Routing In Small-World Network Research Articles

Greedy routing in small-world networks with power-law degrees

In this paper we study decentralized routing in small-world networks that combine a wide variation in node degrees with a notion of spatial embedding. Specifically, we consider a variant of J. Kleinberg’s grid-based small-world model in which (1) the number of long-range edges of each node is not fixed, but is drawn from a power-law probability distribution with exponent parameter \(\alpha \ge 0\) and constant mean, and (2) the long-range edges are considered to be bidirectional for the purposes of routing. This model is motivated by empirical observations indicating that several real networks have degrees that follow a power-law distribution. The measured power-law exponent \(\alpha \) for these networks is often in the range between 2 and 3. For the small-world model we consider, we show that when \(2 < \alpha < 3\) the standard greedy routing algorithm, in which a node forwards the message to its neighbor that is closest to the target in the grid, finishes in an expected number of \(O(\log ^{\alpha -1} n\cdot \log \log n)\) steps, for any source–target pair. This is asymptotically smaller than the \(O(\log ^2 n)\) steps needed in Kleinberg’s original model with the same average degree, and approaches \(O(\log n)\) as \(\alpha \) approaches 2. Further, we show that when \(0\le \alpha < 2\) or \(\alpha \ge 3\) the expected number of steps is \(O(\log ^2 n)\), while for \(\alpha = 2\) it is \(O(\log ^{4/3} n)\). We complement these results with lower bounds that match the upper bounds within at most a \(\log \log n\) factor.

First passage time of multiple Brownian particles on networks with applications

In this article, we study some theoretical and technological problems with relation to multiple Brownian particles on networks. We are especially interested in the behavior of the first arriving Brownian particle when all the Brownian particles start out from the source s simultaneously and head to the destination h randomly. We analyze the first passage time (FPT) Y s h ( z ) and the mean first passage time (MFPT) 〈 Y s h ( z ) 〉 of multiple Brownian particles on complex networks. Equations of Y s h ( z ) and 〈 Y s h ( z ) 〉 are obtained. On a variety of commonly encountered networks, we observe first passage properties of multiple Brownian particles from different aspects. We find that 〈 Y s h ( z ) 〉 drops substantially when particle number z increases at the first stage, and converges to d s h , the distance between the source and the destination when z → ∞ . The distribution of FPT Prob { Y s h ( z ) = t } , t = 0 , 1 , 2 , … is also analyzed in these networks. The distribution curve peaks up towards t = d s h when z increases. Consequently, if particle number z is set appropriately large, the first arriving Brownian particle will go along the shortest or near shortest paths between the source and the destination with high probability. Simulations confirm our analysis. Based on theoretical studies, we also investigate some practical problems using multiple Brownian particles, such as communication on P2P networks, optimal routing in small world networks, phenomenon of asymmetry in scale-free networks, information spreading in social networks, pervasion of viruses on the Internet, and so on. Our analytic and experimental results on multiple Brownian particles provide useful evidence for further understanding and properly tackling these problems.