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We address the problem of selecting a topological design for a network having a single traffic source and uncertain demand at the remaining nodes. Solving the associated fixed charge network flow (FCF) problem requires finding a network design that limits both the fixed costs of establishing links and the variable costs of sending flow to the destinations. In this paper, we discuss how to obtain a sequence of optimal solutions that arise as the demand intensity varies from low levels to high. One of the network design alternatives associated with these solutions will be chosen based upon the dominant selection criteria of the decision maker. We consider both probabilistic and non-probabilistic criteria and compare the network designs associated with each. We show that the entire sequence of optimal solutions can be identified with little more effort than solving a single FCF problem instance. We also provide solution approaches that are relatively efficient and suggest good design alternatives based upon approximations to the optimal sequence.