Directed forests and the constancy of Kemeny’s constant

Abstract

Consider a discrete-time, time-homogeneous Markov chain on states 1, . . . , n whose transition matrix is irreducible. Denote the mean first passage times by m jk , j, k = 1, . . . , n, and stationary distribution vector entries by v k , k = 1, . . . , n. A result of Kemeny reveals that the quantity n k=1 m jk v k , which is the expected number of steps needed to arrive at a randomly chosen destination state starting from state j, issurprisingly-independent of the initial state j. In this note, we consider n k=1 m jk v k from the perspective of algebraic combinatorics and provide an intuitive explanation for its independence on the initial state j. The all minors matrix tree theorem is the key tool employed.

Kemeny's constant for a graph with bridges

“…. , G k are given, maximizing/minimizing κ(G) is equivalent to maximizing/minimizing the second and third summands of the right side of (18).…”

“…It follows that κ(G ′ ) ≤ κ(G). So, we may ignore the third expression on the right side in the formula (18). Therefore, the minimization problem for Kemeny's constant for all possible chains of G 1 , .…”

“…As seen when addressing Question 3.16, we may annihilate the third expression of the right side in (18) by assuming that exactly one vertex from each G i is incident with bridges in B G . Note that for each x ∼ y ∈ B G , W x + W y = m G + 1.…”

“…Note that for each x ∼ y ∈ B G , W x + W y = m G + 1. The minimum of the second expression can be obtained by determining a vertex v whose accessibility index in H is minimum, and letting each bridge in B G join the copies of v in each copy of H. Now we only need to consider the last summand of the right side in (18). Considering the fact that for x ∼ y ∈ B G , W x = W y + 1, W y = W x + 1, and W x + W y = m G − 1, minimizing Kemeny's constant is equivalent to minimizing the following:…”

“…Originally defined in [14] as the expected time to reach a randomly-chosen state from a fixed starting state, this quantity is astonishingly independent of the choice of initial state, and thus is named Kemeny's constant. Since its introduction in [14], this quantity has fascinated a number of researchers interested in providing an intuitive explanation of its constant property (see, for example, [3,9,18,19]). Due to its interpretation in terms of how 'well-mixing' a Markov chain is, it is not surprising that Kemeny's constant has been recently used for a wide range of applications.…”