Mixing and Merging Metric Spaces using Directed Graphs

TL;DR

This paper introduces a new metric for combining multiple metric spaces using directed graphs, explores its properties, and examines its behavior under various graph operations and limits, with applications in error-correcting codes and graph limits.

Contribution

The paper defines a novel metric on product spaces based on directed graphs and analyzes its properties, including behavior under graph operations and in limiting cases.

Findings

The new metric is proven to satisfy metric space properties.

Behavior of the metric under disjoint unions and Cartesian products is characterized.

Connections to error-correcting code distances and graphon limits are established.

Abstract

Let $(X_1,d_1),\dots, (X_N,d_N)$ be metric spaces, where $d_i: X_i \times X_i \rightarrow [0,1]$ is a distance function for $i=1,\dots,N$. Let $\mathcal{X}$ denote the set theoretic product $X_1\times \cdots \times X_N$. Let $\mathcal{G} = \left(\mathcal{V},\mathcal{E}\right)$ be a directed graph with vertex set $\mathcal{V} =\{1,\dots, N\}$, and let $\mathcal{P} = \{p_{ij}\}$ be a collection of weights, where each $p_{ij}\in (0, 1]$ is associated with the edge $(i,j) \in \mathcal{E}$. We introduce the function $d_{\mathcal{X},\mathcal{G},\mathcal{P}}: \mathcal{X}\times \mathcal{X} \to [0,1]$ defined by \begin{align*} d_{\mathcal{X},\mathcal{G},\mathcal{P}}(\mathbf{g},\mathbf{h}) := \left(1 - \frac{1}{N}\sum_{j=1}^N \prod_{i=1}^N \left[1- d_i(g_i,h_i)\right]^{\frac{1}{p_{ji}}} \right), \end{align*} for all $\mathbf{g},\mathbf{h} \in \mathcal{X}$. In this paper we show that $d_{\mathcal{X},\mathcal{G},\mathcal{P}}$ defines a metric space over $\mathcal{X}$. Then we determine how this distance behaves under various graph operations, including disjoint unions and Cartesian products. We investigate two limiting cases: (a) when $d_{\mathcal{X},\mathcal{G},\mathcal{P}}$ is defined over a finite field, leading to a broad generalization of graph-based distances commonly studied in error-correcting code theory; and (b) when the metric is extended to graphons, enabling the measurement of distances in a continuous graph limit setting.