# Product Graph-based Higher Order Contextual Similarities for Inexact Subgraph Matching

## TL;DR

This paper introduces a novel approach to inexact subgraph matching by leveraging higher-order contextual similarities computed via tensor product graphs, resulting in more reliable and discriminative matching solutions.

## Contribution

It proposes a new method using tensor product graphs to incorporate higher-order contextual information into subgraph matching, improving accuracy over traditional pairwise methods.

## Key findings

- Enhanced discriminative power of node and edge similarities.
- Improved subgraph matching accuracy on synthetic and real benchmarks.
- Higher order contextual similarities lead to better approximate solutions.

## Abstract

Many algorithms formulate graph matching as an optimization of an objective function of pairwise quantification of nodes and edges of two graphs to be matched. Pairwise measurements usually consider local attributes but disregard contextual information involved in graph structures. We address this issue by proposing contextual similarities between pairs of nodes. This is done by considering the tensor product graph (TPG) of two graphs to be matched, where each node is an ordered pair of nodes of the operand graphs. Contextual similarities between a pair of nodes are computed by accumulating weighted walks (normalized pairwise similarities) terminating at the corresponding paired node in TPG. Once the contextual similarities are obtained, we formulate subgraph matching as a node and edge selection problem in TPG. We use contextual similarities to construct an objective function and optimize it with a linear programming approach. Since random walk formulation through TPG takes into account higher order information, it is not a surprise that we obtain more reliable similarities and better discrimination among the nodes and edges. Experimental results shown on synthetic as well as real benchmarks illustrate that higher order contextual similarities add discriminating power and allow one to find approximate solutions to the subgraph matching problem.