Is the brain really a small-world network?

It is commonly assumed that the brain is a small-world network (e.g., Sporns and Honey 2006). Indeed, one of the present authors claimed as much 15 years ago (Hilgetag et al. 2000). The small-worldness is believed to be a crucial aspect of efficient brain organization that confers significant advantages in signal processing (e.g., LagoFernandez et al. 2000). 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. 2007), autism (Barttfeld et al. 2011) or schizophrenia spectrum diseases (Liu et al. 2008; Wang et al. 2012; Zalesky et al. 2011). 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’). 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. 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. These features can be mathematically summarized by the smallworld coefficient (Humphries et al. 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. While any network that has a smallworld 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. 1a). Indeed, the estimation of the rewiring probability has been used to directly associate real-world networks with the Watts and Strogatz model (Humphries and Gurney 2008). Incidentally, the small-world coefficient might not faithfully capture the small-world property as originally described by Watts and Strogatz (1998). Therefore, an alternative coefficient has been proposed that compares the clustering of the network to a lattice instead of a random network (Telesford et al. 2011). A large number of empirical network data conform to the small-world features of short paths combined with high clustering, including many neural networks—but do these Electronic supplementary material The online version of this article (doi:10.1007/s00429-015-1035-6) contains supplementary material, which is available to authorized users.