Stationary distribution for different types of graph

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This is a follow-up question to posts:

Stationary distribution for directed graph

The definition of stationary distribution in wikipedia:Steady-state analysis and limiting distributions

Are stationary distributions of graphs with every property(for example directed or undirected, strongly connected or sparse, periodic or aperiodic)proportional to eigenvector corresponding to eigenvalue 1 ?

If not, what is the difference for each case?

I know when the graph is undirected strongly connected and aperiodic, there is a unique stationary distribution equal to the degree of the vertex divided by the overall degree of nodes.But I don't Know how is it for other type of graph.

Thanks

edited Feb 24 at 19:55

asked Feb 24 at 5:43

Fatime
11●3

	This is exactly the definition of a stationary distribution. – Ori Gurel-Gurevich Feb 24 at 7:43
	@Ori Thanks for comment. I think this property is only satisfied for unique stationary distribution.Are you sure it is satisfied for each graph? – FatimeRahman 5 hours ago – Fatime Feb 24 at 17:40
	Please specify your definition of "stationary distribution", if what you mean does not coincide with the definition as an eigenvector. – Günter Rote Feb 24 at 19:36
	@FatimeRahman: in the wikipedia article you linked to, the left hand side has the vector $\pi$ and the right hand side has $T \pi$, where $T$ is the transition matrix. Thus $\pi$ is an eigenvector with eigenvalue 1. When the stationary distribution isn't unique it simply means that the space of eigenvectors with eigenvalue 1 has dimension more than 1. – Ori Gurel-Gurevich Feb 24 at 22:28

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