  Small-world architectures may be implicated in a range of phenomena from
disease propagation to networks of neurons in the cerebral cortex. While most
of the recent attention on small-world networks has focussed on the effect of
introducing disorder/randomness into a regular network, we show that that the
fundamental mechanism behind the small-world phenomenon is not
disorder/randomness, but the presence of connections of many different length
scales. Consequently, in order to explain the small-world phenomenon, we
introduce the concept of multiple scale graphs and then state the multiple
length scale hypothesis. Multiple scale graphs form a unifying conceptual
framework for the study of evolving graphs. Moreover, small-world behavior in
randomly rewired graphs is a consequence of features common to all multiple
scale graphs. To support the multiple length scale hypothesis, novel graph
architectures are introduced that need not be a result of random rewiring of a
regular graph. In each case it is shown that whenever the graph exhibits
small-world behavior, it also has connections of diverse length scales. We also
show that the distribution of the length scales of the new connections is
significantly more important than whether the new connections are long range,
medium range or short range connections.
