Network coherence in the web graphs

• Study network coherence in web graphs with infinite fractal dimension.
• Propose a method to calculate the network coherence.
• The network coherence does not depend on its fractal dimension.
• The scalings of web graphs are larger than those of other studied graphs.

Network coherence is used to characterize the consensus dynamics with additive stochastic disturbances and can be described by Laplacian spectrum. In this paper, we mainly obtain the scalings of network coherence in the web graphs with a special feature that its fractal dimension is infinite. We then investigate the relationship between the scalings and fractal dimension. Based on the structures of web graphs, we obtain the relationships for Laplacian matrix and Laplacian eigenvalues between web graphs and their corresponding equilateral polygons. We also obtain analytical expressions for the sum of the reciprocals and square reciprocals of all nonzero Laplacian eigenvalues. Finally we calculate first and second order coherence and see that the scalings of network coherence with network size N are N   and N3N3, which shows that the scalings are not related to the fractal dimension of web graphs. In addition, the scalings of network coherence in web graphs are larger than those performed on some fractal networks.