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.