We study the linearization of a discrete transportation distance between probability distributions on finite weighted graphs originally due to Maas (``Gradient flows of the entropy for finite {M}arkov chains,'' J. Funct. Anal. 261(8), 2011) which demonstrates various connections to the underlying combinatorial structure of the graph. For a connected graph and a reference density $μ$ on its vertices, our main result is a nonasymptotic local linearization theorem showing that if $ν$ is a small additive perturbation of $μ$ then their squared discrete transportation distance is controlled above and below by the quadratic form of the pseudoinverse of a re-weighted graph Laplacian matrix. When the reference measure is stationary for the simple random walk on the graph, the weights agree with the original graph and this yields the quadratic form $(μ-ν)^\top L_w^\dagger (μ-ν)$, which can be viewed as a form of resistance distance between probability measures. This distance has a number of combinatorial and variational characterizations, including Beckmann and Benamou--Brenier formulas, a dual homogeneous Sobolev norm formula, a spanning $2$-forest formula, and a representation through random walk hitting times. Finally, we show that on the resulting ``resistance manifold,'' the gradient flow of the $χ^2$ functional is the continuous-time random walk and that its geodesic strong convexity modulus equals the spectral gap of the normalized Laplacian. From this geometric vantage point, one recovers the classical fact that the spectral gap of the normalized Laplacian controls the exponential convergence rate of the random walk to stationarity.

翻译：我们研究有限加权图上概率分布之间离散运输距离的线性化问题，该距离最初由Maas提出（“有限马尔可夫链的熵梯度流”，J. Funct. Anal. 261(8), 2011），并展示了与图底层组合结构的多种联系。对于连通图及其顶点上的参考密度$μ$，我们的主要结果是一个非渐近局部线性化定理：若$ν$是$μ$的一个小加性扰动，则它们的平方离散运输距离上下受控于一个加权图拉普拉斯矩阵伪逆的二次型。当参考测度是图上简单随机游走的平稳测度时，权重与原图一致，此时二次型化为$(μ-ν)^\top L_w^\dagger (μ-ν)$，可视为概率测度之间的电阻距离。该距离具有多种组合与变分刻画，包括Beckmann公式和Benamou-Brenier公式、对偶齐次Sobolev范数公式、生成$2$-森林公式以及随机游走击中时间表示。最后，我们证明在由此得到的“电阻流形”上，$χ^2$泛函的梯度流即为连续时间随机游走，且其测地线强凸模等于归一化拉普拉斯矩阵的谱隙。从这一几何视角可重新导出经典结论：归一化拉普拉斯矩阵的谱隙控制着随机游走向平稳分布的指数收敛速率。