Graph Neural Networks (GNNs) have shown considerable effectiveness in a variety of graph learning tasks, particularly those based on the message-passing approach in recent years. However, their performance is often constrained by a limited receptive field, a challenge that becomes more acute in the presence of sparse graphs. In light of the power series, which possesses infinite expansion capabilities, we propose a novel Graph Power Filter Neural Network (GPFN) that enhances node classification by employing a power series graph filter to augment the receptive field. Concretely, our GPFN designs a new way to build a graph filter with an infinite receptive field based on the convergence power series, which can be analyzed in the spectral and spatial domains. Besides, we theoretically prove that our GPFN is a general framework that can integrate any power series and capture long-range dependencies. Finally, experimental results on three datasets demonstrate the superiority of our GPFN over state-of-the-art baselines.

This article presents a design principle of a neural network using Gaussian activation functions, referred to as a Gaussian Potential Function Network (GPFN), and explores the capability of a GPFN in learning a continuous input-output mapping from a given set of teaching patterns. The design principle is highlighted by a Hierarchically Self-Organizing Learning (HSOL) algorithm featuring the automatic recruitment of hidden units under the paradigm of hierarchical learning. A GPFN generates an arbitrary shape of a potential field over the domain of the input space, as an input-output mapping, by synthesizing a number of Gaussian potential functions provided by individual hidden units referred to as Gaussian Potential Function Units (GPFUs). The construction of a GPFN is carried out by the HSOL algorithm which incrementally recruits the minimum necessary number of GPFUs based on the control of the effective radii of individual GPFUs, and trains the locations (mean vectors) and shapes (variances) of individual Gaussian potential functions, as well as their summation weights, based on the Backpropagation algorithm. Simulations were conducted for the demonstration and evaluation of the GPFNs constructed based on the HSOL algorithm for several sets of teaching patterns.