Conviction versus Conformity in a Random-Field Ising Electorate: Antagonistic Ties, Network Density, and a Parameter-Free Prediction

Abstract

We study the zero-temperature random-field Ising model with +-J bond disorder on sparse graphs as a model of an electorate: survey-derived, trimodal fields compete with social couplings, a fraction alpha of which are antagonistic. With exact ground states up to N=180 (over 78000 optimizations, certified except in the densest frustrated cases) and validated heuristics to N=720, we locate the crossover r_h* at which the field carries half of the ground-state energy, close to where half the voters held against their own field have switched (0.93 r_h* for alpha<=0.2). Our main test is a zero-temperature cavity theory solved on the graph ensemble with no fitted parameter: at seven densities not otherwise used, predicted before their exact ground states were computed, it placed r_h* within 10% in 19 of 21 cells (median error 3%). Its errors are unsigned at alpha=0.05 and 0.5 but all negative at alpha=0.2, where it predicts the coupling energy correctly but a crossover that is early and, at high density, too sharp, and where its replica-symmetric solution is unstable in six of seven cells, consistent with replica-symmetry breaking at intermediate frustration. Empirically r_h*~0.91 e_J(0), with e_J(0) the coupling energy of the zero-field ground state, inside a rigorous per-instance factor-of-two window, so the known +-J energy laws carry over: r_h* ~ ^gamma with gamma=1.01-1.09 for alpha=0.05 and 0.50-0.57 for alpha>=0.5 ( =3-28; five graph families agree at matched density). At intermediate frustration r_h* is not a power law: sparse networks accommodate antagonistic ties at the cost of cooperative ones, which stops paying near ~10; the theory reproduces this without adjustment. In weakly frustrated dense networks the ground state is a near-consensus following the sign of the aggregate field in 93-100% of realisations at small r_h, against near-chance agreement in frustrated ones.