Emergent Topology of Optimal Networks for Synchrony

Abstract

Designing high-performing networks requires optimizing for functionality while respecting physical, geometric, or budget constraints. Yet, mathematical and computational tools to design such systems remain limited, particularly for collective dynamics arising from heterogeneous dynamical units. Here, we develop a gradient-based optimization framework to identify synchrony-optimal weighted networks under a constrained coupling budget. The resulting networks exhibit counterintuitive properties: they are sparse, bipartite, elongated, and extremely monophilic (i.e., the neighbors of any node are similar to one another while differing from the node itself). These structural patterns persist across dynamical models ranging from the power-grid swing equations to chaotic Rössler systems, suggesting broad applicability to coupled oscillator technologies. To gain insight, we develop a "constructive" theory for coupled Kuramoto oscillators: a nonlinear differential equation identifies which pairs of nodes are coupled, while a variational principle prescribes the budget allocated to each node. Dynamics unfolding over optimal networks provably lack a synchronization threshold; instead, as the budget exceeds a calculable critical value, the system globally phase-locks, exhibiting critical scaling at the transition. Together, our findings offer design principles for synchrony-dependent technologies with potential applications ranging from microgrids to laser arrays and quantum oscillators.