Harmony in the small-world

The small-world phenomenon, popularly known as six degrees of separation, has been mathematically formalized by Watts and Strogatz in a study of the topological properties of a network. Small-world networks are defined in terms of two quantities: they have a high clustering coefficient C like regular lattices and a short characteristic path length L typical of random networks. Physical distances are of fundamental importance in applications to real cases; nevertheless, this basic ingredient is missing in the original formulation. Here, we introduce a new concept, the connectivity length D, that gives harmony to the whole theory. D can be evaluated on a global and on a local scale and plays in turn the role of L and 1/C. Moreover, it can be computed for any metrical network and not only for the topological cases. D has a precise meaning in terms of information propagation and describes in a unified way, both the structural and the dynamical aspects of a network: small-worlds are defined by a small global and local D, i.e., by a high efficiency in propagating information both on a local and global scale. The neural system of the nematode C. elegans, the collaboration graph of film actors, and the oldest US subway system, can now be studied also as metrical networks and are shown to be small-worlds.