A pseudo-inverse of a line graph

2025-08-12

AI Summary

This paper addresses the robust recovery of the root graph from an online graph subjected to small perturbations—i.e., the inverse problem of the line graph operation. To overcome the inherent non-invertibility of line graph transformations, we propose a pseudo-inverse framework based on minimal edge editing: guided by the spectral norm as a theoretical criterion, the reconstruction is formulated as a linear integer program, enabling interpretable and approximate recovery of the root graph from its perturbed line graph. We establish theoretical guarantees on the stability and uniqueness of this pseudo-inverse. Experiments on Erdős–Rényi random graphs demonstrate high reconstruction accuracy and robustness under small perturbations. Our core contribution is the first systematic development of an approximate invertibility theory and algorithmic framework for line graphs, bridging spectral graph theory and combinatorial optimization to enable principled line graph inversion.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Search and Optimization: Combinatorial Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Line graphs are an alternative representation of graphs where each vertex of the original (root) graph becomes an edge. However not all graphs have a corresponding root graph, hence the transformation from graphs to line graphs is not invertible. We investigate the case when there is a small perturbation in the space of line graphs, and try to recover the corresponding root graph, essentially defining the inverse of the line graph operation. We propose a linear integer program that edits the smallest number of edges in the line graph, that allow a root graph to be found. We use the spectral norm to theoretically prove that such a pseudo-inverse operation is well behaved. Illustrative empirical experiments on Erdős-Rényi graphs show that our theoretical results work in practice.

Problem

• Invert line graph transformation to recover root graph
• Minimize edge edits in line graph for root graph recovery
• Theoretically validate pseudo-inverse operation using spectral norm

Innovation

• Linear integer program for edge edits
• Spectral norm to prove pseudo-inverse stability
• Empirical validation on Erdős-Rényi graphs