Multiple Scales in Small-World Graphs

TL;DR

This paper proposes that the key to small-world phenomena is the presence of multiple connection length scales, not randomness, and introduces multiple scale graphs as a unifying framework for understanding evolving networks.

Contribution

It introduces the concept of multiple scale graphs and the multiple length scale hypothesis, challenging the focus on randomness in explaining small-world properties.

Findings

Small-world behavior is linked to diverse connection length scales.

Distribution of length scales is more crucial than connection range.

Novel graph architectures support the multiple length scale hypothesis.

Abstract

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.