  Cycles are fundamental elements in graph-structured data and have
demonstrated their effectiveness in enhancing graph learning models. To encode
such information into a graph learning framework, prior works often extract a
summary quantity, ranging from the number of cycles to the more sophisticated
persistence diagram summaries. However, more detailed information, such as
which edges are encoded in a cycle, has not yet been used in graph neural
networks. In this paper, we make one step towards addressing this gap, and
propose a structure encoding module, called CycleNet, that encodes cycle
information via edge structure encoding in a permutation invariant manner. To
efficiently encode the space of all cycles, we start with a cycle basis (i.e.,
a minimal set of cycles generating the cycle space) which we compute via the
kernel of the 1-dimensional Hodge Laplacian of the input graph. To guarantee
the encoding is invariant w.r.t. the choice of cycle basis, we encode the cycle
information via the orthogonal projector of the cycle basis, which is inspired
by BasisNet proposed by Lim et al. We also develop a more efficient variant
which however requires that the input graph has a unique shortest cycle basis.
To demonstrate the effectiveness of the proposed module, we provide some
theoretical understandings of its expressive power. Moreover, we show via a
range of experiments that networks enhanced by our CycleNet module perform
better in various benchmarks compared to several existing SOTA models.
