Multiple Scales in Small-World Networks
Small-world architectures may be implicated in a range of phenomena from networks of neurons in the cerebral cortex to social networks and propogation of viruses. Small-world networks are interpolations of regular and random networks that retain the advantages of both regular and random networks by being highly clustered like regular networks and having small average path length between nodes, like random networks. 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 networks and then state the multiple length scale hypothesis. We show that small-world behavior in randomly rewired networks is a consequence of features common to all multiple scale networks. To support the multiple length scale hypothesis, novel network architectures are introduced that need not be a result of random rewiring of a regular network. In each case it is shown that whenever the network 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.