Global triangle estimation based on first edge sampling in large graph streams

Triangle approximate counting has emerged as a prominent issue in graph stream research in the past few years, with applications ranging from social network analy-sis to web topic mining and motif detection in informatics. Many graph stream sam-pling and triangle approximate counting algorithms have been proposed, with the majority of them guaranteeing unbiased estimation. However, they either cannot ensure that the memory overhead or the result's uncertainty is too great due to the use of an excessively large sampling space. In this article, we propose RFES, a set of one-pass stream algorithms for counting the global number of triangles in a fully dynamic graph stream in an unbiased, low-variance, and high-precision manner. RFES has three algorithms: RFESBASE, RFES-IMPR, and RFES-FD, which rep-resent the basic, improved, and complete dynamic versions, respectively. Each algo-rithm is based on our proposed first-edge reservoir sampling method, which shrinks the sampling space while increasing the uncertainty of triangles in the sample. It can deal with fully dynamic data with a lower theoretical estimation variance than state-of-the-art algorithms. A significant number of experimental results demonstrated that our RFES algorithm is more accurate and takes less time. The source codes of RFES can be downloaded from the website: https://github.com/BioLab310/RFES.