Random Walks with Traversal Costs: Variance-Aware Performance Analysis and Network Optimization

Abstract

We introduce weighted Markovian graphs, a random walk model that decouples the transition dynamics of a Markov chain from (random) edge weights representing the cost of traversing each edge. This decoupling allows us to study the accumulated weight along a path independently of the routing behavior. Crucially, we derive closed-form expressions for the mean and variance of weighted first passage times and weighted Kemeny constants, together with their partial derivatives with respect to both the weight and transition matrices. These results hold for both deterministic and stochastic weights with no distributional assumptions. We demonstrate the framework through two applications, highlighting the dual role of variance. In surveillance networks, we introduce the surprise index, a coefficient-of-variation metric quantifying patrol unpredictability, and show how maximizing it yields policies that are both efficient and hard to anticipate. In traffic networks subject to cascading edge failures, we develop a minimal-intervention framework that adjusts speed limits to preserve connectivity under three increasingly flexible regulatory policies.

Figures (3)

• Figure 1: Optimized policies on the $4 \times 4$ grid with deterministic weights. (a) Minimizing variance produces a predictable Hamiltonian cycle with $16$ dominant edges ($\bm P(i,j) > 0.5$), reducing $\mathcal{S}$ by $79.5\%$. (b) Maximizing $\mathcal{S}$ distributes transition probabilities broadly, producing high variance in first-passage times. Edge thickness indicates $\bm P(i,j)$; node shading indicates $\boldsymbol{\pi}_{\mathcal{W}}(i)$.
• Figure 2: Maximum-surprise policy on an $8 \times 8$ grid with non-uniform $\bm \mu$ and stochastic weights ($\mathcal{S}^* = 1.26$). Black squares denote obstacles; darker nodes indicate higher target coverage weight. Green edges have low CV ($< 1$), red edges have high CV ($\geq 1$); edge thickness indicates $\bm P(i,j)$. The policy respects the $2{:}1$ priority ratio on obstacle-adjacent nodes.
• Figure 3: (a) Aggregated improvement in the Weighted Kemeny Constant (left) and Variance (right) for 150 randomly generated geometric graphs with 10 nodes, $k=5$, and connection distances $r = \sqrt{d/(10 \pi)}$, where $d=5$ is the desired average degree. Nodes 2, 5, and 9 were arbitrarily chosen to represent destinations. (b) Comparison of the average change to the stationary distribution for all three policies.

Theorems & Definitions (20)

• Theorem 1: MWFPT formula
• Remark 1
• Theorem 2: Second moment of weighted first passage times
• Lemma 1: Derivative of $\bm \pi_{\cal W}$ with respect to $\bm W$
• Lemma 2: Derivative of First Moment, Mean Weights
• Theorem 3: Derivative of VWFPTs with respect to $\bm W$
• Remark 2
• Lemma 3: Linearity of $K(\bm P, {\cal W})$ in $\bm W$
• Theorem 4: Derivative of $K(\bm P, {\cal W})$ with respect to $\bm W$
• Theorem 5: Derivative of $K_{\cal W} (\bm P, {\cal W})$ with respect to $\bm W$