Evolutionary graph theory beyond pairwise interactions: Higher-order network motifs shape times to fixation in structured populations

To design population topologies that can accelerate rates of solution discovery in directed evolution problems or for evolutionary optimization applications, we must first systematically understand how population structure shapes evolutionary outcome. Using the mathematical formalism of evolutionary graph theory, recent studies have shown how to topologically build networks of population interaction that increase probabilities of fixation of beneficial mutations, at the expense, however, of longer fixation times, which can slow down rates of evolution, under elevated mutation rate. Here we find that moving beyond dyadic interactions in population graphs is fundamental to explain the trade-offs between probabilities and times to fixation of new mutants in the population. We show that higher-order motifs, and in particular three-node structures, allow the tuning of times to fixation, without changes in probabilities of fixation. This gives a near-continuous control over achieving solutions that allow for a wide range of times to fixation. We apply our algorithms and analytic results to two evolutionary optimization problems and show that the rate of solution discovery can be tuned near continuously by adjusting the higher-order topology of the population. We show that the effects of population structure on the rate of evolution critically depend on the optimization landscape and find that decelerators, with longer times to fixation of new mutants, are able to reach the optimal solutions faster than accelerators in complex solution spaces. Our results highlight that no one population topology fits all optimization applications, and we provide analytic and computational tools that allow for the design of networks suitable for each specific task.Author summary: Accelerating the rate of solution discovery has been a long-standing goal in directed evolution and evolutionary optimization problems. One way to accelerate evolutionary search is to design population structures that control the mode and tempo of evolutionary dynamics. Population structure can amplify the action of selection and accelerate rates of evolution, or reversely, suppress selection and slow evolution down. Here we use evolutionary graph theory to systematically study how higher-order motifs in a population’s network of interaction and replacement can shape the velocity of evolution. In particular, we show that by changing the number of triangles (motifs of degree three) in a network, while fixing lower-order structures, we can continuously tune times to fixation of new variants in the population, independently of their probabilities of fixation. This allows us to design population structures that find optimal solutions across a broad spectrum of evolutionary optimization problems. We show that the best solutions need not always be the ones with the smallest times to fixation and discuss what motifs found in real biological networks can teach us about designing highly evolvable artificial ones.