We study the weighted average consensus problem for a gossip network of agents with vector-valued states. For a given matrix-weighted graph, the gossip process is described by a sequence of pairs of adjacent agents communicating and updating their states based on the edge matrix weight. Our key contribution is providing conditions for the convergence of this non-homogeneous Markov process as well as the characterization of its limit set. To this end, we introduce the notion of "$w$-holonomy" of a set of stochastic matrices, which enables the characterization of sequences of gossiping pairs resulting in reaching a desired consensus in a decentralized manner. Stated otherwise, our result characterizes the limiting behavior of infinite products of (non-commuting, possibly with absorbing states) stochastic matrices.

翻译：我们研究了具有向量值状态的智能体广播网络中的加权平均一致性问题。针对给定的矩阵加权图，广播过程由相邻智能体对构成的序列描述，这些智能体基于边矩阵权重进行通信并更新其状态。本文的核心贡献在于给出了这一非齐次马尔可夫过程的收敛条件及其极限集的刻画。为此，我们引入了随机矩阵集合的"$w$-全纯性"概念，这使得刻画以去中心化方式达成期望一致性的广播对序列成为可能。换言之，我们的结果刻画了（可能包含吸收态的）非交换随机矩阵无限乘积的极限行为。