We study the mixing time of random walks on small-world networks modelled as follows: starting with the 2-dimensional periodic grid, each pair of vertices {u,v} with distance d> 1 is added as a “long-range” edge with probability proportional to d -r , where r≥ 0 is a parameter of the model. Kleinberg [33{ studied a close variant of this network model and proved that the (decentralised) routing time is O((log n ) 2 ) when r =2 and n Ω (1) when r≠ 2. Here, we prove that the random walk also undergoes a phase transition at r=2 , but in this case, the phase transition is of a different form. We establish that the mixing time is ϴ (log n) for r< 2, O((log n ) 4 ) for r =2, and n Ω (1) for r> 2.

中文翻译：

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小世界网络上的随机游走

我们研究了小世界网络上随机游走的混合时间，建模如下：从二维周期性网格开始，每对距离 d> 1 的顶点 {u,v} 被添加为“远程”边概率与 d 成正比-r, 其中 r≥ 0 是模型的一个参数。Kleinberg [33{ 研究了该网络模型的一个紧密变体，并证明（分散的）路由时间为 O((logn)2） 什么时候r=2 和 nΩ (1)当 r≠2 时。在这里，我们证明了随机游走也经历了相变r=2，但在这种情况下，相变的形式不同。我们确定混合时间为 ϴ (log n) 对于 r< 2, O((logn)4） 为了r=2，并且n Ω (1)对于 r > 2。