Routing in the Watts and Strogatz Small World Networks Revisited

The study of small-world networks (SWN) has become popular with the growth of the World Wide Web (or simply Web) and more recently with the rising growth of social networking sites such as Facebook. The idea of a small world can be simply described as between any two people there is only a short chain linking them through their acquaintances. SWNs was first studied in the pioneering work of Stanley Milgram who showed experiments forwarding letters that the length of the chain was between five and six. This is also popularly known as "six degrees of separation". The small world phenomena suggest that a graph modeling a small world should have a small diameter. Furthermore, SWNs are not just a random graph but they have structure. A well studied model of a SWN is the one proposed by Watts and Strogatz. We will abbreviate the Watts and Strogatz model as WS-SWN. WS-SWN has the virtue of simplicity, while capturing structure in various networks and possessing low diameter and has been shown to capture the structure real small-world networks. In this paper, we revisit the question of navigability in WS-SWN. Firstly, the WS-SWN makes it easy to construct SWNs with different amounts of clustering making it a useful model for real SWNs. Secondly, WS-SWN is not navigable using pure greedy routing. Our initial experiments suggest that by considering other routing algorithms, WS-SWN may be navigable. It is at least more navigable than might be expected.