Convergence analysis of directed signed networks via an M-matrix approach

ABSTRACT

This paper aims at solving convergence problems on directed signed networks with multiple nodes, where interactions among nodes are described by signed digraphs. The convergence analysis is achieved by matrix-theoretic and graph-theoretic tools, in which M-matrices play a central role. The fundamental digon sign-symmetry assumption upon signed digraphs can be removed with the proposed analysis approach. Furthermore, necessary and sufficient conditions are established for semi-positive and positive stabilities of Laplacian matrices of signed digraphs, respectively. A benefit of this result is that given strong connectivity, a directed signed network can achieve bipartite consensus (or state stability) if and only if the signed digraph associated with it is structurally balanced (or unbalanced). If the interactions between nodes are described by a signed digraph only with spanning trees, a directed signed network can achieve interval bipartite consensus (or state stability) if and only if the signed digraph contains a structurally balanced (or unbalanced) rooted subgraph. Simulations are given to illustrate the developed results by considering signed networks associated with digon sign-unsymmetric signed digraphs.

KEYWORDS:

• Directed signed networks
• stability
• M-matrices
• structural (un)balance
• digon sign-(un)symmetry
• (interval) bipartite consensus

Acknowledgements

The author would like to thank the associate editor and the anonymous referees for their constructive criticisms and insightful suggestions.

Disclosure statement

No potential conflict of interest was reported by the authors

Notes

1. By state stability (or stability if no confusions occur) of signed networks, it means (hereafter) that the states of all agents asymptotically converge to zero. From the Lyapunov stability theory point of view, it is the asymptotic stability of linear systems conducted by the Laplacian matrices of signed graphs.

2. For the case when $\mathcal{G}$ has no rooted cycles, it has exactly one root, i.e. ${\mathcal{G}}_{\mathbf{r}}$ consists of one node. It is a trivial case that ${\mathcal{G}}_{\mathbf{r}}$ is structurally balanced and strongly connected.

Additional information

Funding

This work was supported in part by the National Natural Science Foundation of China (NSFC: [grant number 61473010], [grant number 61573031], [grant number 61327807); in part by the Beijing Natural Science Foundation of China [grant number 4162036]; in part by the Fundamental Research Funds for the Central Universities [grant number YWF-16-BJ-Y-27].