An analysis framework for Turing instability on multigraph networks from the perspective of optimization

The study of Turing patterns has been successfully extended from continuous media to single-layer and, more recently, multigraph networks. However, analyzing Turing instability in multigraph networks remains a challenge, as the existing approximate method relies on restrictive assumptions about dense-network connectivity. To address this limitation, we propose a novel least squares framework by reformulating the stability analysis as an optimization problem, leading to improved approximate conditions for Turing instability in multigraph networks. We validate our framework through numerical simulations, demonstrating its superior effectiveness, particularly in networks with Poisson degree distributions where prior methods fail. More than an analytical tool, this framework is leveraged to showcase how to design Laplacian spectra of a network family to drive Turing instability. Furthermore, we develop greedy algorithms for targeted modifications of edges to induce Turing instability. This work provides a versatile theoretical tool and opens new avenues for engineering pattern formation in multi-layer systems.