Let $G=(V,E)$ be a connected, non-bipartite graph with $n$ nodes and $m$ edges. Consider a random walk of length $t$ on $G$ that starts at a uniform random node.
The sequence of nodes in the random walk is a Markov chain with a probability distribution that the random walk ends at a particular node. The initial distribution is $P_0(v) = 1/n$, $v \in V$. The stationary (equilibrium) distribution is $\pi(v) = \frac{deg(v)}{2m}$.
What is $E(P_t(v_k))$, the average of $P_t$ across all nodes $v_k$ with degree exactly $k$? As described, $E(P_0(v_k)) = 1/n$ and $\lim_{t \rightarrow \infty} E(P_t(v_k)) = \frac{k}{2m}$. But what about for other (finite) values of $t$? Bounds would also be helpful.
$P_t(v_k)$ of course depends on the structure of the graph, but thinking that $E(P_t(v_k))$ does not, or at least has tight bounds.
Thank you for any suggestions or feedback.