A weak average consensus problem is investigated for a distributed multi-agent system with partial flowing nodes (MASPFN), which is composed of subsystem I and subsystem II: subsystem I consists of a fixed agent set, whereas subsystem II consists of a varying agent set in which some agents may join in or quit from the multi-agent system with partial flowing nodes randomly. The weak average consensus refers to the fact that states of all agents in subsystem I approach a common value under the influence of subsystem II. Based on the Lyapunov function and the iterative method, several consensus criteria are obtained. One numerical example shows the reliability of the proposed methods.