Approximate Graph Edit Distance Guided by Bipartite Matching of Bags of Walks

Abstract

The definition of efficient similarity or dissimilarity measures between graphs is a key problem in structural pattern recognition. This problem is nicely addressed by the graph edit distance, which constitutes one of the most flexible graph dissimilarity measure in this field. Unfortunately, the computation of an exact graph edit distance is known to be exponential in the number of nodes. In the early beginning of this decade, an efficient heuristic based on a bipartite assignment algorithm has been proposed to find efficiently a suboptimal solution. This heuristic based on an optimal matching of nodes' neighborhood provides a good approximation of the exact edit distance for graphs with a large number of different labels and a high density. Unfortunately, this heuristic works poorly on unlabeled graphs or graphs with a poor diversity of neighborhoods. In this work we propose to extend this heuristic by considering a mapping of bags of walks centered on each node of both graphs.

Graph edit distance as a quadratic program

“…Table II shows results obtained by different state of the art methods to compute an estimation of the GED. These methods have been tested on four chemoinformatics datasets 1 , in the same way as in previous papers using these datasets [12], [13]. Each dataset is composed of molecular graphs.…”

“…• Bipartite GED based on the Hungarian algorithm adapted to the LSAPE [18], with the substructure attached to each node defined by: incident edges and adjacent nodes [9], [10], a bag of random walks up to 3 edges (line 2) [12], a subgraph up to a radius 3 (line 3) [13].…”

“…• Approximate GED formalized as a QAP (line 4) [17] and as a QAPE (line 5), both solved by the IPFP algorithm with the same initialization provided by [12], since initialization influences the approximation of the GED. Observe that these two last approaches obtain the best approximation results.…”

Approximating Graph Edit Distance Using GNCCP

“…The edit path γ corresponds to an optimal edit path. GED has been widely used by the structural pattern recognition community [15,4,11,12] despite the fact that such distance comes along with several drawbacks. First of all, computing the GED of two graphs requires to find a path having a minimal cost among all possible paths, which is a NP-complete problem [8].…”

“…In practice, due to its high complexity, the computation of an exact GED is intractable for graphs having more than 10 nodes [11,4]. Such a limitation restricts applications of GED on real datasets, hence motivating the graph community to focus on heuristics providing suboptimal solutions of Eq.…”