Multiple Scales in Small-World Graphs

Small-world architectures may be implicated in a range of phenomena from disease propagation to networks of neurons in the cerebral cortex [1-2]. While most of the recent attention on small-world networks has focussed on the effect of introducing disorder/randomness into a regular graph, we show that that the fundamental mechanism behind the small-world phenomenon is not dis-order/randomness, but the presence of edges 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 edges of diverse length scales. We also show that the distribution of the length scales of the new edges is significantly more important than whether the new edges are long range, medium range or short range.