Pinning passivity and bipartite synchronization of fractional signed networks without gauge transformation

Recently, passivity of fractional complex networks has aroused much interest, but the concerned models contain only cooperative relationships and the competitive interaction among individuals is ignored. In this article, a class of fractional complex networks with a signed graph is considered and several conditions are derived to achieve the passivity of fractional signed networks by pinning strategies designed based on M-matrix theory. Particularly, in pinning adaptive control schemes, the pinning nodes are selected only according to the cooperative relationships among nodes and the control gains can regulate automatically to meet the actual demand. In addition, without translating the signed networks into corresponding unsigned networks based on the gauge transformation, some criteria of bipartite synchronization for fractional signed networks are obtained based on the feature of the signed topology and the derived passivity results. The theoretical results are eventually verified by several illustrate examples.