Mixing and merging metric spaces using directed graphs

Let ( X 1 , d 1 ) , … , ( X N , d N ) be metric spaces with d i : X i × X i → [ 0 , 1 ] and let X = ∏ i = 1 N X i . Let G be a directed graph with vertex set [ N ] = { 1 , … , N } and edge set E ( G ) . Given edge weights P = { p i j ∈ ( 0 , 1 ] : ( i , j ) ∈ E ( G ) } , define the closed in-neighborhood B j ( G ) : = { j } ∪ { i : ( i , j ) ∈ E ( G ) } and put p j j = 1 . Define, for g , h ∈ X , d X , G , P ( g , h ) : = 1 − 1 N ∑ j = 1 N ∏ i ∈ B j ( G ) ( 1 − d i ( g i , h i ) ) 1 / p i j . We prove that d X , G , P is a metric and analyze its behavior under graph operations (disjoint unions, Cartesian products). Two quantitative principles govern this family: (i) edge densification (adding edges) increases the distance pointwise and in expectation, with an exact single-edge increment formula and a uniform 1 / N sensitivity bound; and (ii) weight monotonicity : increasing an edge weight p i j decreases the metric. When the coordinate pairs are independent across coordinates, we obtain a closed-form factorization of the mean distance. In the discrete specialization X = F q N with d i ∈ { 0 , 1 } and G acyclic, after replacing G by the oriented reflexive transitive closure G ⪯ of Corollary 1.4 , the unnormalized distance D X , G ⪯ , P : = N d X , G ⪯ , P coincides with the size of the hereditary closure of the support, yielding a poset/digraph sphere-packing bound via the counting of lower ideals. Finally, we give a continuous analog of the construction, yielding a metric on C ( [ 0 , 1 ] , C ) through an explicit integral formula. These results unify and generalize several graph-based metrics used in coding theory while providing new design levers for mean-distance control.