The very small world of the well-connected

Online networks occupy an increasingly larger position in how we acquire information, how we communicate with one another, and how we disseminate information. Frequently, small sets of vertices dominate various graph and statistical properties of these networks and, because of this, they are relevant for structural analysis and efficient algorithms and engineering. For the web overall, and specifically for social linking in blogs and instant messaging, we provide a principled, rigorous study of the properties, the construction, and the utilization of subsets of special vertices in large online networks. We show that graph synopses defined by the importance of vertices provide small, relatively accurate portraits, independent of the importance measure, of the larger underlying graphs and of the important vertices. Furthermore, they can be computed relatively efficiently.