Graph embedding on spheres and its application to visualization of information diffusion data

We address the problem of visualizing structure of undirected graphs that have a value associated with each node into a K-dimensional Euclidean space in such a way that 1) the length of the point vector in this space is equal to the value assigned to the node and 2) nodes that are connected are placed as close as possible to each other in the space and nodes not connected are placed as far apart as possible from each other. The problem is reduced to K-dimensional spherical embedding with a proper objective function. The existing spherical embedding method can handle only a bipartite graph and cannot be used for this purpose. The other graph embedding methods, e.g., multi-dimensional scaling, spring force embedding methods, etc., cannot handle the value constraint and thus are not applicable, either. We propose a very efficient algorithm based on a power iteration that employs the double-centering operations. We apply the method to visualize the information diffusion process over a social network by assigning the node activation time to the node value, and compare the results with the other visualization methods. The results applied to four real world networks indicate that the proposed method can visualize the diffusion dynamics which the other methods cannot and the role of important nodes, e.g. mediator, more naturally than the other methods.