All You Need is Resistance: On the Equivalence of Effective Resistance and Certain Optimal Transport Problems on Graphs

TL;DR

This article makes a bold claim: that the two fields of effective resistance and optimal transport on graphs should be understood as one and the same, up to a choice of $p$, and introduces the parameterized family of $p$-Beckmann distances for probability measures on graphs.

Abstract

The fields of effective resistance and optimal transport on graphs are filled with rich connections to combinatorics, geometry, machine learning, and beyond. In this article we put forth a bold claim: that the two fields should be understood as one and the same, up to a choice of $p$. We make this claim precise by introducing the parameterized family of $p$-Beckmann distances for probability measures on graphs and relate them sharply to certain Wasserstein distances. Then, we break open a suite of results including explicit connections to optimal stopping times and random walks on graphs, graph Sobolev spaces, and a Benamou-Brenier type formula for $2$-Beckmann distance. We further explore empirical implications in the world of unsupervised learning for graph data and propose further study of the usage of these metrics where Wasserstein distance may produce computational bottlenecks.

Figures (7)

• Figure 1: A side-by-side comparison of optimal flows for the $1$-Beckmann and $2$-Beckmann problems on a $4\times 4$ hexagonal lattice graph. The masses of $\mu,\nu$ at each node are rendered proportionally to opacity; and similarly the optimal flow values at each edge are rendered proportionally to opacity. The arrows indicate orientation of the flow value; i.e., $\circ\rightarrow \circ'$ if the optimal flow $J$ satisfies $J(\circ,\circ') >0$ and $\circ\leftarrow \circ'$ if $J(\circ,\circ')<0$.
• Figure 2: (a)-(b) Illustrations of two measures $\mu,\nu$ and their edge cdfs $K_\mu, K_\nu$ for the fixed path graph $P_3$. (c)-(d) Plots of the empirical cdfs $F_{\widehat{\mu}}$ and $F_{\widehat{\nu}}$, as well as their inverses. Note that the shaded regions are reflections of each other; and that for $p=1$ their common area is $\mathcal{B}_1(\mu,\nu)= \mathcal{W}_1(\mu,\nu)$. This also demonstrates the divergence of the metrics for $p>1$.
• Figure 3: Two illustrations of 1000 simulated simple random walks on the dodecahedral graph with given initial distribution, illustrated with node opacity proportional to density; and naïve stopping rule according to the given stopping distribution, illustrated with opposite node color and opacity proportional to density. The edges are dashed only to indicate depth, and edge opacity is proportional to the total number of times the simulated random walks landed on each edge. The mean lengths of paths in (a) (resp. (b)) correspond to $H_n(\mu,\nu)$ (resp. $H_n(\nu,\mu))$.
• Figure 4: An illustration of the preprocessing pipeline for the digits data, with an example from the class of handwritten zeros. The first step is a mass normalization to convert the pixel values into a fixed-sum distribution viewed on the nodes $V$ of the $8\times 8$ lattice graph. The second step is an embedding $\mu\mapsto L^{-1/2}\mu$, such that $\ell_2$ distance in the target corresponds to $2$-Beckmann distance in $\mathcal{P}(V)$. When computing $\mathcal{W}_2$, we omit the final step.
• Figure 5: Using the digits dataset, and for each pair of digit classes, we computed the pairwise $2$-Beckmann and $2$-Wasserstein distances for each pair of samples originating from the respective digit classes (with around 30,000 pairs of distances per pair of digit classes). Within each tile of the grid, we render a scatterplot of the distances over the overall linear regression between $\mathcal{B}_2$ and $\mathcal{W}_2$ for the experiment given by $\mathcal{W}_2\approx 8.446 \mathcal{B}_2$.

Theorems & Definitions (60)

• Theorem 1.1: Informal statement of \ref{['thm:b2-is-resistance']}
• Definition 1.2
• Definition 1.3
• Definition 1.4
• Remark 1.5
• Proposition 1.6
• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Definition 2.4