Tree-Like Structure in Large Social and Information Networks

Abstract

Although large social and information networks are often thought of as having hierarchical or tree-like structure, this assumption is rarely tested. We have performed a detailed empirical analysis of the tree-like properties of realistic informatics graphs using two very different notions of tree-likeness: Gromov's d-hyperbolicity, which is a notion from geometric group theory that measures how tree-like a graph is in terms of its metric structure, and tree decompositions, tools from structural graph theory which measure how tree-like a graph is in terms of its cut structure. Although realistic informatics graphs often do not have meaningful tree-like structure when viewed with respect to the simplest and most popular metrics, e.g., the value of d or the tree width, we conclude that many such graphs do have meaningful tree-like structure when viewed with respect to more refined metrics, e.g., a size-resolved notion of d or a closer analysis of the tree decompositions. We also show that, although these two rigorous notions of tree-likeness capture very different tree-like structures in worst-case, for realistic informatics graphs they empirically identify surprisingly similar structure. We interpret this tree-like structure in terms of the recently-characterized "nested core-periphery" property of large informatics graphs, and we show that the fast and scalable k-core heuristic can be used to identify this tree-like structure.