We study the effect of a specific kind of labor market friction on aggregate unemployment. In our model, unemployed workers search for jobs through a network of firms, the labor flow network (LFN). The lack of an edge between two companies indicates the impossibility of labor flows between them. Thus, any network topology other than the complete graph will have give rise to labor market frictions. Application of the method of random walks on graphs yields analytical solutions for the equilibrium unemployment rate. To accomplish this it is convenient to introduce a new statistic, firm-specific unemployment, in order to derive aggregate unemployment as a function of the structure of LFNs. Using comprehensive employer-employee matched records for the universe of workers and firms in Finland, we characterize LFNs and their influence on aggregate unemployment. We show that, to first order, the structure of the LFN and the correlation between firms' hiring behaviors are responsible for more than half of the equilibrium unemployment rate and unemployment volatility. Our theory provides new micro-foundations for the aggregate matching function and other stylized facts of labor markets, such as the Beveridge curve, wage dispersion, and the employer-size premium. We also develop a new way to estimate the hiring policy of each firm, without the need for data on firm vacancies. Overall, the theory and methods described represent new ways for employer-employee matched records to be used for the study labor dynamics.