Maximizing the sum rate of a wireless network with multiple interfering links is an important and challenging problem in communication systems. This difficult non-convex problem has been approached from both an algorithmic perspective to achieve global optimality (e.g., using d.c. programming) and a relaxation perspective to obtain approximate solutions (e.g., high signal-to-noise-ratio approximation, binary power control, network symmetry, game-theoretic reformulation, etc.). It is generally agreed that 1) the global algorithms suffer from scalability issues and are more appropriate for problems of small instances; and 2) the solutions obtained based on maximizing the instantaneous performance are most likely suboptimal in practical fading wireless networks. In this work, we demonstrate that the sum rate can be efficiently optimized using the tool of stochastic geometry. In particular, we show that the average network sum rate can be derived in closed-form, taking into account both the random spatial distribution of the transmitters and the random Nakagami channel fading. An optimal contention density is further derived, which indicates the optimal number of supportable concurrent transmissions that attains the maximal sum rate. We discuss several applications of the derived results in interference-limited wireless systems.