Characteristic Functions on Graphs: Birds of a Feather, from Statistical Descriptors to Parametric Models

In this paper, we propose a flexible notion of characteristic functions\ndefined on graph vertices to describe the distribution of vertex features at\nmultiple scales. We introduce FEATHER, a computationally efficient algorithm to\ncalculate a specific variant of these characteristic functions where the\nprobability weights of the characteristic function are defined as the\ntransition probabilities of random walks. We argue that features extracted by\nthis procedure are useful for node level machine learning tasks. We discuss the\npooling of these node representations, resulting in compact descriptors of\ngraphs that can serve as features for graph classification algorithms. We\nanalytically prove that FEATHER describes isomorphic graphs with the same\nrepresentation and exhibits robustness to data corruption. Using the node\nfeature characteristic functions we define parametric models where evaluation\npoints of the functions are learned parameters of supervised classifiers.\nExperiments on real world large datasets show that our proposed algorithm\ncreates high quality representations, performs transfer learning efficiently,\nexhibits robustness to hyperparameter changes, and scales linearly with the\ninput size.\n