From Invariance to Symmetry Breaking in FIM-Aware Cooperative Heterogeneous Agent Networks

We recast cooperative localization and scheduling in heterogeneous multi-agent systems through the lens of symmetry and symmetry breaking. On the geometric side, the Fisher Information Matrix (FIM) objective is invariant to rigid Euclidean transformations of the global frame, while its maximization admits symmetric optimal sensor formations; on the algorithmic side, heterogeneity and task constraints break permutation symmetry across agents, requiring policies that are sensitive to role asymmetries. We model communication as a random graph and quantify structural symmetry via topology metrics (average path length, clustering, betweenness) and graph automorphism-related indices, connecting these to estimation uncertainty. We then design a hybrid reward for reinforcement learning (RL) that is equivariant to agent relabeling within roles yet intentionally introduces asymmetry through distance/FIM terms to avoid degenerate symmetric configurations with poor observability. Simulations show that (i) symmetry-aware, FIM-optimized path planning reduces localization error versus symmetric but non-informative placements; and (ii) controlled symmetry breaking in policy learning improves robustness and data rate–reward trade-offs over baselines. Our results position symmetry/asymmetry as first-class design principles that unify estimation-theoretic invariances with learning-based coordination in complex heterogeneous networks. Under DDPG training, the total data rate (SDR) reaches 6.63±0.97 and the average reward per step (ARPS) is −80.70±6.94, representing improvements of approximately 11.8% over the baseline (5.93±3.51) and 11.1% over SAC (5.97±2.66), respectively. The network’s mean shortest-path length is L=1.721, and the average betweenness centrality of the coordination nodes is ≈0.098. Moreover, the FIM-optimized path-planning strategy achieves the lowest localization error among all evaluated policies.