  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-worlds 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 the 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 term of information propagation and describes in an
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 on a global scale.
The neural system of the nematode C. elegans, the collaboration graph of film
actors, and the oldest U.S. subway system, can now be studied also as metrical
networks and are shown to be small-worlds.
