The Coherence and Properties Analysis of Balanced $2^{p}$-Ary Tree Networks

This study aims to analyze the coherence and properties in generalized Bethe tree networks within a uniform noise distribution. We handle two distinct categories of consensus algorithms within the balanced
$2^{p}$
-ary trees, denoted as
$B_{k}^{p}$
, designed for both leaderless and leader-follower networks. First, we derive the precise expressions for two cases of consensus algorithms based on the regular network topology. Second, we investigate an in-depth analysis of network properties from three perspectives: (1) Kirchhoff index of balanced
$2^{p}$
-ary tree networks. (2) Global average first-passage time of balanced
$2^{p}$
-ary tree networks. (3) The robustness of network coherence. Finally, we further obtain exact solutions for the first two perspectives, gaining insights into the impacts of network parameters on consensus. We reveal: (1) Single-leader networks show better coherence than leaderless networks. (2) Different network parameters result in different consensus distribution patterns, indicating that the topology structures of complex networks significantly influence the robustness of network coherence.