We investigate the network topology convergence in a peer-to-peer (P2P) network system, where the goal of the system is to maximize live-streaming performance. The P2P system constructs a gradient overlay topology, characterized by a directed graph, where each node prefers neighbors containing higher utility values such that paths of increasing utilities emerge in the network topology. The gradient overlay network is built using gossiping and a preference function that samples nodes from a uniform random peer sampling service. Conditions for convergence to a gradient topology is derived, including the expected convergence time, and a threshold on the churn rate is provided for a gradient topology to emerge. Finally, a live-streaming video distribution experiment illustrates the benefits of constructing and utilizing the gradient topology for information dissemination in P2P systems.