Consider a sequential process where agents have individual values at every possible step. A planner is in charge of selecting steps and distributing the accumulated aggregate values among a number of agents. We model this process by a directed network, whereby each edge is associated with a vector of individual values. This model applies to several new and existing problems, e.g. developing a connected public facility and distributing total values received by surrounding districts, selecting a long-term production project and sharing final profits among partners of a firm, or choosing a machine schedule to serve different tasks and distributing total benefits among task owners. Herein, we provide the first axiomatic study on path selection and value-sharing in networks. We consider four sets of axioms from different perspectives, including those related to (1) the sequential consistency of value-sharing; (2) the monotonicity of value-sharing with respect to technology improvements; (3) the independence of value-sharing with respect to certain network transformations; and (4) the robust implementation of the efficient path selection when the planner has no information about network configuration. Surprisingly, these four disparate sets of axioms characterize similar classes of solutions, namely selecting an efficient path(s) and assigning to each agent a share of total values that is independent of individual values. Furthermore, we characterize more general solutions that depend on individual values.