  This paper aims at addressing distributed averaging problems for signed
networks in the presence of general directed topologies that are represented by
signed digraphs. A new class of improved Laplacian potential functions is
proposed by presenting two notions of any signed digraph: induced unsigned
digraph and mirror (undirected) signed graph, based on which two distributed
averaging protocols are designed using the nearest neighbor rules. It is shown
that with any of the designed protocols, signed-average consensus
(respectively, state stability) can be achieved if and only if the associated
signed digraph of signed network is structurally balanced (respectively,
unbalanced), regardless of whether weight balance is satisfied or not. Further,
improved Laplacian potential functions can be exploited to solve fixed-time
consensus problems of signed networks with directed topologies, in which a
nonlinear distributed protocol is proposed to ensure the bipartite consensus or
state stability within a fixed time. Additionally, the convergence analyses of
directed signed networks can be implemented with the Lyapunov stability
analysis method, which is realized by revealing the tight relationship between
convergence behaviors of directed signed networks and properties of improved
Laplacian potential functions. Illustrative examples are presented to
demonstrate the validity of our theoretical results for directed signed
networks.
