On defining Kemeny's constant for non-backtracking random walks

TL;DR

This paper develops vertex-space definitions of Kemeny’s constant for non-backtracking random walks on graphs by leveraging non-backtracking first hitting times and a vertex-space projection of the edge-space fundamental matrix. It proves that the two NB definitions coincide for graphs with locally uniform return time (in particular edge-transitive graphs) and explores the gap in general graphs via an error term. The authors provide closed-form or tractable expressions for the NB Kemeny constant across several graph families (including $K_n$, $C_nC_k$, cycle barbells, complete bipartite graphs, and pinwheels), consistently finding NB values smaller than their simple-random-walk counterparts. They also discuss computational cost advantages of the trace-based fundamental-matrix approach and pose several conjectures and open questions about extremal structures and possible universal bounds. The work advances understanding of non-backtracking dynamics and offers practical tools for comparing NBW behavior to SRW across graph classes.

Abstract

We propose two possible definitions for a version of Kemeny's constant of a graph based on non-backtracking random walks (in place of the usual simple random walk). We show that these two definitions coincide for edge-transitive graphs, and give a condition generalizing edge-transitive for which equality holds, and investigate by how much they can differ in general. We compute our non-backtracking Kemeny's constant for several families of graphs.

Figures (6)

• Figure 1: Two graphs that satisfy Condition \ref{['cond: locally uniform return time']} but are not edge-transitive.
• Figure 2: A numerical comparison on all connected graphs on 6-9 vertices between values of $\widehat{\mathcal{K}}_{v}^{nb}(G) = \tr{Z^{(nb)}_v} - 1$ and $\mathcal{K}_{v}^{nb}(G) = \pi^\top M^{(nb)}_v\pi$. Of note, these two values never seem to significantly stray from each other. Moreover, it suggests the relation $\tr{Z^{(nb)}_v} - 1 \leq \pi^\top M^{(nb)}_v\pi$. All computations done using SageMath.
• Figure 3: A numerical comparison of values of the SRW Kemeny's constant $\mathcal{K}_v(G)$ and the NB Kemeny's constant $\mathcal{K}_{v}^{nb}(G) = \pi^\top M^{(nb)}_v\pi$ on all graphs with $|V(G)| \in \{6, 7, 8, 9\}$. Of interest, the NB value is strictly less than the SRW counterpart in each case.
• Figure 4: The graph denoted $C_8C_3$, an example of the family $C_nC_k$. This graph is formed by taking the $8$-cycle and adding an additional edge to create a cycle of length $3$. This graph is conjectured to minimize $\mathcal{K}^{(nb)}_v(G)$ for all $G$ with $|V(G)|=8$. In general, we conjecture that $C_nC_3$ minimizes $\mathcal{K}^{(nb)}_v(G)$ with $|V(G)| = n$.
• Figure 5: The graph CB(3, 5, 6).

Theorems & Definitions (53)

• Definition 2.1
• Definition 3.1
• Example 3.1
• Definition 3.3
• Proposition 3.4
• proof
• Definition 3.6
• Proposition 3.7
• Proposition 3.8
• proof