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

D Eppstein, ES Spiro
### 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 $\Phi(\sqrt{m})$ 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.

Publication

*Journal of Graph Algorithms and Applications*