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We study a discrete-time, infinite-horizon process of partition formation. We introduce two cost functions for maintaining connections within a network, allowing the cost of a link between two players to depend on the composition of each player's neighborhood. Stable partitions are characterized when players interact on a star network, a complete network, and on balanced complete bipartite networks. Finally, we apply the model to Zachary's karate club network, providing an explanation for why two specific individuals can be viewed as group leaders. Numerical simulations on random networks further illustrate the process and underscore the theoretical intractability of the problem. (With Elena Parilina)