We consider the problem of efficient network formation in a distributed fashion. Network formation is modeled as an evolutionary process, where agents can form and sever unidirectional links and derive direct and indirect benefits from these links. We formulate and analyze an evolutionary model in which each agent's choices depend on its own previous links and benefits, and link selections are subject to random perturbations. Agents reinforce the establishment of a link if it was beneficial in the past, and suppress it otherwise. We illustrate the flexibility of the model to incorporate various design criteria, including dynamic cost functions that reflect link establishment and maintenance, and distance-dependent benefit functions. We show that the evolutionary process assigns positive probability to the emergence of multiple stable configurations (i.e., strict Nash networks), which need not emerge under alternative processes such as best-reply dynamics. We analyze the specific case of so-called frictionless benefit flow, and show that a single agent can reinforce the emergence of an efficient network through an enhanced evolutionary process known as dynamic reinforcement.