Fixation times on directed graphs

Computing the rate of evolution in spatially structured populations is difficult. A key quantity is the fixation time of a single mutant with relative reproduction rate r which invades a population of residents. We say that the fixation time is "fast" if it is at most a polynomial function in terms of the population size N. Here we study fixation times of advantageous mutants (r > 1) and neutral mutants (r = 1) on directed graphs, which are those graphs that have at least some one-way connections. We obtain three main results. First, we prove that for any directed graph the fixation time is fast, provided that r is sufficiently large. Second, we construct an efficient algorithm that gives an upper bound for the fixation time for any graph and any r ≥ 1. Third, we identify a broad class of directed graphs with fast fixation times for any r ≥ 1. This class includes previously studied amplifiers of selection, such as Superstars and Metafunnels. We also show that on some graphs the fixation time is not a monotonically declining function of r; in particular, neutral fixation can occur faster than fixation for small selective advantages.

The speed of neutral evolution on graphs

The speed of evolution on structured populations is crucial for biological and social systems. The likelihood of invasion is key for evolutionary stability. But it makes little sense if it takes long. It is far from known what population structure slows down evolution. We investigate the absorption time of a single neutral mutant for all the 112 non-isomorphic undirected graphs of size 6. We find that about three-quarters of the graphs have an absorption time close to that of the complete graph, less than one-third are accelerators, and more than two-thirds are decelerators. Surprisingly, determining whether a graph has a long absorption time is too complicated to be captured by the joint degree distribution. Via the largest sojourn time, we find that echo-chamber-like graphs, which consist of two homogeneous graphs connected by few sparse links, are likely to slow down absorption. These results are robust for large graphs, mutation patterns as well as evolutionary processes. This work serves as a benchmark for timing evolution with complex interactions, and fosters the understanding of polarization in opinion formation.

Wald’s martingale and the conditional distributions of absorption time in the Moran process

Many models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moranbirth–death process, where that quantity is the number of mutants in a population. In such processes, we are often interested in their absorption (i.e. fixation) probabilities and the conditional distributions of absorption time. Those conditional time distributions can be very difficult to calculate, even for relatively simple processes like the Moran birth–death model. Instead of considering the time to absorption, we consider a closely related quantity: the number of mutant population size changes before absorption. We use Wald’s martingale to obtain the conditional characteristic functions of that quantity in the Moran process. Our expressions are novel, analytical and exact, and their parameter dependence is explicit. We use our results to approximate the conditional characteristic functions of absorption time. We state the conditions under which that approximation is particularly accurate. Martingales are an elegant framework to solve principal problems of evolutionary stochastic processes. They do not require us to evaluate recursion relations, so when they are applicable, we can quickly and tractably obtain absorption probabilities and times of evolutionary models.

Population structure determines the tradeoff between fixation probability and fixation time

The rate of biological evolution depends on the fixation probability and on the fixation time of new mutants. Intensive research has focused on identifying population structures that augment the fixation probability of advantageous mutants. But these amplifiers of natural selection typically increase fixation time. Here we study population structures that achieve a tradeoff between fixation probability and time. First, we show that no amplifiers can have an asymptotically lower absorption time than the well-mixed population. Then we design population structures that substantially augment the fixation probability with just a minor increase in fixation time. Finally, we show that those structures enable higher effective rate of evolution than the well-mixed population provided that the rate of generating advantageous mutants is relatively low. Our work sheds light on how population structure affects the rate of evolution. Moreover, our structures could be useful for lab-based, medical, or industrial applications of evolutionary optimization.

Environmental fitness heterogeneity in the Moran process

Many mathematical models of evolution assume that all individuals experience the same environment. Here, we study the Moran process in heterogeneous environments. The population is of finite size with two competing types, which are exposed to a fixed number of environmental conditions. Reproductive rate is determined by both the type and the environment. We first calculate the condition for selection to favour the mutant relative to the resident wild-type. In large populations, the mutant is favoured if and only if the mutant's spatial average reproductive rate exceeds that of the resident. But environmental heterogeneity elucidates an interesting asymmetry between the mutant and the resident. Specifically, mutant heterogeneity suppresses its fixation probability; if this heterogeneity is strong enough, it can even completely offset the effects of selection (including in large populations). By contrast, resident heterogeneity has no effect on a mutant's fixation probability in large populations and can amplify it in small populations.

Martingales and the fixation probability of high-dimensional evolutionary graphs

A principal problem of evolutionary graph theory is to find the probability that an initial mutant population will fix on a graph, i.e. that the mutants will eventually replace the indigenous population. This problem is particularly difficult when the dimensionality of a graph is high. Martingales can yield compact and exact expressions for the fixation probability of an evolutionary graph. Crucially, the tractability of martingales does not necessarily depend on the dimensionality of a graph. We will use martingales to obtain the exact fixation probability of graphs with high dimensionality, specifically k-partite graphs (or 'circular flows') and megastars (or 'superstars'). To do so, we require that the edges of the graph permit mutants to reproduce in one direction and indigenous in the other. The resultant expressions for fixation probabilities explicitly show their dependence on the parameters that describe the graph structure, and on the starting position(s) of the initial mutant population. In particular, we will investigate the effect of funneling on the fixation probability of k-partite graphs, as well as the effect of placing an initial mutant in different partitions. These are the first exact and explicit results reported for the fixation probability of evolutionary graphs with dimensionality greater than 2, that are valid over all parameter space. It might be possible to extend these results to obtain fixation probabilities of high-dimensional evolutionary graphs with undirected or directed connections. Martingales are a formidable theoretical tool that can solve fundamental problems in evolutionary graph theory, often within a few lines of straightforward mathematics.

Exploring the topological sources of robustness against invasion in biological and technological networks

For a network, the accomplishment of its functions despite perturbations is called robustness. Although this property has been extensively studied, in most cases, the network is modified by removing nodes. In our approach, it is no longer perturbed by site percolation, but evolves after site invasion. The process transforming resident/healthy nodes into invader/mutant/diseased nodes is described by the Moran model. We explore the sources of robustness (or its counterpart, the propensity to spread favourable innovations) of the US high-voltage power grid network, the Internet2 academic network, and the C. elegans connectome. We compare them to three modular and non-modular benchmark networks, and samples of one thousand random networks with the same degree distribution. It is found that, contrary to what happens with networks of small order, fixation probability and robustness are poorly correlated with most of standard statistics, but they depend strongly on the degree distribution. While community detection techniques are able to detect the existence of a central core in Internet2, they are not effective in detecting hierarchical structures whose topological complexity arises from the repetition of a few rules. Box counting dimension and Rent’s rule are applied to show a subtle trade-off between topological and wiring complexity.

Most Undirected Random Graphs Are Amplifiers of Selection for Birth-Death Dynamics, but Suppressors of Selection for Death-Birth Dynamics

We analyze evolutionary dynamics on graphs, where the nodes represent individuals of a population. The links of a node describe which other individuals can be displaced by the offspring of the individual on that node. Amplifiers of selection are graphs for which the fixation probability is increased for advantageous mutants and decreased for disadvantageous mutants. A few examples of such amplifiers have been developed, but so far it is unclear how many such structures exist and how to construct them. Here, we show that almost any undirected random graph is an amplifier of selection for Birth-death updating, where an individual is selected to reproduce with probability proportional to its fitness and one of its neighbors is replaced by that offspring at random. If we instead focus on death-Birth updating, in which a random individual is removed and its neighbors compete for the empty spot, then the same ensemble of graphs consists of almost only suppressors of selection for which the fixation probability is decreased for advantageous mutants and increased for disadvantageous mutants. Thus, the impact of population structure on evolutionary dynamics is a subtle issue that will depend on seemingly minor details of the underlying evolutionary process.

Evolutionary dynamics describes the spread of individuals with different features within a population. This spreading process can be strongly influenced by the population structure—if a highly successful individual can only displace a few neighbors, it may take more time to spread than an individual that can displace all other individuals from the population. In particular, a population structure can also amplify the evolutionary success of a type. We show that almost all random population structures lead to such an amplification. However, if we change a presumably minor detail of the evolutionary model, almost all random population structures have the opposite effect and suppress the evolutionary success of a type. Thus, it is crucial to consider the underlying assumptions of such models when discussing their possible implications for real biological systems.

Amplifiers of selection

When a new mutant arises in a population, there is a probability it outcompetes the residents and fixes. The structure of the population can affect this fixation probability. Suppressing population structures reduce the difference between two competing variants, while amplifying population structures enhance the difference. Suppressors are ubiquitous and easy to construct, but amplifiers for the large population limit are more elusive and only a few examples have been discovered. Whether or not a population structure is an amplifier of selection depends on the probability distribution for the placement of the invading mutant. First, we prove that there exist only bounded amplifiers for adversarial placement—that is, for arbitrary initial conditions. Next, we show that the Star population structure, which is known to amplify for mutants placed uniformly at random, does not amplify for mutants that arise through reproduction and are therefore placed proportional to the temperatures of the vertices. Finally, we construct population structures that amplify for all mutational events that arise through reproduction, uniformly at random, or through some combination of the two.

Universality of fixation probabilities in randomly structured populations

The stage of evolution is the population of reproducing individuals. The structure of the population is known to affect the dynamics and outcome of evolutionary processes, but analytical results for generic random structures have been lacking. The most general result so far, the isothermal theorem, assumes the propensity for change in each position is exactly the same, but realistic biological structures are always subject to variation and noise. We consider a finite population under constant selection whose structure is given by a variety of weighted, directed, random graphs; vertices represent individuals and edges interactions between individuals. By establishing a robustness result for the isothermal theorem and using large deviation estimates to understand the typical structure of random graphs, we prove that for a generalization of the Erdős-Rényi model, the fixation probability of an invading mutant is approximately the same as that of a mutant of equal fitness in a well-mixed population with high probability. Simulations of perturbed lattices, small-world networks, and scale-free networks behave similarly. We conjecture that the fixation probability in a well-mixed population, (1 − r⁻¹)/(1 − r⁻ⁿ), is universal: for many random graph models, the fixation probability approaches the above function uniformly as the graphs become large.