Geometric graph matching and similarity: a probabilistic approach

Abstract

Finding common structures is vital for many graph-based applications, such as road network analysis, pattern recognition, or drug discovery. Such a task is formalized as the inexact graph matching problem, which is known to be NP-hard. Several graph matching algorithms have been proposed to find approximate solutions. However, such algorithms still face many problems in terms of memory consumption, runtime, and tolerance to changes in graph structure or labels. In this paper, we propose a solution to the inexact graph matching problem for geometric graphs in 2D space. Geometric graphs provide a suitable modeling framework for applications like the above, where vertices are located in some 2D space. The main idea of our approach is to formalize the graph matching problem in a maximum likelihood estimation framework. Then, the expectation maximization technique is used to estimate the match between two graphs. We propose a novel density function that estimates the similarity between the vertices of different graphs. It is computed based on both 1) the spatial properties of a vertex and its direct neighbors, and 2) the shortest paths that connect a vertex to other vertices in a graph. To guarantee scalability, we propose to compute the density function based on the properties of sub-structures of the graph. Using representative geometric graphs from several application domains, we show that our approach outperforms existing graph matching algorithms in terms of matching quality, runtime, and memory consumption.