Property analysis and coherence dynamics for tree-symmetric networks with noise disturbance.

This article focuses on analyzing the leaderless and leader–follower coherence of tree-symmetric networks, which are networks structured with a central vertex connected to multiple subgraphs. It derives exact analytical expressions for coherence based on the Laplacian eigenvalues of these networks and compares the effects of leader presence and network parameters on coherence in noisy environments. The study finds that coherence increases with the number of vertices in each subgraph before stabilizing, while leader–follower coherence is independent of the number of subgraphs but sensitive to vertex count. Additionally, the paper applies the Laplacian spectrum to calculate related network properties such as the Kirchhoff index, global mean-first passage time, average path length, and the number of spanning trees, highlighting implications for distributed systems and blockchain technology.