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

TL;DR

This paper introduces a dynamic data structure for efficiently maintaining triangle counts and subgraph statistics in graphs, leveraging the h-index to optimize performance, with applications in social network analysis.

Contribution

It presents a novel data structure that maintains triangle counts and the h-index in dynamic graphs with improved efficiency, especially for power-law degree distributions.

Findings

The data structure operates in O(h) time per update.

It can maintain the h-index and high-degree vertices in constant time.

Performance is better than static algorithms on real-world networks.

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 Theta(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.