Emergent Optimal Frustration Density in Signed Relational Networks

We investigate how signed relational structure shapes emergent geometric organization in complex networks. Using signed small-world graphs with a tunable fraction of negative edges, we introduce a frustration-sensitive cancellation index that quantifies global incompatibility arising from opposing relational contributions. Across numerical experiments, global frustration exhibits a robust non-monotonic dependence on negative-edge density: frustration is maximized at an intermediate, sharply localized sign density rather than increasing monotonically. A refined scan yields an optimal frustration density q* ≈ 0.4946 ± 0.0044 (bootstrap std ≈ 0.0037 at N=200), robust to coarse-graining and stable under resampling. Repeating the analysis for N=150–300 shows no statistically significant finite-size drift within uncertainty; a constant model provides the most stable description over this range. These results indicate an emergent global frustration scale characteristic of a glassy crossover regime, distinct from a critical phase transition.