In this work, we place the problem of group leader election in a context "inside the network," meaning that participants in the election process are network switches/routers, rather than hosts. A robust solution to the problem, called the Network Leader Election (NLE) protocol, is proposed for use in networks based on link-state routing (LSR). The protocol is robust, for it achieves leadership consensus in the presence of adverse events, such as leader failures and network partitioning. The correctness of the protocol can be proved formally. A simulation study reveals that the NLE protocol incurs low overhead in handling leader failures and in group creation. In addition, it is shown how important network functions, including hierarchical routing, address resolution, and multicast core management, can benefit from the NLE protocol.