Node similarity is a fundamental problem in graph analytics. However, node similarity between nodes in different graphs (inter-graph nodes) has not received a lot of attention yet. The inter-graph node similarity is important in learning a new graph based on the knowledge of an existing graph (transfer learning on graphs) and has applications in biological, communication, and social networks. In this paper, we propose a novel distance function for measuring inter-graph node similarity with edit distance, called NED. In NED, two nodes are compared according to their local neighborhood structures which are represented as unordered k-adjacent trees, without relying on labels or other assumptions. Since the computation problem of tree edit distance on unordered trees is NP-Complete, we propose a modified tree edit distance, called TED*, for comparing neighborhood trees. TED* is a metric distance, as the original tree edit distance, but more importantly, TED* is polynomially computable. As a metric distance, NED admits efficient indexing, provides interpretable results, and shows to perform better than existing approaches on a number of data analysis tasks, including graph de-anonymization. Finally, the efficiency and effectiveness of NED are empirically demonstrated using real-world graphs.
Another node similarity for inter-graph nodes based only on the network structure is proposed by @cite_11 which is called HITS-based similarity. In HITS-based similarity, all pairs of nodes between two graphs are virtually connected. The similarity between a pair of inter-graph nodes is calculated using the following similarity matrix: where @math and @math are the adjacency matrices of the two graphs and @math is the similarity matrix in the @math iteration.
@cite_11
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We introduce a concept of similarity between vertices of directed graphs. Let GA and GB be two directed graphs with, respectively, nA and nB vertices. We define an nB nA similarity matrix S whose real entry sij expresses how similar vertex j (in GA) is to vertex i (in GB): we say that sij is their similarity score. The similarity matrix can be obtained as the limit of the normalized even iterates of Sk+1 = BSkAT + BTSkA, where A and B are adjacency matrices of the graphs and S0 is a matrix whose entries are all equal to 1. In the special case where GA = GB = G, the matrix S is square and the score sij is the similarity score between the vertices i and j of G. We point out that Kleinberg's "hub and authority" method to identify web-pages relevant to a given query can be viewed as a special case of our definition in the case where one of the graphs has two vertices and a unique directed edge between them. In analogy to Kleinberg, we show that our similarity scores are given by the components of a dominant eigenvector of a nonnegative matrix. Potential applications of our similarity concept are numerous. We illustrate an application for the automatic extraction of synonyms in a monolingual dictionary.