MR-Metric Spaces: Theory and Applications in Weighted Graphs, Expander Graphs, and Fixed-Point Theorems

This paper introduces the concept of MR-metric spaces, a generalization of traditional metric spaces that operates on triples of points rather than pairs. We define MR-metrics and establish their fundamental properties, including non-negativity, identity, symmetry, and a generalized triangle inequality with a constant \( R > 1 \). Three main theorems are presented: (1) a construction of MR-metrics on weighted graphs via minimal spanning subtrees, with applications in network design and VLSI circuit optimization; (2) a set-valued fixed-point theorem for contractions in MR-metric spaces, applied to distributed consensus and fault-tolerant systems; and (3) an MR-metric based on coupling times in expander graphs, with implications for distributed storage and decentralized machine learning. The results are supported by rigorous proofs, illustrative examples, and performance analyses demonstrating practical advantages over traditional methods.