Edit Distance From Graph Spectra

This paper is concerned with computing graph edit distance. One of the criticisms that can be leveled at existing methods for computing graph edit distance is that it lacks the formality and rigour of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that standard string edit distance techniques can be used. To do this we use graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We pose the problem of graph-matching as maximum a posteriori probability alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression for the edit costs. We compute the edit distance by finding the sequence of string edit operations which minimise the cost of the path traversing the edit lattice. The edit costs are defined in terms of the a posteriori probability of visiting a site on the lattice. We demonstrate the method with results on a data-set of Delaunay graphs.