The number of fixed mutations accumulated in an evolving population often
displays a variance that is significantly larger than the mean (the
overdispersed molecular clock). By examining a generic evolutionary process on
a neutral network of high-fitness genotypes, we establish a formalism for
computing all cumulants of the full probability distribution of accumulated
mutations in terms of graph properties of the neutral network, and use the
formalism to prove overdispersion of the molecular clock. We further show that
significant overdispersion arises naturally in evolution when the neutral
network is highly sparse, exhibits large global fluctuations in neutrality, and
small local fluctuations in neutrality. The results are also relevant for
elucidating the topological structure of a neutral network from empirical
measurements of the substitution process.Comment: 10 page