# Graph Comparison via the Non-backtracking Spectrum

## TL;DR

This paper introduces a new graph comparison metric based on the non-backtracking spectrum, enabling reliable comparison of graphs of different sizes and types, with applications in classification tasks.

## Contribution

It proposes a novel spectral metric using the non-backtracking operator, addressing limitations of existing methods in graph comparison and scalability.

## Key findings

- The metric effectively compares graphs of different sizes.
- Watts-Strogatz graphs form a manifold in the spectral embedding.
- The metric improves graph classification performance.

## Abstract

The comparison of graphs is a vitally important, yet difficult task which arises across a number of diverse research areas including biological and social networks. There have been a number of approaches to define graph distance however often these are not metrics (rendering standard data-mining techniques infeasible), or are computationally infeasible for large graphs. In this work we define a new metric based on the spectrum of the non-backtracking graph operator and show that it can not only be used to compare graphs generated through different mechanisms, but can reliably compare graphs of varying size. We observe that the family of Watts-Strogatz graphs lie on a manifold in the non-backtracking spectral embedding and show how this metric can be used in a standard classification problem of empirical graphs.