Exact numerical calculation of fixation probability and time on graphs

Abstract

The Moran process on graphs is a popular model to study the dynamics of evolution in a spatially structured population. Exact analytical solutions for the fixation probability and time of a new mutant have been found for only a few classes of graphs so far. Simulations are time-expensive and many realizations are necessary, as the variance of the fixation times is high. We present an algorithm that numerically computes these quantities for arbitrary small graphs by an approach based on the transition matrix. The advantage over simulations is that the calculation has to be executed only once. Building the transition matrix is automated by our algorithm. This enables a fast and interactive study of different graph structures and their effect on fixation probability and time. We provide a fast implementation in C with this note [11]. Our code is very flexible, as it can handle two different update mechanisms (Birth-death or death-Birth), as well as arbitrary directed or undirected graphs.

Methods for approximating stochastic evolutionary dynamics on graphs

“…We start with a representation of the stochastic evolutionary process using a master equation [11], from which we develop exact equations describing individual node probabilities. We then apply ideas for approximating the master equation based around developing hierarchies of moment equations.…”

“…Important quantities of interest such as the exact fixation probability and time can, in principle, be obtained by solving the discrete-time difference equations of the underlying stochastic model [11], although this is only feasible for very small populations unless there are simplifying symmetries. Individual-based stochastic simulations [3,21] provide numerically accurate representations of the evolutionary process on arbitrary graphs but have limited scope for generating conceptual insights into the dynamics on their own.…”

Should tissue structure suppress or amplify selection to minimize cancer risk?

“…We study this by numerically calculating the fixation probability on an amplifier and suppressor graph of size 10. We use standard methods based on the transition matrix, which we generate from the adjacency matrix of the graph [13,38,39]. The transition matrix and the vector of fixation probabilities form a linear system of equations which can be solved for the fixation probabilities.…”

“…All fixation probabilities have been calculated numerically based on the method described in [13,39]. This approach is based on the numerical evaluation of the transition matrix of the Markov process associated with a graph, which naively scales with the graph size N as 2 N × 2 N .…”

“…This approach is based on the numerical evaluation of the transition matrix of the Markov process associated with a graph, which naively scales with the graph size N as 2 N × 2 N . This allows us to obtain numerically exact results, but restricts the analysis to relatively small graphs (currently, for our implementation [39] up to 23 nodes).…”