Configuration-Induced Metrics on Finite Relational Structures

AbstractThis study investigates the structural consequences of averaging graph-theoretic distances over finite, constraint-closed families of admissible relational configurations. While the averaging operation itself is elementary, we prove that it gives rise to sharp rigidity, non-reconstructibility, and minimality phenomena that do not follow from general properties of metric spaces. In particular, distinct configuration families may induce identical averaged metrics while admitting no configuration-wise correspondence. The metric depends essentially on configuration multiplicity rather than on any distinguished relational realization. In particular, it cannot, in general, be reduced to a single-graph metric, a probabilistic ensemble, a spectral construction, or an order-theoretic framework. We analyse how admissibility constraints influence the induced geometry and show that restricting the configuration space produces a monotonic increase of distances, while enlarging admissibility leads to stabilization under explicit uniform bounds. Under stated conditions, families of such metrics are shown to be pre-compact in the Gromov–Hausdorff sense. Explicit finite constructions demonstrate non-degeneracy and verify that configuration families with identical graph isomorphism classes may induce inequivalent metrics. The results identify a distinct class of discrete metrics determined by admissible configuration families rather than individual relational realizations, providing a self-contained and combinatorial framework for studying geometry induced by constrained relational variability. Keywords: Discrete metric spaces Configuration spaces Graph distance Relational structures Combinatorial geometry