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Laplacian spectrum of a family of recursive trees and its applications in network coherence

Many of the topological and dynamical properties of a network are related to its Laplacian spectrum; these properties include network diameter, Kirchhoff index, and mean first-passage time. This paper investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. We choose a family of uniform recursive trees as our model, and propose a method to calculate the first- and second-order network coherence. Using the tree structures, we identify a relationship between the Laplacian matrix and Laplacian eigenvalues. We then derive the exact solutions for the reciprocals and square reciprocals of all nonzero Laplacian eigenvalues. We also obtain the scalings of network coherence with network size. The scalings of network coherence of the studied trees are smaller than those of Vicsek fractals and are not related to its fractal dimension.