Fairness-Aware Opinion Seeding in Undirected Signed Friedkin–Johnsen Networks

Opinion-maximization methods often optimize an aggregate network response and may therefore distribute target-aligned intervention gains unevenly across groups. This issue is especially important in signed networks, where cooperative and antagonistic relations can make the same positive opinion seed increase the equilibrium opinion of one group while decreasing that of another. We study fairness-aware internal-opinion seeding in undirected signed Friedkin–Johnsen networks with heterogeneous anchoring strengths. The objective combines the whole-network average opinion gain with the minimum group-average gain through a tunable fairness parameter. We show that the equilibrium response of any seed set decomposes additively into candidate-wise group-gain vectors. This representation yields an equivalent mixed-integer linear program for exact optimization on tractable instances and an adjoint formulation that evaluates all candidate gains using only one shifted signed-Laplacian solve per group. Because the global averaging vector is a size-weighted combination of the group averaging vectors, no additional global solve is required. We further introduce a dynamically maintained Pareto frontier based on componentwise dominance and establish conditions under which pruning preserves the underlying greedy sequence. Theoretical results establish well-posedness and stability of the signed equilibrium, correctness of the gain and adjoint formulations, sufficient conditions for monotonicity, non-submodularity in general, residual-based numerical error bounds, and computational complexity. Experiments on 14 signed networks ranging from hundreds to more than four million nodes show that the adjoint implementation scales most robustly among the tested methods. The results further demonstrate that the fairness parameter and heterogeneous anchoring strengths can substantially affect both the selected seed sets and the fairness–efficiency balance. Against structural, random, and target-aware Top-K baselines, the proposed greedy method remains broadly competitive, and on small full-candidate instances it matches the solver-certified mixed-integer linear program (MILP) optimum in five of six tested configurations. Overall, the framework provides a scalable approach to group-aware opinion intervention in signed networks while explicitly distinguishing opinion seeding from clamped leader selection.