The h-Index of a Graph and Its Application to Dynamic Subgraph Statistics

We describe a data structure that maintains the number of triangles in a dynamic undirected graph, subject to insertions and deletions of edges and of degree-zero vertices. More generally it can be used to maintain the number of copies of each possible three-vertex subgraph in time O(h) per update, where h is the h-index of the graph, the maximum number such that the graph contains h vertices of degree at least h. We also show how to maintain the h-index itself, and a collection of h high-degree vertices in the graph, in constant time per update. Our data structure has applications in social network analysis using the exponential random graph model (ERGM); its bound of O(h) time per edge is never worse than the Θ( √ m) time per edge necessary to list all triangles in a static graph, and is strictly better for graphs obeying a power law degree distribution. In order to better understand the behavior of the h-index statistic and its implications for the performance of our algorithms, we also study the behavior of the h-index on a set of 136 real-world networks. Submitted: December 2011 Accepted: August 2012 Final: August 2012 Published: August 2012 Article type: Regular paper Communicated by: S. Albers This work was supported in part by NSF grants 0830403 and 1217322 and by the Office of Naval Research under grant N00014-08-1-1015. A preliminary version of these results was presented at the Algorithms and Data Structures Symposium (WADS), 2009. E-mail addresses: [email protected] (David Eppstein) [email protected] (Emma S. Spiro)