Motivated by the structure of social networks, this paper initiates a study of distributed algorithms in networks that exhibit a core-periphery structure. Such networks contain two distinct groups of nodes: a large and sparse, group identified as the periphery, which is loosely organized around a small, and densely connected group identified as the core. We identify four basic properties that are relevant to the interplay between core and periphery. For each of these properties, we propose a corresponding axiom that captures the behavior expected of a social network based on a core-periphery structure. We then address their usefulness for distributed computation, by considering a nontrivial algorithmic task of significance in both the distributed systems world and the social networks world, namely, the distributed construction of a minimum-weight spanning tree.