Quantifying Community Evolution in Developer Social Networks: Proof of Indices' Properties

The document provides the proof to properties of community evolution indices including community split and shrink in paper: Liang Wang, Ying Li, Jierui Zhang, and Xianping Tao. 2022. Quantifying Community Evolution in Developer Social Networks. In Proceedings of the30th ACM Joint European Software Engineering Conference and Symposiumon the Foundations of Software Engineering (ESEC/FSE 22), November 14 - 18, 2022, Singapore, Singapore. ACM, New York, NY, USA, 12 pages. Proof to properties of community merge and expand is similar.