In this paper, we propose a flexible notion of characteristic functions defined on graph vertices to describe the distribution of vertex features at multiple scales. We introduce FEATHER, a computationally efficient algorithm to calculate a specific variant of these characteristic functions where the probability weights of the characteristic function are defined as the transition probabilities of random walks. We argue that features extracted by this procedure are useful for node level machine learning tasks. We discuss the pooling of these node representations, resulting in compact descriptors of graphs that can serve as features for graph classification algorithms. We analytically prove that FEATHER describes isomorphic graphs with the same representation and exhibits robustness to data corruption. Using the node feature characteristic functions we define parametric models where evaluation points of the functions are learned parameters of supervised classifiers. Experiments on real world large datasets show that our proposed algorithm creates high quality representations, performs transfer learning efficiently, exhibits robustness to hyperparameter changes, and scales linearly with the input size.