Network coherence on the weighted treelike network

The study of complex networks has gained much interest. In particular, network coherence is a current hot topic. In this paper, we construct the weighted treelike network with weight factor to be our model. Although some properties have been revealed in weighted treelike networks, studies on their network coherence are still less and remain a challenge. In order to enrich the research of weighted networks, the first-order network coherence is investigated in this paper. We investigates consensus dynamics in a linear dynamical system with additive stochastic disturbances, which is characterized as network coherence by the Laplacian spectrum. Based on the tree structures, we identify a recursive relationship of its eigenvalues at two successive generations of Laplacian matrix. We then compute and derive the exact solutions for the sum of reciprocals of all nonzero Laplacian eigenvalues. The obtained results show that the scalings of first-order coherence with network size obey three laws along with the range of the weight factor. All results in this paper can help us get deeper understanding about the effect of link weight on the properties and functions of networks.