The systems you are looking for belong to the class of small-world networks, introduced in 1998 by Watts and Strogatz. Quite generally, a small-world network is defined to be a network where the typical distance between two randomly chosen nodes (the number of steps required) grows proportionally to the logarithm of the number of nodes in the network.

Wikipedia provides a good starting point for exploration:

<A HREF="https://en.wikipedia.org/wiki/Small-world_network">Small-world networks</A>

<A HREF="https://en.wikipedia.org/wiki/Small-world_experiment">Small-world experiments</A>

Specifically related to the Erdős number is this online lecture by John Barrow:

<A HREF="http://www.youtube.com/watch?v=F5ZsM5OijYs"> Erdős Numbers: A mathematical example of 'small world' networks</A>