When Does Triadic Balance Reduce to Pairwise Order?

Abstract

Structural-balance interactions are naturally defined on triads, but when pair affinities are themselves generated by continuous node attributes, it is not clear whether the resulting interaction remains genuinely higher order. Here we address this question for agents represented by unit vectors in an attribute space, with pair affinity given by their scalar product and a Heider product assigned to each triangle. For planar attributes, we show that the triadic interaction reduces exactly, at finite system size, to a pairwise Curie--Weiss XY model of the doubled angles. The ordered phase is therefore nematic and undergoes a continuous mean-field $O(2)$ transition at $k_{\mathrm B}T_c=J/8$. The same reduction reveals a local $\mathbb Z_2$ degeneracy: each nematically ordered state corresponds to $2^N$ polar assignments, whose binomial multiplicities make two nearly equal opposite groups typical at low temperature. Monte Carlo simulations support the exact critical temperature, the mean-field Binder-cumulant value, and the predicted complete-graph finite-size scaling. Finally, a tensor representation of the interaction shows that the planar reduction is dimension specific: in two dimensions the cubic invariant is constrained by lower-order invariants, whereas for $d\geq3$ it is generically independent. These results identify when attribute-induced triadic balance is effectively pairwise and when genuinely higher-order structure can remain.