Title: A Simple and General Graph Neural Network with Stochastic Message Passing. Abstract: Graph neural networks (GNNs) are emerging machine learning models on graphs. One key property behind the expressiveness of existing GNNs is that the learned node representations are permutation-equivariant. Though being a desirable property for certain tasks, however, permutation-equivariance prevents GNNs from being proximity-aware, i.e., preserving the walk-based proximities between pairs of nodes, which is another critical property for graph analytical tasks. On the other hand, some variants of GNNs are proposed to preserve node proximities, but they fail to maintain permutation-equivariance. How to empower GNNs to be proximity aware while maintaining permutation-equivariance remains an open problem. In this paper, we propose Stochastic Message Passing (SMP), a general and simple GNN to maintain both proximity-awareness and permutation-equivariance properties. Specifically, we augment the existing GNNs with stochastic node representations learned to preserve node proximities. Though seemingly simple, we prove that such a mechanism can enable GNNs to preserve node proximities in theory while maintaining permutation-equivariance with certain parametrization. Extensive experimental results demonstrate the effectiveness and efficiency of SMP for tasks including node classification and link prediction.