Complex real-world phenomena are often modeled as dynamical systems on networks. In many cases of interest, the spectrum of the underlying graph Laplacian sets the system stability and ultimately shapes the matter or information flow. This motivates devising suitable strategies, with rigorous mathematical foundation, to generate Laplacians that possess prescribed spectra. In this paper, we show that a weighted Laplacian can be constructed so as to exactly realize a desired complex spectrum. The method configures as a non trivial generalization of existing recipes which assume the spectra to be real. Applications of the proposed technique to (i) a network of Stuart–Landau oscillators and (ii) to the Kuramoto model are discussed. Synchronization can be enforced by assuming a properly engineered, signed and weighted, adjacency matrix to rule the pattern of pairing interactions.

复杂的现实世界现象通常被建模为网络上的动态系统。在许多感兴趣的情况下，基础图拉普拉斯算子的频谱设置了系统稳定性，并最终塑造了事物或信息流。这激发了在严格的数学基础上设计合适的策略来生成具有规定光谱的拉普拉斯算子的能力。在本文中，我们表明可以构造加权拉普拉斯算子以精确实现所需的复数光谱。该方法配置为现有配方的非平凡概括，这些配方假定光谱是真实的。讨论了所提出的技术在（i）Stuart-Landau振荡器网络和（ii）在Kuramoto模型中的应用。可以通过假设经过适当设计，签名和加权的邻接矩阵来控制配对交互的方式来实施同步。