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Biological invasion, climate change and urbanization are presently altering species compositions worldwide (McKinney and Lockwood 1999, McKinney 2006, Qian and Ricklefs 2006, Magurran et al. 2015, Gossner et al. 2016, Newbold et al. 2018), while changes in community composition are generally associated with higher community similarities and losses of community distinctiveness (McKinney and Lockwood 1999, Olden et al. 2004). The underlying processes of homogenization are still unclear and especially lack theoretical underpinnings (Olden and Poff 2003, Rosenblad and Sax 2017). It was theoretically and empirically proposed that similarity changes are determined by different events (Olden and Poff 2003, Villéger and Brosse 2012, Rosenblad and Sax 2017). For example, the introduction of the same species to two sites would result in an increase in similarity, while the extinction of same native species from two sites would lead to a decrease in similarity. In complement to earlier works on the mathematical description of homogenization (Olden and Poff 2003, Villéger and Brosse 2012). Rosenblad and Sax (2017) provided a concise framework of six types of introduction and extinction events with formula derivations to qualify the relative importance of these events on changes in community similarity. They found that the effects of these events vary according to event type and initial similarity. However, their conclusions mainly derive from simplified assumptions of species richness and number of events, which prevent us from understanding the precise relationships among initial similarity, different event types and total number of species. Through mathematical derivation and simulations (Fig. 1), we thus aim to: 1) combine previously identified determinants of change in similarity into a single comprehensive formula; 2) illustrate how these determinants interact with each other; and 3) highlight the important role of gradient in species richness in affecting the magnitude of similarity changes. [ABSTRACT FROM AUTHOR]