LGC: Long-range Modeling on Molecular Graphs with Legendre-basis

Molecular graphs have emerged as a focal point of research interest in recent years. Polymers such as polyamides and proteins displaying various sizes, structures and functions constitute as suitable set of data to test the long-range modeling capabilities of graph learning models. Recent literature points to spectral graph neural networks as a potential candidate to tackle the limitations showcased by message-passing frameworks and graph transformer models. These networks bridge scalability and long-range modeling capacity. The majority of current research has focused heavily on Chebyshev, Bernstein, and monomial polynomial basis functions. In this work, we investigate the Legendre polynomial, an underexplored basis function, to provide insights into its ability to model non-local information. It is observed that off-the-shelf Legendre polynomial based spectral GNNs displays comparable performance to the current state-of-the-art models. We conduct layer-wise Jacobian analysis of Legendre polynomial to show unstable signal propagation at higher-orders. To address this limitation, we reform the standard Legendre propagation with Cayley transformation to achieve a stable variation, named Legendre Graph Convolution (LGC). Across several synthetic and real-world benchmarks the proposed LGC model portrays performance in line with the state-of-the-art models.