We study a non-cooperative model of unilateral network formation. Derks et al. [2008b] prove the existence of local-Nash and global-Nash networks for games with payoff functions that satisfy a framework of axiomatic properties. These properties are inspired by the one-way flow model, which is characterized by the following payoff structure. Each agent pays a cost for each formed link and receives profits from being connect to other agents. In this paper we fully characterize the payoff functions in the one-way flow model that satisfy all properties. We show that under certain conditions, payoff functions with heterogeneous link costs and heterogeneous profits satisfy all properties for which the existence of local-Nash networks is proved. Furthermore, we show that all payoff functions with owner-homogeneous link costs and heterogeneous profits satisfy all properties, and therefore imply the existence of global-Nash networks.