In this paper, we use tools from network theory to trace the properties of the matching function to the structure of granular connections between applicants and firms. We link seemingly disparate parts of the literature and recover existing functional forms as special cases. Our overarching message is that structure counts. For rich structures, captured by non-random networks, the matching function depends on whole sets rather than just the sizes of the two sides of the market. For less rich―random network―structures it depends on the sizes of the two sides and a few structural parameters. Structures characterized by greater asymmetries reduce the matching function’s efficacy, while denser structures can have ambiguous effects on it. For the special case of the Erdos-Renyi network, we show that the way the network varies with the sizes of the two sides of the market determines if the matching function exhibits constant returns to scale, or even if it is of a specific functional form, such as CES.