The paper examines the dynamic formation and stochastic evolution of networks where individuals' payoffs depend on the network structure. Individuals form and sever links based on the improvement the resulting network offers them relative to the current network, creating a sequence of networks called an 'improving path'. The paper analyzes the properties of improving paths, including the existence of pairwise stable networks and cycles. It introduces a stochastic evolutionary process where, in addition to intended changes, there is a small probability of unintended changes or errors. The evolutionary process selects from among statically stable networks and cycles, and the paper shows that in some cases, inefficient networks can be evolutionarily stable even though efficient ones are statically stable. The paper applies these results to matching problems, demonstrating that in certain contexts, evolutionarily stable networks coincide with core stable networks, achieving efficiency. The analysis is based on the concept of improving paths and the idea of resistance, which tracks how many mutations are needed to move from one network to another. The paper provides conditions for the absence of cycles and discusses the implications for network stability and efficiency.The paper examines the dynamic formation and stochastic evolution of networks where individuals' payoffs depend on the network structure. Individuals form and sever links based on the improvement the resulting network offers them relative to the current network, creating a sequence of networks called an 'improving path'. The paper analyzes the properties of improving paths, including the existence of pairwise stable networks and cycles. It introduces a stochastic evolutionary process where, in addition to intended changes, there is a small probability of unintended changes or errors. The evolutionary process selects from among statically stable networks and cycles, and the paper shows that in some cases, inefficient networks can be evolutionarily stable even though efficient ones are statically stable. The paper applies these results to matching problems, demonstrating that in certain contexts, evolutionarily stable networks coincide with core stable networks, achieving efficiency. The analysis is based on the concept of improving paths and the idea of resistance, which tracks how many mutations are needed to move from one network to another. The paper provides conditions for the absence of cycles and discusses the implications for network stability and efficiency.