Navigable Small-World Networks with Few Random Bits

Westudy Small-World graphs in the perspective of their use in the development of efficient as well as easy to implement network infrastructures. Our analysis starts from the Small- World model proposed by Kleinberg: a grid network augmented with directed long-range random links. The choices of the long-range links are independent from one node to another. In this setting greedy routing and some of its variants have been analyzed and shown to produce paths of polylogarithmic expected length. We start from asking whether all the randomness, used in Kleinberg's model for establishing the long-range contacts of the nodes, is indeed necessary to assure the existence of short paths. In order to deal with the above question, we impose (stringent) restrictions on the choice of long-range links and we show that such restrictions do not increase the average path length of greedy routing and its variations. We are able to decrease the number of random bits, required to establish each node's long-range link to O(log log n) on a network of size n. Diminishing the randomness in the choice of random links has several benefits; in particular, it implies an increase in the clustering of the graph, thus increasing the resilience of the network.