Graph partitions and the controllability of directed signed networks

This paper studies the controllability problem of signed networks which is presented by weighted and directed signed graphs. Graph partitions such as structural balance and almost equitable partitions (AEPs) are studied. We generalize the definition of AEPs to any graphs, directed or undirected, signed or unsigned, with or without edge weights. Based on AEP theory, a graph-theoretic necessary condition is proposed for the controllability of directed signed networks and an algorithm is given for the computation of the coarsest partition. Besides, the upper bound on the controllable subspace is derived when the system is uncontrollable.