Before revision: Exploration of the emerging patterns of mutants in a finite wild-type group on spatial structure is an attracting topic. The mutation is assumed to appear with a low probability. Many studies have investigated the expansion and invasion chances for a few mutants under different games on spatial structures. It is shown that different games could lead to different evolving patterns. There are rules generalized for when the process favors the mutants, where many of the rules indicate that clustering plays a heavy role in the expansion of mutants.
After revision: Cooperation is ubiquitous across all levels of biological systems ranging from microbial communities to human societies. It, however, seemingly contradicts the evolutionary theory, since cooperators are exploited by free-riders and thus are disfavored by natural selection. Many studies based on evolutionary game theory have tried to solve the puzzle and figure out the reason why cooperation exists and how it emerges. Network reciprocity is one of the mechanisms to promote cooperation, where nodes refer to individuals and links refer to social relationships. The spatial arrangement of mutant individuals, which refers to the clustering of mutants, plays a key role in network reciprocity. Besides, many other mechanisms supporting cooperation suggest that the clustering of mutants plays an important role in the rules indicate that clustering plays a heavy role in the expansion of mutants.