  We consider a stochastic individual-based model for the evolution of a
haploid, asexually reproducing population. The space of possible traits is
given by the vertices of a (possibly directed) finite graph $G=(V,E)$. The
evolution of the population is driven by births, deaths, competition, and
mutations along the edges of $G$. We are interested in the large population
limit under a mutation rate $\mu_K$ given by a negative power of the carrying
capacity $K$ of the system: $\mu_K=K^{-1/\alpha},\alpha>0$. This results in
several mutant traits being present at the same time and competing for invading
the resident population. We describe the time evolution of the orders of
magnitude of each sub-population on the $\log K$ time scale, as $K$ tends to
infinity. Using techniques developed in [Champagnat, M\'el\'eard, Tran, 2019]
we show that these are piecewise affine continuous functions, whose slopes are
given by an algorithm describing the changes in the fitness landscape due to
the succession of new resident or emergent types. This work generalises [Kraut,
Bovier, 2019] to the stochastic setting, and Theorem 3.2 of [Bovier, Coquille,
Smadi, 2018] to any finite mutation graph. We illustrate our theorem by a
series of examples describing surprising phenomena arising from the geometry of
the graph and/or the rate of mutations.
