Abstract
We study the of a family of recursive trees with novel features that include the initial states controlled by a parameter. The problem in a linear system with additive noises is characterized as , which is defined by a Laplacian spectrum. Based on the structures of our recursive treelike model, we obtain the recursive relationships for Laplacian eigenvalues in two successive steps and further derive the exact solutions of first- and second-order coherences, which are calculated by the sum and square sum of the reciprocal of all nonzero Laplacian eigenvalues. For a large network size , the scalings of the first- and second-order coherences are ln and $, respectively. The smaller the number of initial nodes, the better the bears. Finally, we numerically investigate the relationship between and , showing that the first- and second-order coherences increase with the increase of at approximately exponential and linear rates, respectively.
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