The Mutual Information between Graphs

The estimation of mutual information between graphs has been an elusive problem until the formulation of graph matching in terms of manifold alignment. Then, graphs are mapped to multi-dimensional sets of points through structural preserving embeddings. Point-wise alignment algorithms can be exploited in this context to re-cast graph matching in terms of point matching. Unfortunately, the potentially high dimensionality of the point-sets points encompass the development of mutual information means that bypass entropy estimation. These methods must be deployed to render the estimation of mutual information computationally tractable. In this paper the novel contribution is to show how manifold alignment can be combined with copula-based entropy estimators to efficiently estimate the mutual information between graphs. We compare the empirical copula with an Archimedean copula (the independent one) in terms of retrieval/recall after graph comparison. Our experiments show that mutual information built in both choices improves significantly state-of-the art divergences.