A Spectral Theory of Distortion in LLM Graph Reconstruction: Sharp Bounds and Empirical Characterization

Abstract

Evaluations of graph reconstruction by language models typically report a single aggregate distance between the original and the reconstructed graph. We prove that for the Wasserstein distance between Laplacian spectra such a summary is bracketed by two edge counts, the net change in edge number from below and the symmetric difference from above, each scaled by $2/n$ where $n$ is the number of vertices. The bracket is sharp: its two ends coincide exactly when the reconstruction only adds edges or only deletes them, and on that class the distance is a rescaled edge count that says nothing about which edges changed. When the ends differ, the residual between the distance and the lower end is positive only if the reconstruction both invented and lost edges, which turns it into a certificate of mixed editing computable from the reported summaries alone. We characterize these regimes in 135 reconstructions produced by three open-weight models over 45 synthetic graphs. Seventy-seven outputs are one-sided and 29 mixed outputs have $X>0$, including cases where edge count is exactly preserved while nineteen edges were simultaneously invented and lost. The three models differ in editing policy, ranging from copying the input to attempting completion at the cost of large hallucination volume, a distinction that aggregate distortion does not reveal.