Mixing and Merging Metric Spaces using Directed Graphs

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.