Abstract:

This study investigates the consensus problem, a fundamental issue in distributed systems and network control. Consensus studies have traditionally focused on unweighted networks, overlooking the impact of edge weights in real-world networks. However, networks such as transportation systems, social networks, and power networks exhibit significant weighted properties, and unweighted models fail to fully capture their complex interactions. To address this issuse, this study examines a cluster of pseudo-fractal-weighted networks to determine how edge weights affect consensus. The Laplacian matrix is used to establish a relationship between the Kirchhoff indices and network consensus, providing an in-depth analysis of consensus behavior in weighted networks. Through the calculation of recursive relations for various indices across iterations, precise formulas for key quantities such as the multiplicative Kirchhoff index, additive Kirchhoff index, Kirchhoff index, and network coherence are derived. A numerical analysis shows that as the network size increases, consensus in weighted networks converges to a constant, indicating greater resistance to external noise.

Key words: weighted network, resistance distance, Kirchhoff index, consensus problem, pseudo-fractal network

摘要：

一致性问题是分布式系统和网络控制中的一个基本问题。传统上，关于一致性的研究主要集中于无权网络，忽略了网络中边的权重影响。然而，现实中的网络，如交通网络、社交网络和电力网络，往往具有显著的加权特性，单纯依赖无权网络模型无法充分刻画其中的复杂交互行为。为了将权重影响纳入网络分析，重点研究了一类伪分形加权网络簇，探讨了边的权重对网络一致性的影响。利用拉普拉斯矩阵构建基尔霍夫指标与网络一致性的关系，深入分析一致性问题在加权网络中的表现。通过计算在相邻迭代间不同指标的递推关系，推导出相关指标的精确计算公式，包括基尔霍夫指标、加法基尔霍夫指标、乘法基尔霍夫指标和网络一致性等重要量的计算公式。数值分析表明，随着网络规模的增长，加权网络中的一致性逐渐收敛为常数，网络能够更好地抵抗外部噪声的影响。

关键词: 加权网络, 电阻距离, 基尔霍夫指标, 一致性问题, 伪分形网络