Emergent Topology of Optimal Networks for Synchrony

Real-world networks, whether shaped by evolution or intelligent design, are typically optimized for functionality while adhering to physical, geometric, or budget constraints. Yet tools to identify such structures remain limited. We develop a gradient-based optimization framework to identify synchrony-optimal weighted networks under a constrained coupling budget. The resulting networks exhibit counterintuitive features: 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 findings are matched by"constructive"theory: 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. Our results have implications for power grids, neuromorphic computing, and other coupled oscillator technologies.