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.
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