Tensor-space joint optimisation of topology-role configuration in multi-agent systems

Topology-role coupling is central to multi-agent formation control, yet its combinatorial complexity makes analytical treatment challenging. We propose a three-tensor framework that separately encodes graph structure, time-varying channel states, and role assignments. Their contraction yields a coupling metric equal to the mean of the non-trivial Laplacian eigenvalues, which serves as a tractable joint optimisation objective. Within this framework, the optimal leader emerges intrinsically from the tensor algebra, without recourse to heuristic rules or external designation. Optimisation is performed via spectral gradient flow restricted to a Tucker-compressed subspace, using incremental updates and warm starts from previous solutions. Under mild assumptions, Tucker compression guarantees polynomial complexity, and the gradient flow is proven to converge. Simulations demonstrate substantially better scaling than exhaustive combinatorial search as agent numbers grow. The framework unifies multi-agent control, graph theory, and tensor algebra, providing a principled foundation for scalable coordination in networked systems. Here, the authors develop a tensor-based framework for jointly optimizing communication topology and role assignment in multi-agent systems. They establish theoretical guarantees for the proposed optimization and demonstrate improved scalability and performance over combinatorial approaches.