If you do a random walk on an undirected, connected graph, is the stationary distribution for the probability that you have just traversed edge $e$ uniform over all edges no matter what the graph looks like?
[Math] Stationary distribution of random walk on a graph
graph theoryprobabilityrandom walk
Best Answer
The stationary measure of a vertex is proportional to its degree. Let $G$ be a graph with $e$ edges. Then for any vertex the stationary probability measure is $$\pi(x) = \deg(x)/2e.$$ To see it is stationary, note $p(x,y) = 1/\deg(x)$ if $x$ is incident to $y$. Hence $$ \sum_{x} \pi(x) p(x,y) = \sum_{x\sim y} \deg(x)\frac{1}{2e\deg(x)} = \deg(y)/2e = \pi(y).$$