strogatz ( 1998), who derived small-world networks from regular networks by including a small proportion of random network shortcuts. While the colloquial idea of a small, interconnected world has a long tradition (e.g., Klemperer 1938), the present concept of small-world features of networks is frequently associated with the Milgram experiment (Milgram 1967) that demonstrated surprisingly short paths across social networks (‘six degrees of separation’). 2011) or schizophrenia spectrum diseases (Liu et al. Correspondingly, the small-world organization is deemed essential for healthy brain function, as alterations of small-world features are observed in patient groups with Alzheimer’s disease (Stam et al.
The small-worldness is believed to be a crucial aspect of efficient brain organization that confers significant advantages in signal processing (e.g., Lago-Fernández et al. Indeed, one of the present authors claimed as much 15 years ago (Hilgetag et al. It is commonly assumed that the brain is a small-world network (e.g., Sporns and Honey 2006).
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