The deformed consensus protocol

This paper studies a generalization of the standard continuous-time consensus protocol, obtained by replacing the Laplacian matrix of the communication graph with the so-called deformed Laplacian  . The deformed Laplacian is a second-degree matrix polynomial in the real variable ss which reduces to the standard Laplacian for ss equal to unity. The stability properties of the ensuing deformed consensus protocol   are studied in terms of parameter ss for some special families of undirected and directed graphs, and for arbitrary graph topologies by leveraging the spectral theory of quadratic eigenvalue problems. Examples and simulation results are provided to illustrate our theoretical findings.