The paper examines the dynamic formation and stochastic evolution of social and economic networks, focusing on how individuals form or sever links based on the improvement in their payoffs. It introduces the concept of an "improving path," a sequence of networks where each network differs by one link from the previous one, and each change benefits the involved individuals. The paper shows that improving paths can lead to pairwise stable networks or cycles, where networks are repeatedly visited. The study also considers a stochastic evolutionary process where small errors or mutations can occur, leading to deviations from improving paths. This process is modeled as a Markov chain, and the paper analyzes how the evolutionary process selects among statically stable networks and cycles. It demonstrates that even if efficient networks are statically stable, the evolutionary process may select inefficient ones due to the influence of mutations. The paper applies these results to matching problems, showing that evolutionarily stable networks can coincide with core stable networks, achieving efficiency. It also discusses the conditions under which cycles exist and the role of resistance in determining which networks are visited most frequently. The paper highlights the importance of pairwise stability and the conditions under which cycles or stable states emerge. It concludes that the evolutionary process can lead to stable states or cycles, depending on the network structure and the incentives of individuals. The analysis provides insights into the long-term behavior of network formation in dynamic settings.The paper examines the dynamic formation and stochastic evolution of social and economic networks, focusing on how individuals form or sever links based on the improvement in their payoffs. It introduces the concept of an "improving path," a sequence of networks where each network differs by one link from the previous one, and each change benefits the involved individuals. The paper shows that improving paths can lead to pairwise stable networks or cycles, where networks are repeatedly visited. The study also considers a stochastic evolutionary process where small errors or mutations can occur, leading to deviations from improving paths. This process is modeled as a Markov chain, and the paper analyzes how the evolutionary process selects among statically stable networks and cycles. It demonstrates that even if efficient networks are statically stable, the evolutionary process may select inefficient ones due to the influence of mutations. The paper applies these results to matching problems, showing that evolutionarily stable networks can coincide with core stable networks, achieving efficiency. It also discusses the conditions under which cycles exist and the role of resistance in determining which networks are visited most frequently. The paper highlights the importance of pairwise stability and the conditions under which cycles or stable states emerge. It concludes that the evolutionary process can lead to stable states or cycles, depending on the network structure and the incentives of individuals. The analysis provides insights into the long-term behavior of network formation in dynamic settings.