概率论 (Summer 2013)/Problem Set 5

Problem 2

Let [math]\displaystyle{ G(V,E) }[/math] be an undirected connected graph with maximum degree [math]\displaystyle{ \Delta }[/math].

• Design an efficient, time reversible, ergodic random walk on [math]\displaystyle{ G }[/math] whose stationary distribution is the uniform distribution.
• Let [math]\displaystyle{ \pi }[/math] be an arbitrary distribution on [math]\displaystyle{ V }[/math] such that [math]\displaystyle{ \pi(v)\gt 0 }[/math] for all [math]\displaystyle{ v\in V }[/math]. Design a time reversible, ergodic random walk on [math]\displaystyle{ G }[/math] whose stationary distribution is [math]\displaystyle{ \pi }[/math].