Fixation dynamics on multilayer networks

TL;DR

It is shown mathematically and numerically that two-layer networks are suppressors of selection, which means that they suppress the effects of the different fitness values among the different types on the final outcomes of the evolutionary dynamics (called fixation probability) relative to the constituent one-layer networks.

Abstract

Network structure has a large impact on constant-selection evolutionary dynamics, with which multiple types of fitness (i.e., strength) compete on the network. Here we study constant-selection dynamics on two-layer networks in which the fitness of a node in one layer affects that in the other layer, under birth-death processes and uniform initialization, which are commonly assumed. We show mathematically and numerically that two-layer networks are suppressors of selection, which means that they suppress the effects of the different fitness values among the different types on the final outcomes of the evolutionary dynamics (called fixation probability) relative to the constituent one-layer networks. In fact, many two-layer networks are suppressors of selection relative to the most basic baseline, the Moran process. This result is in stark contrast with the results for conventional one-layer networks for which most networks are amplifiers of selection.

Figures (9)

• Figure 1: An example of a two-layer network. Each individual occupies a replica node in layer 1 and the corresponding replica node in layer 2, as indicated by dashed lines. A resident replica node and a mutant replica node are shown in blue and red, respectively.
• Figure 2: Fixation probability for coupled complete graphs under models 1 and 2. (a) $N=6$. (b) $N=30$. The insets to the left within each panel magnify the results for $r$ values less than and close to $r=1$. Those to the right within each panel magnify the results for $r$ values greater than and close to $r=1$.
• Figure 3: Fixation probability for two-layer networks composed of a complete graph layer and a star graph layer under model 1. (a) $N=6$. (b) $N=30$.
• Figure 4: Fixation probability for coupled star graphs under model 1. We compare the results for the coupled star graphs, shown in blue, with those for the Moran process, shown in black, and those for the single-layer star graphs, shown in green. (a) $N=6$. (b) $N=30$. The inset in (b) magnifies the result for $r$ values less than and close to $r=1$.
• Figure 5: Fixation probability for two-layer networks composed of a complete graph layer and a complete bipartite graph layer under model 1. (a) $N=6$ with $K_{3,3}$. (b) $N=6$ with $K_{2,4}$. (c) $N=20$ with $K_{10,10}$. (d) $N=20$ with $K_{7,13}$. In panels (a) and (c), the results for the complete graph layer are close to those for the complete bipartite graph layer such that the blue lines are almost hidden behind the green lines.

Theorems & Definitions (17)

• Theorem 1
• proof
• Remark 2
• Lemma 3
• proof
• Remark 4
• Theorem 5
• proof
• Lemma 6
• Theorem 7