Du Mingjun, Ma Baoli, Meng Deyuan
IEEE Trans Cybern. 2019 Nov;49(11):4029-4041. doi: 10.1109/TCYB.2018.2857854. Epub 2018 Sep 10.
This paper focuses on characterizing edge dynamics of signed networks subject to both cooperative and antagonistic interactions and copes with the state convergence problems of the resulting edge systems. To represent the two competitive classes of interactions that emerge in signed networks, signed digraphs are adopted and the relevant edge Laplacian matrices are introduced, with which an edge-based distributed protocol is presented. The relation between the edge Laplacian matrix and the structural balance (or unbalance) of a signed digraph is disclosed by taking advantage of properties of undirected cycles. Further, it is shown that for a signed network, the state of its edge system converges to a constant vector, regardless of whether its associated signed digraph is structurally balanced or unbalanced. This result does not need to impose the assumption upon the digon sign-symmetry of the signed digraph that is generally required by the node-based distributed protocols. In particular, the state convergence results of edges can be exploited to handle traditional bipartite consensus problems for the nodes of signed networks. Simulation examples are given to illustrate the effectiveness of the edge-based analysis method proposed for signed networks.
本文着重刻画受合作与对抗交互作用影响的带符号网络的边动态特性，并解决由此产生的边系统的状态收敛问题。为了表示带符号网络中出现的两类竞争性交互作用，采用了带符号有向图并引入了相关的边拉普拉斯矩阵，据此提出了一种基于边的分布式协议。利用无向环的性质揭示了边拉普拉斯矩阵与带符号有向图的结构平衡（或不平衡）之间的关系。此外，研究表明，对于一个带符号网络，其边系统的状态会收敛到一个常向量，无论其相关的带符号有向图在结构上是平衡还是不平衡。该结果无需对带符号有向图的双节点符号对称性施加基于节点的分布式协议通常所需的假设。特别地，边的状态收敛结果可用于处理带符号网络节点的传统二分共识问题。给出了仿真示例以说明所提出的针对带符号网络的基于边的分析方法的有效性。
