We consider the power optimization problem of maximizing the sum rate of a symmetric network of interfering links in Gaussian noise. All transmitters have an average transmit power constraint, the same for all transmitters. This problem has application to DSL, as well as wireless networks. We solve this nonconvex problem by indentifying some underlying convex structure. In particular, we characterize the maximum sum rate of the network, and show that there are essentially two possible states at the optimal solution depending on the cross-gain (√ε) between the links, and/or the average power constraint: the first is a wideband (WB) state, in which all links interfere with each other, and the second is a frequency division multiplexing (FDM) state, in which all links operate in orthogonal frequency bands. The FDM state is optimal if the cross-gain between the links is above 1/√2. If √ε < 1/√2, then FDM is still optimal provided the SNR of the links is sufficiently high. With √ε < 1/√2, the WB state occurs when the SNR is low, but as we increase the SNR from low to high, there is a smooth transition from the WB state to the FDM state: For intermediate SNR values, the optimal configuration is a mixture, with some fraction of the bandwidth in the WB state, and the other fraction in the FDM state. We also consider an alternative formulation in which the power is mandated to be frequency flat. In this formulation, the optimal configuration is either all links at full power, or just one link at full power. In this setting, there is an abrupt phase transition between these two states.