Redefining the Graph Edit Distance

Francesc Serratosa

doi:10.1007/s42979-021-00792-5

Abstract

Graph edit distance has been used since 1983 to compare objects in machine learning when these objects are represented by attributed graphs instead of vectors. In these cases, the graph edit distance is usually applied to deduce a distance between attributed graphs. This distance is defined as the minimum amount of edit operations (deletion, insertion and substitution of nodes and edges) needed to transform a graph into another. Since now, it has been stated that the distance properties have to be applied [(1) non-negativity (2) symmetry (3) identity and (4) triangle inequality] to the involved edit operations in the process of computing the graph edit distance to make the graph edit distance a metric. In this paper, we show that there is no need to impose the triangle inequality in each edit operation. This is an important finding since in pattern recognition applications, the classification ratio usually maximizes in the edit operation combinations (deletion, insertion and substitution of nodes and edges) that the triangle inequality is not fulfilled.

Highlights

Several kinds of problems have been used to model attributed graphs for more than 4 decades [1,2,3]
Theorem 1 can be applied to strings and trees and we have demonstrated that the tree edit distance and the string edit distance do not need to fulfil the triangle inequality between their edit operations to be considered a metric
We realize that the classification ratio is maximized at the value of the insertion and deletion costs that are much smaller than half of the maximum value of the substitution cost, which is the point that the triangle inequality begins not to be fulfilled

Summary

Introduction

Several kinds of problems have been used to model attributed graphs for more than 4 decades [1,2,3]. In this case, if the function to compare attributed graphs is not a metric, the representative graph is not properly defined (the construction of the representative is carried out by several comparisons between the graphs in the set) and there is a quality reduction of the pattern recognition task. If the function to compare attributed graphs is not a metric, the representative graph is not properly defined (the construction of the representative is carried out by several comparisons between the graphs in the set) and there is a quality reduction of the pattern recognition task For this reason, in the previously commented examples, the graph edit distance parameters are set such as the graph edit distance becomes a metric. For a specific comparison of graph matching methods, we refer to [32]

Graph Edit Distance

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Practical Experiment

Discussion and Conclusion