Laplacian spectrum constrains collective performance enhancement.

Abstract

In the network model, the interactions between nodes represent the transmission of information, energy, or substance. Strengthening interactions to enhance collective dynamical performance has garnered extensive interest. However, the specific degree to which collective performance is enhanced is a critical issue that has yet to be clarified. This work reveals that, although interactions between individuals are gradually strengthening, the enhancement of collective dynamical performance has a ceiling, referred to as dynamical potential, which coincidentally relies on the network Laplacian eigen-spectrum. Different interaction groupings correspond to different dynamical potentials, which determine diverse enhancement degrees. Additionally, two fascinating phenomena-potential preservation and potential jump-are observed in a network. It inspires a further step in network control and optimization, for both conventional node-pair interactions and higher-order interactions.