Laplacian Flows in Complex-valued Directed Networks: Analysis, Design, and Consensus

Abstract: In the interdisciplinary field of network science, a complex-valued network, with edges assigned complex weights, provides a more nuanced representation of relationships by capturing both the magnitude and phase of interactions. Additionally, an important application of this setting arises in distribution power grids. Motivated by the richer framework, we study the necessary and sufficient conditions for achieving consensus in both strongly and weakly connected digraphs. The paper establishes that complex-valued Laplacian flows converge to consensus subject to an additional constraint termed as real dominance which relies on the phase angles of the edge weights. Our approach builds on the complex Perron-Frobenius properties to study the spectral properties of the Laplacian and its relation to graphical conditions. Finally, we propose modified flows that guarantee consensus even if the original network does not converge to consensus. Additionally, we explore diffusion in complex-valued networks as a dual process of consensus and simulate our results on synthetic and real-world networks.