  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 expansion
of mutants. However, the clustering of mutants and the game dynamics are
typically coupled. It is still unclear how the clustering of mutants alone
alters the evolutionary dynamics. To this end, we employ a minimal model with
frequency independent fitness on a circle. It disentangles the clustering of
mutants from game dynamics. The distance between two mutants on the circle is
adopted as a natural indicator for the clustering of mutants or assortment. We
find that the assortment is an amplifier of the selection for the connected
mutants compared with the separated ones. Nevertheless, as mutants are
separated, the more dispersed mutants are, the greater the chance of invasion
is. It gives rise to the non-monotonic effect of clustering, which is
counterintuitive. On the other hand, we find that less assortative mutants
speed up fixation. Our model shows that the clustering of mutants plays a
non-trivial role in fixation, which has emerged even if the game interaction is
absent.
