  A wealth of evidence shows that real world networks are endowed with the
small-world property i.e., that the maximal distance between any two of their
nodes scales logarithmically rather than linearly with their size. In addition,
most social networks are organized so that no individual is more than six
connections apart from any other, an empirical regularity known as the six
degrees of separation. Why social networks have this ultra-small world
organization, whereby the graph's diameter is independent of the network size
over several orders of magnitude, is still unknown. We show that the 'six
degrees of separation' are the property featured by the equilibrium state of
any network where individuals weigh between their aspiration to improve their
centrality and the costs incurred in forming and maintaining connections. We
show, moreover, that the emergence of such a regularity is compatible with all
other features, such as clustering and scale-freeness, that normally
characterize the structure of social networks. Thus, our results show how
simple evolutionary rules of the kind traditionally associated with human
cooperation and altruism can also account for the emergence of one of the most
intriguing attributes of social networks.
