In this paper, a new mathematical framework is proposed for maximizing the self-cleaning capacity (SCC) of drinking water distribution systems by controlling the diurnal peak flow velocities in the pipes under normal operation. This is achieved through an optimal change of the network connectivity (topology). This paper proposes an efficient algorithm for the network analysis of valve closures, which allows enforcing favorable changes in the flow velocities for maximizing the SCC by determining an optimal set of links to isolate in the forming of a more branched network, while concurrently satisfying the hydraulic and regulatory pressure constraints at the demand nodes. Multiple stochastic demands from an end-use demand model are generated to test the robustness in the improved SCC for the modified network connectivity under changing demand. An operational network model is used to demonstrate the efficacy of the proposed approach.