DOI: 10.7155/jgaa.00273
The h-Index of a Graph and its Application to Dynamic Subgraph Statistics
David Eppstein and Emma S. Spiro
Vol. 16, no. 2, pp. 543-567, 2012. Regular paper

Abstract 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.
Communicated by Susanne Albers

Journal of Graph Algorithms and Applications