Critical Asymmetric Ball Twins in Directed Graphs: Structural Characterization, Reciprocity Effects, and Asymptotic Scaling

In asymmetric (quasi-metric) spaces, the forward closed balls B⁺(x, d(x,y)) and B⁺(y, d(y,x)) centered at distinct points x,y with their respective critical radii generally differ.We study pairs where these balls coincide, termed critical asymmetric ball twins (CABT), and prove complete characterization theorems for fundamental graph families.We establish exact CABT density formulas:(i) complete graphs Kₙ achieve density 1 for all n;(ii) star graphs Sₙ have density 1 − 2/n, where the 2(n−1) non-CABT pairs are exactly the hub–leaf pairs;(iii) wheel graphs Wₙ have density 1 − 4/n, with the 4(n−1) non-CABT pairs consisting of hub–rim and adjacent-rim pairs;(iv) cycles satisfy ρ(Cₙ) = 1/(n−1) for even n and 2/(n−1) for odd n, with CABT pairs being exactly the diametrically opposite vertices;(v) paths satisfy ρ(Pₙ) = 2/n(n−1), with only the two endpoint pairs being CABT.A structural dichotomy theorem shows hub-dominated graphs are asymptotically CABT-dense (ρ → 1) while sparse families are CABT-free (ρ → 0).In random directed graphs G(n,p,α) with controllable reciprocity α ∈ 0,1, CABT density exhibits strong nonlinear dependence, well-approximated by a cubic polynomial (R² = 0.995).A confound-control experiment with fixed arc count isolates the pure reciprocity effect, revealing that reciprocity monotonically increases CABT density; the non-monotonic U-shaped curve observed in the original model is driven by the joint variation of reciprocity and edge density.Statistical validation via ANOVA (F = 81.9, p < 10⁻⁶), Kruskal–Wallis (H = 483.6, p < 10⁻⁶), and 10,000-iteration bootstrap confirms robustness.We provide a polynomial-time detection algorithm using prefix-ball signatures with O(1) per-pair verification, and connect CABT to domain-theoretic formal balls and metric identification theory.