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Mobile edge computing (MEC) deploys edge servers at the base station in the proximity of users to provide cloud computing-like computing and storage functionalities, which can achieve applications' low latency requirement at the network edge. The edge server network (ESN), constituted by edge servers in an area and the links between them, can host app vendors' services for serving nearby users. Many existing studies have demonstrated that a high ESN density allows for high service performance because edge servers can communicate and share resources with each other effectively over the ESN. However, in the real-world MEC environment, constructing a high-density ESN may incur high construction costs. The trade-off between construction cost and network density plays a vital role in the design of an ESN. Unfortunately, existing studies of MEC have commonly and simply assumed the densities of the ESNs in their experiments. In this paper, we make the first attempt to study the design of cost-effective ESNs with the aim to trade off between the network construction cost and the network density. We model this novel Edge Server Network Design (ESND) problem as a constrained optimization problem and prove its NP-hardness. ESND-O as an optimal approach is proposed based on integer programming to solve small-scale ESND problems. Another approximation approach named ESND-A is designed to solve large-scale ESND problems efficiently. We conduct extensive experiments to test the performance of ESND-O and ESND-A on a real-world dataset, and the experimental results demonstrate their effectiveness and efficiency against four representative approaches.