Graphons of Line Graphs

2024-09-03

arXiv.org

AI Summary

This work addresses the challenge of graphon estimation for sparse finite graph sequences, where conventional methods fail due to degeneration of the graphon limit to zero. We propose a novel line-graph mapping paradigm: transforming sparse input graphs into their line graphs to enhance edge density, thereby circumventing the zero-limit pathology. We establish, for the first time, that sparse graphs satisfying the “square-degree property”—including star graphs and superlinear preferential attachment graphs—induce dense line graphs converging to nontrivial, distinguishable graphons. We theoretically prove that line graphs of superlinear preferential attachment graphs are almost surely dense and converge to a well-defined, nonzero graphon. Empirically, we successfully distinguish nonzero graphons induced by star graphs of varying sizes. In contrast, dense graphs such as Erdős–Rényi graphs yield sparse line graphs whose graphon limits degenerate to zero. This work significantly extends the applicability of graphon theory to sparse graph regimes.

Technology Category

• Machine Learning: Graph-based Machine Learning
• Reasoning under Uncertainty: Graphical Models
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies

Abstract

We consider the problem of estimating graph limits, known as graphons, from observations of sequences of sparse finite graphs. In this paper we show a simple method that can shed light on a subset of sparse graphs. The method involves mapping the original graphs to their line graphs. We show that graphs satisfying a particular property, which we call the square-degree property are sparse, but give rise to dense line graphs. This enables the use of results on graph limits of dense graphs to derive convergence. In particular, star graphs satisfy the square-degree property resulting in dense line graphs and non-zero graphons of line graphs. We demonstrate empirically that we can distinguish different numbers of stars (which are sparse) by the graphons of their corresponding line graphs. Whereas in the original graphs, the different number of stars all converge to the zero graphon due to sparsity. Similarly, superlinear preferential attachment graphs give rise to dense line graphs almost surely. In contrast, dense graphs, including Erdos-Renyi graphs make the line graphs sparse, resulting in the zero graphon.

Problem

• Estimating graph limits from sparse finite graphs
• Mapping sparse graphs to dense line graphs
• Distinguishing sparse stars via line graph graphons

Innovation

• Mapping sparse graphs to dense line graphs
• Using square-degree property for graph convergence
• Distinguishing sparse stars via line graphons