This paper proposes a new simulation algorithm for analyzing large power distribution networks, modeled as linear RLC circuits, based on a novel partial random walk concept. The random walk simulation method has been shown to be an efficient way to solve for voltages of small number of nodes in a large power distribution network, but the algorithm becomes expensive to solve for voltages of nodes that are more than a few with high accuracy. In this paper, we combine direct methods like LU factorization with the random walk concept to solve power distribution networks when voltage waveforms from a large number of nodes are required. We extend the random walk algorithm to deal with general RLC networks and show that Norton companion models for capacitors and self-inductors are more amenable for transient analysis by using random walks than Thevenin companion models. We also show that by nodal analysis (NA) formulation for all the voltage sources, LU-based direct simulations of subcircuits can be speeded up. Experimental results demonstrate that the resulting algorithm, called partial random walk (PRW), has significant advantages over the existing random walk method especially when the VDD/GND nodes are sparse and accuracy requirement is high.