Coherence analysis and Laplacian energy of recursive trees with controlled initial states

We study the consensus of a family of recursive trees with novel features that include the initial states controlled by a parameter. The consensus problem in a linear system with additive noises is characterized as network coherence, 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 N , the scalings of the first- and second-order coherences are ln N and N , respectively. The smaller the number of initial nodes, the better the consensus bears. Finally, we numerically investigate the relationship between network coherence and Laplacian energy, showing that the first- and second-order coherences increase with the increase of Laplacian energy at approximately exponential and linear rates, respectively.