Graph edit distance from spectral seriation

Abstract

Abstract-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 they lack some of the formality and rigor of the computation of string edit distance. Hence, our aim is to convert graphs to string sequences so that string matching techniques can be used. To do this, we use a graph spectral seriation method to convert the adjacency matrix into a string or sequence order. We show how the serial ordering can be established using the leading eigenvector of the graph adjacency matrix. We pose the problem of graph-matching as a maximum a posteriori probability (MAP) alignment of the seriation sequences for pairs of graphs. This treatment leads to an expression in which the edit cost is the negative logarithm of the a posteriori sequence alignment probability. We compute the edit distance by finding the sequence of string edit operations which minimizes the cost of the path traversing the edit lattice. The edit costs are determined by the components of the leading eigenvectors of the adjacency matrix and by the edge densities of the graphs being matched. We demonstrate the utility of the edit distance on a number of graph clustering problems.

Graph Matching Based on Node Signatures

“…In addition, the results of Dunn index and the Rand index show that the clustering using our method obtains a better separation of the graphs into compact clusters. The time consumed by our method is 39.14% less than the Umeyama one (see Table. Secondly, we have compared our method with the GED from spectral seriation [2], the graph histograms [14] and the graph probing [12]. The experiments consist on applying the previous tests (MDS and MST) on a database derived from COIL-100 [20] which contains different views of 3D objects.…”

“…2, we can note that contrary to our method the first two classes are merged for the three methods (spectral seriation, graph histograms and graph probing). Each of these approaches uses a global description to represent graphs: the probing [12] and the graph histograms [14] methods represent each graph with only one vector, and the spectral seriation method [2] uses a string representation for graphs. Therefore, these global descriptions can not distinct differences when the graphs share similar global characteristics but not local.…”

“…Equivalently, Lopresti [12] gives an equivalence test procedure that allows to quantify the similarity between graphs. Other methods based on spectral approaches [2,3,16], give an elegant matrix representation for graphs that ensure an approximate solutions for graphs matching in polynomial time. Among the pioneering works related to graph matching using the spectral techniques we quote the paper of Umeyama [3], in which the Weighted Graph Isomorphism Problem is addressed by an eigendecomposition.…”

“…The proposed algorithm is validated within images clustering and content-based image retrieval applications. We have compared our results with the Umeyama method [3], the graph edit distance from spectral seriation [2], the graph histograms distance [14], and the graph probing technique [12] (section 3). Finally, in section 4, some conclusions are drawn.…”

Attributed Graph Matching Using Local Descriptions

“…Then, the graph is handled as one entity which can be only one vector [6], a matrix [7] or a string [15]. In few previous work, the concept of node signature has been introduced in [20,19,17], here the node signatures have been computed by making use of spectral approach, decomposition approach and random walks approach.…”