arXiv subjects

Martin A. Nowak

Publications and source records attributed to Martin A. Nowak.

At least 19 recordsLinked to original sources

arXiv2026

Mixed updating in structured populations

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---the individuals on adjacent vertices of the graph---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 on an adjacent vertex. Here we study mixed updating between those two scenarios. In each time step, with probability $δ$ the update is dB and with the 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 that 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 $δ$.

q-bio.PE↗

arXiv2026

Fixation location in structured populations

In stochastic evolutionary dynamics, the replacement of an existing genotype or cultural trait by a newly introduced mutant is typically characterized by the quantities of fixation probability and fixation time. But in a structured population, the disappearance of a lineage occurs at a specific place. For evolutionary dynamics on graphs, we define the fixation location as the node occupied by the last wild-type individual immediately before mutant fixation. Conditional on fixation, this location is described by a probability distribution over the nodes of the graph. We study the fixation location for neutral evolution, for the colonization process, and, more generally, for constant selection on small graphs, cycles, tori, random graphs, and island populations. We find that the distribution of the fixation location is often highly nonuniform, depends strongly on the graph structure and the selection strength, and can differ sharply even when classical fixation statistics are similar. For many graphs, some nodes can never be fixation locations. Our results identify fixation location as a fundamental aspect of evolutionary dynamics and suggest new ways to understand, monitor, and potentially mitigate extinction events in biological and social settings.

q-bio.PE↗

arXiv2026

The evolutionary advantage of replacers in the Moran process

Evolution occurs in populations of reproducing individuals. In stochastic descriptions of evolutionary dynamics, such as the Moran process, individuals are chosen randomly for birth and for death. If the same type is chosen for both steps, then the reproductive event is wasted, because the composition of the population remains unchanged. Here we introduce a new phenotype, which we call a replacer. Replacers are efficient competitors. When a replacer is chosen for reproduction, the offspring will always replace an individual of another type (if available). We determine the selective advantage of replacers in well-mixed populations and on one-dimensional lattices. We find that being a replacer substantially boosts the fixation probability of neutral and deleterious mutants. In particular, fixation probability of a single neutral replacer who invades a well-mixed population of size $N$ is of the order of $1/\sqrt N$ rather than the standard $1/N$. Even more importantly, replacers are much better protected against invasions once they have reached fixation. Therefore, replacers dominate the mutation selection equilibrium even if the phenotype of being a replacer comes at a substantial cost: curiously, for large population size and small mutation rate the relative reproductive rate of a successful replacer can be as low as $1/e$.

q-bio.PE↗

arXiv2026

The Mixed Birth-death/death-Birth Moran Process

We study evolutionary dynamics on graphs in which each step consists of one birth and one death, also known as the Moran processes. There are two types of individuals: residents with fitness $1$ and mutants with fitness $r$. Two standard update rules are used in the literature. In Birth-death (Bd), a vertex is chosen to reproduce proportional to fitness, and one of its neighbors is selected uniformly at random to be replaced by the offspring. In death-Birth (dB), a vertex is chosen uniformly to die, and then one of its neighbors is chosen, proportional to fitness, to place an offspring into the vacancy. We formalize and study a unified model, the $λ$-mixed Moran process, in which each step is independently a Bd step with probability $λ\in [0,1]$ and a dB step otherwise. We analyze this mixed process for undirected, connected graphs. As an interesting special case, we show at $λ=1/2$, for any graph that the fixation probability when $r=1$ with a single mutant initially on the graph is exactly $1/n$, and also at $λ=1/2$ that the absorption time for any $r$ is $O_r(n^4)$. We also show results for graphs that are "almost regular," in a manner defined in the paper. We use this to show that for suitable random graphs from $G \sim G(n,p)$ and fixed $r>1$, with high probability over the choice of graph, the absorption time is $O_r(n^4)$, the fixation probability is $Ω_r(n^{-2})$, and we can approximate the fixation probability in polynomial time. Another special case is when the graph has only two distinct degree values $\{d_1, d_2\}$ with $d_1 \leq d_2$. For those graphs, we give exact formulas for fixation probabilities when $r = 1$ and any $λ$, and establish an absorption time of $O_r(n^4 α^4)$ for all $λ$, where $α= d_2 / d_1$. We also provide explicit formulas for the star and cycle under any $r$ or $λ$.

math.PR↗

arXiv2025

Maintaining diversity in structured populations

We examine population structures for their ability to maintain diversity in neutral evolution. We use the general framework of evolutionary graph theory and consider birth-death (bd) and death-birth (db) updating. The population is of size $N$. Initially all individuals represent different types. The basic question is: what is the time $T_N$ until one type takes over the population? This time is known as consensus time in computer science and as total coalescent time in evolutionary biology. For the complete graph, it is known that $T_N$ is quadratic in $N$ for db and bd. For the cycle, we prove that $T_N$ is cubic in $N$ for db and bd. For the star, we prove that $T_N$ is cubic for bd and quasilinear ($N\log N$) for db. For the double star, we show that $T_N$ is quartic for bd. We derive upper and lower bounds for all undirected graphs for bd and db. We also show the Pareto front of graphs (of size $N=8$) that maintain diversity the longest for bd and db. Further, we show that some graphs that quickly homogenize can maintain high levels of diversity longer than graphs that slowly homogenize. For directed graphs, we give simple contracting star-like structures that have superexponential time scales for maintaining diversity.

q-bio.PE↗

arXiv2024

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 $r\ge 1$. Third, we identify a broad class of directed graphs with fast fixation times for any $r\ge 1$. 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.

q-bio.PE↗

arXiv2024

Conditional cooperation with longer memory

Direct reciprocity is a wide-spread mechanism for evolution of cooperation. In repeated interactions, players can condition their behavior on previous outcomes. A well known approach is given by reactive strategies, which respond to the co-player's previous move. Here we extend reactive strategies to longer memories. A reactive-$n$ strategy takes into account the sequence of the last $n$ moves of the co-player. A reactive-$n$ counting strategy records how often the co-player has cooperated during the last $n$ rounds. We derive an algorithm to identify all partner strategies among reactive-$n$ strategies. We give explicit conditions for all partner strategies among reactive-2, reactive-3 strategies, and reactive-$n$ counting strategies. Partner strategies are those that ensure mutual cooperation without exploitation. We perform evolutionary simulations and find that longer memory increases the average cooperation rate for reactive-$n$ strategies but not for reactive counting strategies. Paying attention to the sequence of moves is necessary for reaping the advantages of longer memory.

cs.GT↗

arXiv2023

Trust based attachment

In social systems subject to indirect reciprocity, a positive reputation is key for increasing one's likelihood of future positive interactions. The flow of gossip can amplify the impact of a person's actions on their reputation depending on how widely it spreads across the social network, which leads to a percolation problem. To quantify this notion, we calculate the expected number of individuals, the "audience", who find out about a particular interaction. For a potential donor, a larger audience constitutes higher reputational stakes, and thus a higher incentive, to perform "good" actions in line with current social norms. For a receiver, a larger audience therefore increases the trust that the partner will be cooperative. This idea can be used for an algorithm that generates social networks, which we call trust based attachment (TBA). TBA produces graphs that share crucial quantitative properties with real-world networks, such as high clustering, small-world behavior, and power law degree distributions. We also show that TBA can be approximated by simple friend-of-friend routines based on triadic closure, which are known to be highly effective at generating realistic social network structures. Therefore, our work provides a new justification for triadic closure in social contexts based on notions of trust, gossip, and social information spread. These factors are thus identified as potential significant influences on how humans form social ties.

cs.SI↗

arXiv2023

Mutation enhances cooperation in direct reciprocity

Direct reciprocity is a powerful mechanism for evolution of cooperation based on repeated interactions between the same individuals. But high levels of cooperation evolve only if the benefit-to-cost ratio exceeds a certain threshold that depends on memory length. For the best-explored case of one-round memory, that threshold is two. Here we report that intermediate mutation rates lead to high levels of cooperation, even if the benefit-to-cost ratio is only marginally above one, and even if individuals only use a minimum of past information. This surprising observation is caused by two effects. First, mutation generates diversity which undermines the evolutionary stability of defectors. Second, mutation leads to diverse communities of cooperators that are more resilient than homogeneous ones. This finding is relevant because many real world opportunities for cooperation have small benefit-to-cost ratios, which are between one and two, and we describe how direct reciprocity can attain cooperation in such settings. Our result can be interpreted as showing that diversity, rather than uniformity, promotes evolution of cooperation.

q-bio.PE↗

arXiv2021

Natural selection of mutants that modify population structure

Evolution occurs in populations of reproducing individuals. It is well known that population structure can affect evolutionary dynamics. Traditionally, natural selection is studied between mutants that differ in reproductive rate, but are subject to the same population structure. Here we study how natural selection acts on mutants that have the same reproductive rate, but experience different population structures. In our framework, mutation alters population structure, which is given by a graph that specifies the dispersal of offspring. Reproduction can be either genetic or cultural. Competing mutants disperse their offspring on different graphs. A more connected graph implies higher motility. We show that enhanced motility tends to increase an invader's fixation probability, but there are interesting exceptions. For island models, we show that the magnitude of the effect depends crucially on the exact layout of the additional links. Finally, we show that for low-dimensional lattices, the effect of altered motility is comparable to that of altered fitness: in the limit of large population size, the invader's fixation probability is either constant or exponentially small, depending on whether it is more or less motile than the resident.

q-bio.PE↗

arXiv2020

Infection dynamics of COVID-19 virus under lockdown and reopening

Motivated by COVID-19, we develop and analyze a simple stochastic model for a disease spread in human population. We track how the number of infected and critically ill people develops over time in order to estimate the demand that is imposed on the hospital system. To keep this demand under control, we consider a class of simple policies for slowing down and reopening the society and we compare their efficiency in mitigating the spread of the virus from several different points of view. We find that in order to avoid overwhelming of the hospital system, a policy must impose a harsh lockdown or it must react swiftly (or both). While reacting swiftly is universally beneficial, being harsh pays off only when the country is patient about reopening and when the neighboring countries coordinate their mitigation efforts. Our work highlights the importance of acting decisively when closing down and the importance of patience and coordination between neighboring countries when reopening.

q-bio.PE↗

arXiv2020

The Moran process on 2-chromatic graphs

Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where "properly colored" means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring.

q-bio.PE↗

arXiv2020

Turbulent coherent structures and early life below the Kolmogorov scale

A great number of biological organisms live in aqueous environments. Major evolutionary transitions, including the emergence of life itself, likely occurred in such environments. While the chemical aspects of the role of water in biology are well-studied, the effects of water's physical characteristics on evolutionary events, such as the control of population structure via its rich transport properties, are less clear. Evolutionary transitions such as the emergence of the first cells and of multicellularity, require cooperation among groups of individuals. However, evolution of cooperation faces challenges in unstructured "well-mixed" populations, as parasites quickly overwhelm cooperators. Models that assume population structure to promote cooperation envision such structure to arise from spatial "lattice" models (e.g. surface bound individuals) or compartmentalization models, often realized as protocells. Here we study the effect of turbulent motions in spatial models, and propose that coherent structures, i.e. flow patterns which trap fluid and arise naturally in turbulent flows, may serve many of the properties associated with compartments--collocalization, division, and merging--and thought to play a key role in the origins of life and other evolutionary transitions. These results suggest that group selection models may be applicable with fewer physical and chemical constraints than previously thought, and apply much more widely in aqueous environments.

q-bio.PE↗

arXiv2020

Social goods dilemmas in heterogeneous societies

Prosocial behaviors are encountered in the donation game, the prisoner's dilemma, relaxed social dilemmas, and public goods games. Many studies assume that the population structure is homogeneous, meaning all individuals have the same number of interaction partners, or that the social good is of one particular type. Here, we explore general evolutionary dynamics for arbitrary spatial structures and social goods. We find that heterogeneous networks, wherein some individuals have many more interaction partners than others, can enhance the evolution of prosocial behaviors. However, they often accumulate most of the benefits in the hands of a few highly-connected individuals, while many others receive low or negative payoff. Surprisingly, selection can favor producers of social goods even if the total costs exceed the total benefits. In summary, heterogeneous structures have the ability to strongly promote the emergence of prosocial behaviors, but they also create the possibility of generating large inequality.

q-bio.PE↗

arXiv2019

Evolutionary dynamics with game transitions

The environment has a strong influence on a population's evolutionary dynamics. Driven by both intrinsic and external factors, the environment is subject to continual change in nature. To capture an ever-changing environment, we consider a model of evolutionary dynamics with game transitions, where individuals' behaviors together with the games they play in one time step influence the games to be played next time step. Within this model, we study the evolution of cooperation in structured populations and find a simple rule: weak selection favors cooperation over defection if the ratio of the benefit provided by an altruistic behavior, $b$, to the corresponding cost, $c$, exceeds $k-k'$, where $k$ is the average number of neighbors of an individual and $k'$ captures the effects of the game transitions. Even if cooperation cannot be favored in each individual game, allowing for a transition to a relatively valuable game after mutual cooperation and to a less valuable game after defection can result in a favorable outcome for cooperation. In particular, small variations in different games being played can promote cooperation markedly. Our results suggest that simple game transitions can serve as a mechanism for supporting prosocial behaviors in highly-connected populations.

q-bio.PE↗

arXiv2019

Limits on amplifiers of natural selection under death-Birth updating

The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran process, and computer-assisted exhaustive searches have failed to discover amplification. In this work we resolve this disparity, by showing that any amplification under death-Birth updating is necessarily \emph{bounded} and \emph{transient}. Our boundedness result states that even if a population structure does amplify selection, the resulting fixation probability is close to that of the well-mixed population. Our transience result states that for any population structure there exists a threshold $r^*$ such that the population structure ceases to amplify selection if the mutant fitness advantage $r$ is larger than $r^\star$. Finally, we also extend the above results to $δ$-death-Birth updating, which is a combination of Birth-death and death-Birth updating. On the positive side, we identify population structures that maintain amplification for a wide range of values $r$ and $δ$. These results demonstrate that amplification of natural selection depends on the specific mechanisms of the evolutionary process.

q-bio.PE↗

arXiv2019

Evolutionary Games on Isothermal Graphs

Population structure affects the outcome of natural selection. Static population structures can be described by graphs, where individuals occupy the nodes, and interactions occur along the edges. General conditions for evolutionary success on any weighted graph were recently derived, for weak selection, in terms of coalescence times of random walks. Here we show that for a special class of graphs, the conditions for success take a particularly simple form, in which all effects of graph structure are described by the graph's "effective degree"---a measure of the effective number of neighbors per individual. This result holds for all weighted graphs that are isothermal, meaning that the sum of edge weights is the same at each node. Isothermal graphs encompass a wide variety of underlying topologies, and arise naturally from supposing that each individual devotes the same amount of time to interaction. Cooperative behavior is favored on a large isothermal graph if the benefit-to-cost ratio exceeds the effective degree. We relate the effective degree of a graph to its spectral gap, thereby providing a link between evolutionary dynamics and the theory of expander graphs. As a surprising example, we report graphs of infinite average degree that are nonetheless highly conducive for promoting cooperation.

q-bio.PE↗

arXiv2019

Reactive learning strategies for iterated games

In an iterated game between two players, there is much interest in characterizing the set of feasible payoffs for both players when one player uses a fixed strategy and the other player is free to switch. Such characterizations have led to extortionists, equalizers, partners, and rivals. Most of those studies use memory-one strategies, which specify the probabilities to take actions depending on the outcome of the previous round. Here, we consider "reactive learning strategies," which gradually modify their propensity to take certain actions based on past actions of the opponent. Every linear reactive learning strategy, $\mathbf{p}^{\ast}$, corresponds to a memory one-strategy, $\mathbf{p}$, and vice versa. We prove that for evaluating the region of feasible payoffs against a memory-one strategy, $\mathcal{C}\left(\mathbf{p}\right)$, we need to check its performance against at most $11$ other strategies. Thus, $\mathcal{C}\left(\mathbf{p}\right)$ is the convex hull in $\mathbb{R}^{2}$ of at most $11$ points. Furthermore, if $\mathbf{p}$ is a memory-one strategy, with feasible payoff region $\mathcal{C}\left(\mathbf{p}\right)$, and $\mathbf{p}^{\ast}$ is the corresponding reactive learning strategy, with feasible payoff region $\mathcal{C}\left(\mathbf{p}^{\ast}\right)$, then $\mathcal{C}\left(\mathbf{p}^{\ast}\right)$ is a subset of $\mathcal{C}\left(\mathbf{p}\right)$. Reactive learning strategies are therefore powerful tools in restricting the outcomes of iterated games.

q-bio.PE↗