Near Optimal Routing for Small-World Networks with Augmented Local   Awareness
  In order to investigate the routing aspects of small-world networks, Kleinberg 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 $O(\log^2 n)$ expected number of hops, where $n$ denotes the number of nodes in the network. Martel and Nguyen have found that the expected diameter of Kleinberg's small-world networks is $\Theta(\log n)$. 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? We extend Kleinberg's model and add three augmented local links for each node: two of which are connected to nodes chosen randomly and uniformly within $\log^2 n$ Mahattan distance, and the third one is connected to a node chosen randomly and uniformly within $\log n$ Mahattan distance. We show that if each node is aware of $O(\log n)$ 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 $O(\log n \log \log n)$ 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 $O(\log^2 n)$ bits of routing information on each node, thus they are scalable with the network size. 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. 
Jianyang Zeng, Wen-Jing Hsu and Jiangdian Wang