Recent advancements in bipartite consensus, a scenario where agents are divided into two disjoint sets with agents in the same set agreeing on a certain value and those in different sets agreeing on opposite or specifically related values, have highlighted its potential applications across various fields. Traditional research typically relies on the presence of a positive-negative spanning tree, which limits the practical applicability of bipartite consensus. This study relaxes that assumption by allowing for weak connectivity within the network, where paths can be weighted by semidefinite matrices. By exploring the algebraic constraints imposed by positive-negative trees and semidefinite paths, we derive sufficient conditions for achieving bipartite consensus. Our theoretical findings are validated through numerical results.

翻译：近年来，二分一致性研究取得了显著进展。在该场景中，智能体被划分为两个互不相交的集合，同一集合内的智能体对某个值达成一致，而不同集合间的智能体则对相反值或特定关联值达成一致。这种特性凸显了其在多个领域的潜在应用价值。传统研究通常依赖于正负生成树的存在，这限制了二分一致性的实际应用范围。本研究通过放宽该假设，允许网络中存在弱连通性，即路径可由半正定矩阵加权。通过分析正负树与半正定路径所施加的代数约束，我们推导出实现二分一致性的充分条件。数值实验结果验证了本研究的理论发现。