Amplifiers for the Moran Process

The Moran process, as studied by Lieberman, Hauert, and Nowak, is a randomised algorithm modelling the spread of genetic mutations in populations. The algorithm runs on an underlying graph where individuals correspond to vertices. Initially, one vertex (chosen uniformly at random) possesses a mutation, with fitness r > 1. All other individuals have fitness 1. During each step of the algorithm, an individual is chosen with probability proportional to its fitness, and its state (mutant or nonmutant) is passed on to an out-neighbour which is chosen uniformly at random. If the underlying graph is strongly connected, then the algorithm will eventually reach fixation, in which all individuals are mutants, or extinction, in which no individuals are mutants. An infinite family of directed graphs is said to be strongly amplifying if, for every r > 1, the extinction probability tends to 0 as the number of vertices increases. A formal definition is provided in the article. Strong amplification is a rather surprising property—it means that in such graphs, the fixation probability of a uniformly placed initial mutant tends to 1 even though the initial mutant only has a fixed selective advantage of r > 1 (independently of n). The name “strongly amplifying” comes from the fact that this selective advantage is “amplified.” Strong amplifiers have received quite a bit of attention, and Lieberman et al. proposed two potentially strongly amplifying families—superstars and metafunnels. Heuristic arguments have been published, arguing that there are infinite families of superstars that are strongly amplifying. The same has been claimed for metafunnels. In this article, we give the first rigorous proof that there is an infinite family of directed graphs that is strongly amplifying. We call the graphs in the family “megastars.” When the algorithm is run on an n-vertex graph in this family, starting with a uniformly chosen mutant, the extinction probability is roughly n^(−1/2) (up to logarithmic factors). We prove that all infinite families of superstars and metafunnels have larger extinction probabilities (as a function of n). Finally, we prove that our analysis of megastars is fairly tight—there is no infinite family of megastars such that the Moran algorithm gives a smaller extinction probability (up to logarithmic factors). Also, we provide a counterexample which clarifies the literature concerning the isothermal theorem of Lieberman et al

Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating

Populations evolve by accumulating advantageous mutations. Every population has some spatial structure that can be modeled by an underlying network. The network then influences the probability that new advantageous mutations fixate. Amplifiers of selection are networks that increase the fixation probability of advantageous mutants, as compared to the unstructured fully-connected network. Whether or not a network is an amplifier depends on the choice of the random process that governs the evolutionary dynamics. Two popular choices are Moran process with Birth-death updating and Moran process with death-Birth updating. %Moran process has two popular versions called Birth-death updating and death-Birth updating. Interestingly, while some networks are amplifiers under Birth-death updating and other networks are amplifiers under death-Birth updating, no network is known to function as an amplifier under both types of updating simultaneously. In this work, we identify networks that act as amplifiers of selection under both versions of the Moran process. The amplifiers are robust, modular, and increase fixation probability for any mutant fitness advantage in a range $r\in(1,1.2)$. To complement this positive result, we also prove that for certain quantities closely related to fixation probability, it is impossible to improve them simultaneously for both versions of the Moran process. Together, our results highlight how the two versions of the Moran process differ and what they have in common

Randomised Algorithms on Networks

Networks form an indispensable part of our lives. In particular, computer networks have ranked amongst the most influential networks in recent times. In such an ever-evolving and fast growing network, the primary concern is to understand and analyse different aspects of the network behaviour, such as the quality of service and efficient information propagation. It is also desirable to predict the behaviour of a large computer network if, for example, one of the computers is infected by a virus. In all of the aforementioned cases, we need protocols that are able to make local decisions and handle the dynamic changes in the network topology. Here, randomised algorithms are preferred because many deterministic algorithms often require a central control. In this thesis, we investigate three network-based randomised algorithms, threshold load balancing with weighted tasks, the pull-Moran process and the coalescing-branching random walk. Each of these algorithms has extensive applicability within networks and computational complexity within computer science. In this thesis we investigate threshold-based load balancing protocols. We introduce a generalisation of protocols in [2, 3] to weighted tasks. This thesis also analyses an evolutionary-based process called the death-birth update, defined here as the Pull-Moran process. We show that a class of strong universal amplifiers does not exist for the Pull-Moran process. We show that any class of selective amplifiers in the (standard) Moran process is a class of selective suppressors under the Pull-Moran process. We then introduce a class of selective amplifiers called Punk graphs. Finally, we improve the broadcasting time of the coalescing-branching (COBRA) walk analysed in [4], for random regular graphs. Here, we look into the COBRA approach as a randomised rumour spreading protocol

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 bounded and 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⋆. 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

Amplifiers and Suppressors of Selection for the Moran Process on Undirected Graphs

We consider the classic Moran process modeling the spread of genetic mutations, as extended to structured populations by Lieberman et al. (Nature, 2005). In this process, individuals are the vertices of a connected graph G. Initially, there is a single mutant vertex, chosen uniformly at random. In each step, a random vertex is selected for reproduction with a probability proportional to its fitness: mutants have fitness r > 1, while non-mutants have fitness 1. The vertex chosen to reproduce places a copy of itself to a uniformly random neighbor in G, replacing the individual that was there. The process ends when the mutation either reaches fixation (i.e., all vertices are mutants), or gets extinct. The principal quantity of interest is the probability with which each of the two outcomes occurs. A problem that has received significant attention recently concerns the existence of families of graphs, called strong amplifiers of selection, for which the fixation probability tends to 1 as the order n of the graph increases, and the existence of strong suppressors of selection, for which this probability tends to 0. For the case of directed graphs, it is known that both strong amplifiers and suppressors exist. For the case of undirected graphs, however, the problem has remained open, and the general belief has been that neither strong amplifiers nor suppressors exist. In this paper we disprove this belief, by providing the first examples of such graphs. The strong amplifier we present has fixation probability 1 − ˜ O(n^ −1/3), and the strong suppressor has fixation probability ˜O(n^−1/4). Both graph constructions are surprisingly simple. We also prove a general upper bound of 1 − ˜Ω(n^−1/3) on the fixation probability of any undirected graph. Hence, our strong amplifier is existentially optimal

Strong Amplifiers of Natural Selection: Proofs

We consider the modified Moran process on graphs to study the spread of genetic and cultural mutations on structured populations. An initial mutant arises either spontaneously (aka \emph{uniform initialization}), or during reproduction (aka \emph{temperature initialization}) in a population of $n$ individuals, and has a fixed fitness advantage $r>1$ over the residents of the population. The fixation probability is the probability that the mutant takes over the entire population. Graphs that ensure fixation probability of~1 in the limit of infinite populations are called \emph{strong amplifiers}. Previously, only a few examples of strong amplifiers were known for uniform initialization, whereas no strong amplifiers were known for temperature initialization. In this work, we study necessary and sufficient conditions for strong amplification, and prove negative and positive results. We show that for temperature initialization, graphs that are unweighted and/or self-loop-free have fixation probability upper-bounded by $1-1/f(r)$, where $f(r)$ is a function linear in $r$. Similarly, we show that for uniform initialization, bounded-degree graphs that are unweighted and/or self-loop-free have fixation probability upper-bounded by $1-1/g(r,c)$, where $c$ is the degree bound and $g(r,c)$ a function linear in $r$. Our main positive result complements these negative results, and is as follows: every family of undirected graphs with (i)~self loops and (ii)~diameter bounded by $n^{1-\epsilon}$, for some fixed $\epsilon>0$, can be assigned weights that makes it a strong amplifier, both for uniform and temperature initialization

Counterintuitive properties of the fixation time in network-structured populations

Evolutionary dynamics on graphs can lead to many interesting and counterintuitive findings. We study the Moran process, a discrete time birth-death process, that describes the invasion of a mutant type into a population of wild-type individuals. Remarkably, the fixation probability of a single mutant is the same on all regular networks. But non-regular networks can increase or decrease the fixation probability. While the time until fixation formally depends on the same transition probabilities as the fixation probabilities, there is no obvious relation between them. For example, an amplifier of selection, which increases the fixation probability and thus decreases the number of mutations needed until one of them is successful, can at the same time slow down the process of fixation. Based on small networks, we show analytically that (i) the time to fixation can decrease when links are removed from the network and (ii) the node providing the best starting conditions in terms of the shortest fixation time depends on the fitness of the mutant. Our results are obtained analytically on small networks, but numerical simulations show that they are qualitatively valid even in much larger populations

Suppressors of selection

Inspired by recent works on evolutionary graph theory, an area of growing interest in mathematical and computational biology, we present the first known examples of undirected structures acting as suppressors of selection for any fitness value $r > 1$. This means that the average fixation probability of an advantageous mutant or invader individual placed at some node is strictly less than that of this individual placed in a well-mixed population. This leads the way to study more robust structures less prone to invasion, contrary to what happens with the amplifiers of selection where the fixation probability is increased on average for advantageous invader individuals. A few families of amplifiers are known, although some effort was required to prove it. Here, we use computer aided techniques to find an exact analytical expression of the fixation probability for some graphs of small order (equal to $6$, $8$ and $10$) proving that selection is effectively reduced for $r > 1$. Some numerical experiments using Monte Carlo methods are also performed for larger graphs.Comment: New title, improved presentation, and further examples. Supporting Information is also include