Structural Similarity: Spectral Methods for Relaxed Blockmodeling

In this paper we propose the concept of structural similarity as a relaxation of blockmodeling in social network analysis. Most previous approaches attempt to relax the constraints on partitions, for instance, that of being a structural or regular equivalence to being approximately structural or regular, respectively. In contrast, our approach is to relax the partitions themselves: structural similarities yield similarity values instead of equivalence or non-equivalence of actors, while strictly obeying the requirement made for exact regular equivalences. Structural similarities are based on a vector space interpretation and yield efficient spectral methods that, in a more restrictive manner, have been successfully applied to difficult combinatorial problems such as graph coloring. While traditional blockmodeling approaches have to rely on local search heuristics, our framework yields algorithms that are provably optimal for specific data-generation models. Furthermore, the stability of structural similarities can be well characterized making them suitable for the analysis of noisy or dynamically changing network data.