Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)In finite populations the action of neutral mutations is balanced by genetic drift, leading to a stationary distribution of alleles that displays a transition between two different behaviors. For small mutation rates most individuals will carry the same allele at equilibrium, whereas for high mutation rates of the alleles will be randomly distributed with frequencies close to one half for a biallelic gene. For well-mixed haploid populatiorls the mutation threshold is mu(C) = 1/2N, where N is the population size. In this paper we study how spatial structure affects this mutation threshold. Specifically, we study the stationary allele distribution for populations placed on regular networks where connected nodes represent potential mating partners. We show that the mutation threshold is sensitive to spatial structure only if the number of potential mates is very small. In this limit, the mutation threshold decreases substantially, increasing the diversity of the population at considerably low mutation rates. Defining k(c) as the degree of the network for which the mutation threshold drops to half of its value in well-mixed populations we show that k(c) grows slowly as a function of the population size, following a power law. Our calculations and simulations are based on the Moran model and on a mapping between the Moran model with mutations and the voter model with opinion makers. (C) 2016 Elsevier Ltd. All rights reserved.402917Sao Paulo Research Foundation (FAPESP) [2014/04036-2, 2014/10470-7]CNPqFundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)