Kernel-based embeddings for large graphs with centrality constraints

Complex phenomena involving pairwise interactions in natural and man-made settings can be well-represented by networks. Besides statistical and computational analyses on such networks, visualization plays a crucial role towards effectively conveying 'at-a-glance' structural properties such as node hierarchy. However, most graph embedding algorithms developed for network visualization are ill-equipped to cope with the sheer volume of data generated by modern networks that encompass online social interactions, the Internet, or the world-wide web. Motivated by the emergence of nonlinear manifold learning approaches for dimensionality reduction, this paper puts forth a novel scheme for embedding graphs using kernel matrices defined on graphs. In particular, a kernelized version of local linear embedding is devised for computation of reconstruction weights. Unlike contemporary approaches, the developed embedding algorithm entails low-cost, parallelizable, and closed-form updates that can easily scale to big network data. Furthermore, it turns out that inclusion of embedding constraints to emphasize centrality structure can be accomplished at minimal extra computational cost. Experimental results on Watts-Strogatz small-world networks demonstrate the efficacy of the novel approach.