Abstract: We study the impact of the distance between two hubs on network coherence defined by Laplacian eigenvalues. Network coherence is a measure of the extent of consensus in a linear system with additive noise. To obtain an exact determination of coherence based on the distance, we choose a family of tree networks with two hubs controlled by two parameters. Using the tree’s regular structure, we obtain analytical expressions of the coherences with regard to network parameters and the network size. We then demonstrate that a shorter distance and a larger difference in the degrees of the two hubs lead to a higher coherence. With the same network size and distance, the best coherence occurs in the tree with the largest difference in the hub’s degrees. Finally, we establish a correlation between network coherence and average path length and find that they behave linearly.

Key words: Consensus; Coherence; Distance; Average path length

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