A voter-model representation of the multi-allelic Moran process: Exact stationary distributions and diversity...

We extend the equivalence between the two-allele Moran process and the two-candidate voter model in a finite population to the multi-allele Moran process with parent-independent mutation and a multi-candidate voter model. This exact mapping yields a concise derivation of the finite-population Dirichlet-multinomial stationary distribution for the labeled allele-count vector in fully mixed populations of size n with an arbitrary number m of alleles. It also gives the critical mutation threshold u_c=1/(m+2n-2), making its dependence on m explicit. We then extend the analysis to randomly connected populations modeled by Erdős-Rényi (ER) interaction networks. Under an ER mean-field reduction, the Moran stationary distribution and the corresponding diversity threshold u_c remain unchanged and coincide with the fully mixed results, even though the associated voter model exhibits amplified zealot influence on sparse random networks. Finally, simulations on several sparse and structured network topologies, including Erdős-Rényi, scale-free, ring, and small-world networks, show that structural heterogeneity can substantially reshape the stationary allele distribution and, consequently, the level of genetic diversity. More broadly, the correspondence developed in this paper turns the voter-model perspective into a tractable framework for analyzing the multi-allele Moran process, yielding the stationary law, the critical mutation threshold, and new insight into how network structure shapes stationary diversity.