Near optimal routing in a small-world network with augmented local awareness

Abstract

In order to investigate the routing aspects of small-world networks, Kleinberg [13] proposes a network model based on a d-dimensional lattice with long-range links chosen at random according to the d-harmonic distribution, Kleinberg shows that the greedy routing algorithm by using only local information performs in O(lg2 n) expected number of hops, where n denotes the number of nodes in the network. Martel and Nguyen [17] have found that the expected diameter of Kleinberg's small-world networks is θ(lg n). Thus a question arises naturally: Can we improve the routing algorithms to match the diameter of the networks while keeping the amount of information stored on each node as small as possible? Existing approaches for improving the routing performance in the small-world networks include: (1) Increasing the number of long-range links [2, 15]; (2) Exploring more nodes before making routing decisions [14]; (3) Increasing the local awareness for each node [10, 17]. However, all these approaches can only achieve O((lg n)1+ε) expected number of hops, where ε > 0 denotes a constant. We extend Kleinberg's model and add two augmented local links for each node, which are connected to nodes chosen randomly and uniformly within lg2 n Mahattan distance. Our investigation shows that these augmented local connections can make small-world networks more navigable. We show that if each node is aware of O(lg n) number of neighbors via the augmented local links, there exist both non-oblivious and oblivious algorithms that can route messages between any pair of nodes in O(lg n lg lg n) expected number of hops, which is a near optimal routing complexity and outperforms the other related results for routing in Kleinberg's small-world networks. Our schemes keep only O(lg 2 n) bits of routing information on each node, thus they are scalable with the network size. Our results imply that the awareness of O(lg n) nodes through augmented links is more efficient for routing than via the local links [10, 17]. Besides adding new light to the studies of social networks, our results may also find applications in the design of large-scale distributed networks, such as peer-to-peer systems, in the same spirit of Symphony [15]. © Springer-Verlag Berlin Heidelberg 2005.