Colonization times in Moran process on graphs

Abstract

Moran Birth-death process is a standard stochastic process that is used to model natural selection in spatially structured populations. A newly occurring mutation that invades a population of residents can either fixate on the whole population or it can go extinct due to random drift. The duration of the process depends not only on the total population size $n$, but also on the spatial structure of the population. In this work, we consider the Moran process with a single type of individuals who invade and colonize an otherwise empty environment. Mathematically, this corresponds to the setting where the residents have zero reproduction rate, thus they never reproduce. We present two main contributions. First, in contrast to the Moran process in which residents do reproduce, we show that the colonization time is always at most a polynomial function of the population size $n$. Namely, we show that colonization always takes at most $\frac12n^3-\frac12n^2$ expected steps, and for each $n$, we exactly identify the unique slowest spatial structure where it takes exactly that many steps. Moreover, we establish a stronger bound of roughly $n^{2.5}$ steps for spatial structures that contain only two-way connections and an even stronger bound of roughly $n^2$ steps for lattice-like spatial structures. Second, we discuss various complications that one faces when attempting to measure fixation times and colonization times in spatially structured populations, and we propose to measure the real duration of the process, rather than counting the steps of the classic Moran process.

Figures (11)

• Figure 1: Moran Birth-death process on a spatial structure.a, The spatial structure is given as a network (graph), where nodes represent sites and arrows represent possible migration patterns. Nodes occupied by mutants are green (here $u$ and $v$). b, In each step of the Moran process, first, a random node is selected for reproduction, and then the offspring migrates along a random outgoing edge. Here, the offspring of $u$ migrated to $w$.
• Figure 2: Colonization times on directed graphs.a, In the complete graph $\operatorname{K}_n$, each two nodes are connected by a two-way edge. In the total order graph $\operatorname{TO}_n$, the nodes are arranged left to right and all edges going left to right are included. The backward graph $\operatorname{B}_n$ consists of a directed path going left to right, plus all one-way edges going in the opposite direction, right to left. b, For each $n$, the backward graph $\operatorname{B}_n$ (blue) is the graph with maximal colonization time. We have $\operatorname{T}(\operatorname{B}_n)=\frac{1}{2}n^3-\frac{1}{2}n^2$. The shortest possible colonization time is of the order of $n\log n$ steps, which is achieved for the complete graph $\operatorname{K}_n$ (orange). For the total order graph $\operatorname{TO}_n$ (red) we have $\operatorname{T}(\operatorname{TO}_n)= \Theta(n^2)$. Here the lines show the proved analytical results, the dots show the simulations, and the axes are log-scale.
• Figure 3: Colonization times on undirected graphs.a, In the star graph $\operatorname{S}_n$, one node is the center, and all the other nodes are connected to it by a two-way edge. The double star graph $\operatorname{D}_{2k}$ is obtained by joining the centers of two star graphs $\operatorname{S}_{k}$ using a two-way edge. b, The proved upper bound for undirected graphs is $4n^2\sqrt{n}+o(n^2\sqrt{n})$. Here we plot the function $4n^2\sqrt{n}$ with blue color. The graph with the largest colonization time we found is the star graph $\operatorname{S}_n$ (green). Again the shortest possible colonization time is of the order of $n\log n$ steps, which is achieved for the complete graph $\operatorname{K}_n$ (orange). For the double star graph $\operatorname{D}_n$ (yellow) we have $\operatorname{T}(\operatorname{D}_n)= \Theta(n^2\log{n})$. For the cycle $\operatorname{C}_n$ (red) we have $\operatorname{T}(\operatorname{C}_n)=\Theta(n^2)$. Here the lines show the proved analytical results, the dots show the simulations, and the axes are log-scale.
• Figure 4: Starting with more mutants causes more classic Moran steps.a, The lollipop graph $\operatorname{L}_n$ consists of $\sqrt n$ nodes arranged along a directed path, and the remaining nodes in a fully connected cluster (here $n=16$). b, The classic fixation time on a lollipop graph $\operatorname{L}_n$ with two different initializations: either a single mutant at the start of the path (blue), or additionally all mutants in the fully connected cluster (orange). For $r>2.2$ the first initialization leads to fewer classic Moran steps, despite having a strict subset of nodes that are initially mutants. Here $n=1600$, $r\in[2,10]$, and at least $10^3$ simulations per data point.
• Figure 5: One step of the Moran process.

Theorems & Definitions (38)

• Theorem 1: General upper bound
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Definition 1
• Definition 2: Classic Moran process