  This paper studies deterministic consensus networks with discrete-time
dynamics under persistent flows and non-reciprocal agent interactions. An arc
describing the interaction strength between two agents is said to be persistent
if its weight function has an infinite $l_1$ norm. We discuss two balance
conditions on the interactions between agents which generalize the arc-balance
and cut-balance conditions in the literature respectively. The proposed
conditions require that such a balance should be satisfied over each time
window of a fixed length instead of at each time instant. We prove that in both
cases global consensus is reached if and only if the persistent graph, which
consists of all the persistent arcs, contains a directed spanning tree. The
convergence rates of the system to consensus are also provided in terms of the
interactions between agents having taken place. The results are obtained under
a weak condition without assuming the existence of a positive lower bound of
all the nonzero weights of arcs and are compared with the existing results.
Illustrative examples are provided to show the critical importance of the
nontrivial lower boundedness of the self-confidence of the agents.
