增强局部感知的小世界网络近优路由算法
Near Optimal Routing for Small-World Networks with Augmented Local Awareness
摘要：
为了研究小世界网络的路由问题，Kleinberg提出了一个基于$d$维格的网络模型，该格中的长距离链路是根据$d$ -调和分布随机选择的。Kleinberg证明了仅使用本地信息的贪婪路由算法在$O(\log^2 n)$期望跳数中执行，其中$n$表示网络中的节点数。Martel和Nguyen发现Kleinberg的小世界网络的预期直径是$Θ(\log n)$。因此，一个问题自然出现了：我们是否可以改进路由算法，以匹配网络的直径，同时保持每个节点上存储的信息量尽可能小？我们扩展了Kleinberg的模型，并为每个节点增加了三个增强的本地链接：其中两个连接到随机选择的节点，均匀的$\log^2 n$的Mahattan距离，第三个连接到随机选择的节点，均匀的$\log n$ Mahattan距离。我们发现，如果每个节点都知道$O(\log n)$通过增强本地链路的邻居数，有两个不经意和不经意的算法，可以路由消息之间的任何一对节点在$O(\log n \log \log n)$预期的跳数，这是一个接近最优的路由复杂性和优于其他相关结果在Kleinberg的小世界网络路由。我们的计划只保留$O(\log^2 n)$比特的路由信息在每个节点上，因此，他们是可扩展的网络规模。除了为社交网络的研究增添新的亮点外，我们的结果也可能在大规模分布式网络的设计中找到应用，例如对等系统，同样的交响乐精神。
英文摘要：
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 $Θ(\log n)$. 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.
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