Mixed updating in structured populations

Abstract

Evolutionary graph theory (EGT) studies the effect of population structure on evolutionary dynamics. The vertices of the graph represent the $N$ individuals. The edges denote interactions for competitive replacement. Two standard update rules are death-Birth (dB) and Birth-death (Bd). Under dB, an individual is chosen uniformly at random to die, and its neighbors compete to fill the vacancy proportional to their fitness. Under Bd, an individual is chosen for reproduction proportional to fitness, and its offspring replaces a randomly chosen neighbor. Here we study mixed updating between those two scenarios. In each time step, with probability $δ$ the update is dB and with remaining probability it is Bd. We study fixation probabilities and times as functions of $δ$ under neutral evolution and constant selection. Despite the fact that fixation probabilities and times can be increasing, decreasing, or non-monotonic in $δ$, we prove nearly all unweighted undirected graphs have short fixation times and provide an efficient algorithm to estimate their fixation probabilities. Finally, we prove exact formulas for fixation probabilities on cycles, stars, and more complex structures and classify their sensitivities to $δ$.

Figures (7)

• Figure 1: Mixed $\delta$-updating on a graph. Initially, $N$ individuals are present in the population with one mutant. At each time step, a death-Birth step occurs with probability $\delta$. With remaining probability, a Birth-death step occurs. The thick arrows on the edges of the graph are oriented such that the arrowheads indicate the possible location(s) of death; the tails indicate the possible location(s) of birth for each time step.
• Figure 2: Fixation probabilities, $\rho$, of various graphs under mixed $\delta$-updating for $N=7$ in neutral evolution. In both figures, the dashed line at $1/N$ is the fixation probability of the Moran process (well-mixed) in neutral evolution. For each graph depicted, the small star symbol on the differently shaded vertex indicates the initial mutant location. a,i) A star graph with the initial mutant location at the center. Relative to $\rho$ under the unbiased process, Bd-biased processes have a lower $\rho$ but dB-biased processes have a higher $\rho$. a,ii) A star graph with the initial mutant location at a leaf. Relative to $\rho$ under the unbiased process, Bd-biased processes have a higher $\rho$ but dB-biased processes have a lower $\rho$. a,iii) A graph composed of a star and two additional edges. Relative to $\rho$ under the unbiased process, both pure Bd and pure dB processes have a lower $\rho$. However, panel b, and iv) shows that the graph from a, iii) has higher $\rho$ than the unbiased process when $\delta$ is slightly smaller than $1/2$. This example shows that $\rho$ can be non-monotonic in $\delta$. Further, it is not possible that both pure Bd and pure dB have higher $\rho$ than the unbiased process (see \ref{['thm:both-pure-larger-than-unbiased-not-possible']}).
• Figure 3: Examples of $(d_1,d_2)$-bidegreed graph families (a-f) and their sensitivities in neutral evolution to initial mutant location at $\delta=0,1$ (g-j). Vertices with degree $d_1$ are represented as circles whereas vertices with degree $d_2\geq d_1$ are represented as squares. For the examples depicted, $d_2$ is strictly greater than $d_1$, but it is also possible for $d_2=d_1$ per \ref{['def:bidegreed']}. Let $n_1$ denote the number of vertices with degree $d_1$ and $n_2$ denote the number of vertices with degree $d_2$. The value $\rho_i(\delta)$ is the derivative of $\mathsf{fp}_{r=1}^\delta(G, \{u\})$ with respect to $\delta$ where $u$ is a vertex with degree $d_i$. a, a star graph, $S_N$; the $n_2=1$ center vertex has degree $d_2=6$ while the $n_1=6$ periphery vertices have degree $d_1=1$ for a total of $N=n_1+n_2=6+1=7$ vertices. b, a path graph, $P_N$; the intermediate vertices have degree $2$ while the endpoint vertices have degree $1$. c, a bipartite-biregular graph; the vertices on the left side have degree $\ell=3$ while the vertices on the right side have degree $r=2$; there are $N=10$ vertices. d, a fan graph $F_N$; the center vertex has degree $n=2b=6$ while each of the $2$ vertices on the $b$ blades have degree $2$, for a total of $N=2b+1=7$ vertices. e, a wheel graph $W_N$; the center vertex has degree $n=8$ while the $n$ vertices on the spoke ends have degree $3$, for a total of $N=n+1=9$ vertices. f, a book graph $B_N$; each of the $p=3$ pages has $2$ page corners with degree $2$; the $2$ seam endpoints have degree $p+1$, for a total of $N=2p+2=8$ vertices. g, A contour plot in the $(\eta,\kappa)$ space where $\eta=n_1/n_2$ and $\kappa=d_2/d_1$ are dimensionless parameters. The lines with shapes denote the values of $\eta=n_1/n_2$ and $\kappa=d_2/d_1$ for the various bidegreed graphs in a-f. Note that the path graph in our examples since it has $n_1\leq n_2$ for $N\geq 4$; thus as $N$ increases, $\eta$ decreases. The intensity of the colorbar denotes the magnitude of $\rho'_1(\delta=0)$, h, $\rho'_2(\delta=0)$, i, $\rho'_1(\delta=1)$, and j, $\rho'_2(\delta=1)$.
• Figure 4: Fixation probabilities on a star graph with $N=10$ vertices in neutral evolution. The various curves represent the various values of $\delta$. The dashed lines are the theoretical fixation probabilities from \ref{['thm:stars']}. The translucent regions represent the $95\%$ confidence intervals for the average of $10^4$ simulations of the mixed $\delta$-updating process; a, Fixation probabilities as a function of $\delta$ when starting with a mutant in the center. b, Same as a) except a mutant is initially on a leaf.
• Figure 5: a, Fixation times, $T$, of various graphs under mixed $\delta$-updating for $N=7$ in neutral evolution. For each graph depicted, the small star symbol on the differently shaded vertex indicates the initial mutant location. i) The fixation time increases and then decreases relative to $\delta$. ii) The fixation time decreases and then increases relative to $\delta$. iii) The fixation time always increases relative to $\delta$. iv) The fixation time always decreases relative to $\delta$.

Theorems & Definitions (22)

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• proof : Proof of \ref{['thm:bidegreed']}
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