Spectral Convergence of Sparse Bipartite Graphs via the Line-Graphon
Pure point spectral convergence is generally unavailable for sparse graph sequences, where Laplacian eigenvalues may remain unstable under graph growth. This poses challenges for analyzing spectral operators underlying many graph-based learning methods, including graph neural networks. We identify a class of sparse bipartite graph sequences for which eigenvalue-level spectral convergence can be recovered. Focusing on graphs with strong degree heterogeneity, we exploit the convergence properties of their line-graph sequences. Using a spectral correspondence between bipartite graph Laplacians and line-graph adjacency operators, we transfer convergence guarantees from dense line-graphs back to the original sparse graphs. This framework yields eigenvalue-level spectral convergence for a class of deterministic sparse graph sequences. We further analyze superlinear preferential attachment trees, establishing convergence to an infinite-dimensional limit operator. Numerical experiments empirically support our spectral convergence results.