TY - THES
AB - In this thesis we study certain mathematical aspects of evolution. The two primary forces that drive an evolutionary process are mutation and selection. Mutation generates new variants in a population. Selection chooses among the variants depending on the reproductive rates of individuals. Evolutionary processes are intrinsically random – a new mutation that is initially present in the population at low frequency can go extinct, even if it confers a reproductive advantage. The overall rate of evolution is largely determined by two quantities: the probability that an invading advantageous mutation spreads through the population (called fixation probability) and the time until it does so (called fixation time). Both those quantities crucially depend not only on the strength of the invading mutation but also on the population structure. In this thesis, we aim to understand how the underlying population structure affects the overall rate of evolution. Specifically, we study population structures that increase the fixation probability of advantageous mutants (called amplifiers of selection). Broadly speaking, our results are of three different types: We present various strong amplifiers, we identify regimes under which only limited amplification is feasible, and we propose population structures that provide different tradeoffs between high fixation probability and short fixation time.
AU - Tkadlec, Josef
ID - 7196
TI - A role of graphs in evolutionary processes
ER -
TY - JOUR
AB - 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.
AU - Tkadlec, Josef
AU - Pavlogiannis, Andreas
AU - Chatterjee, Krishnendu
AU - Nowak, Martin A.
ID - 7212
JF - PLoS computational biology
TI - Limits on amplifiers of natural selection under death-Birth updating
VL - 16
ER -
TY - JOUR
AB - 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.
AU - Tkadlec, Josef
AU - Pavlogiannis, Andreas
AU - Chatterjee, Krishnendu
AU - Nowak, Martin A.
ID - 7210
JF - Communications Biology
SN - 2399-3642
TI - Population structure determines the tradeoff between fixation probability and fixation time
VL - 2
ER -
TY - GEN
AB - The input to the token swapping problem is a graph with vertices v1, v2, . . . , vn, and n tokens with labels 1,2, . . . , n, one on each vertex. The goal is to get token i to vertex vi for all i= 1, . . . , n using a minimum number of swaps, where a swap exchanges the tokens on the endpoints of an edge.Token swapping on a tree, also known as “sorting with a transposition tree,” is not known to be in P nor NP-complete. We present some partial results:
1. An optimum swap sequence may need to perform a swap on a leaf vertex that has the correct token (a “happy leaf”), disproving a conjecture of Vaughan.
2. Any algorithm that fixes happy leaves—as all known approximation algorithms for the problem do—has approximation factor at least 4/3. Furthermore, the two best-known 2-approximation algorithms have approximation factor exactly 2.
3. A generalized problem—weighted coloured token swapping—is NP-complete on trees, but solvable in polynomial time on paths and stars. In this version, tokens and vertices have colours, and colours have weights. The goal is to get every token to a vertex of the same colour, and the cost of a swap is the sum of the weights of the two tokens involved.
AU - Biniaz, Ahmad
AU - Jain, Kshitij
AU - Lubiw, Anna
AU - Masárová, Zuzana
AU - Miltzow, Tillmann
AU - Mondal, Debajyoti
AU - Naredla, Anurag Murty
AU - Tkadlec, Josef
AU - Turcotte, Alexi
ID - 7950
T2 - arXiv
TI - Token swapping on trees
ER -
TY - JOUR
AB - We consider a class of students learning a language from a teacher. The situation can be interpreted as a group of child learners receiving input from the linguistic environment. The teacher provides sample sentences. The students try to learn the grammar from the teacher. In addition to just listening to the teacher, the students can also communicate with each other. The students hold hypotheses about the grammar and change them if they receive counter evidence. The process stops when all students have converged to the correct grammar. We study how the time to convergence depends on the structure of the classroom by introducing and evaluating various complexity measures. We find that structured communication between students, although potentially introducing confusion, can greatly reduce some of the complexity measures. Our theory can also be interpreted as applying to the scientific process, where nature is the teacher and the scientists are the students.
AU - Ibsen-Jensen, Rasmus
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 198
IS - 140
JF - Journal of the Royal Society Interface
TI - Language acquisition with communication between learners
VL - 15
ER -
TY - JOUR
AB - Indirect reciprocity explores how humans act when their reputation is at stake, and which social norms they use to assess the actions of others. A crucial question in indirect reciprocity is which social norms can maintain stable cooperation in a society. Past research has highlighted eight such norms, called “leading-eight” strategies. This past research, however, is based on the assumption that all relevant information about other population members is publicly available and that everyone agrees on who is good or bad. Instead, here we explore the reputation dynamics when information is private and noisy. We show that under these conditions, most leading-eight strategies fail to evolve. Those leading-eight strategies that do evolve are unable to sustain full cooperation.Indirect reciprocity is a mechanism for cooperation based on shared moral systems and individual reputations. It assumes that members of a community routinely observe and assess each other and that they use this information to decide who is good or bad, and who deserves cooperation. When information is transmitted publicly, such that all community members agree on each other’s reputation, previous research has highlighted eight crucial moral systems. These “leading-eight” strategies can maintain cooperation and resist invasion by defectors. However, in real populations individuals often hold their own private views of others. Once two individuals disagree about their opinion of some third party, they may also see its subsequent actions in a different light. Their opinions may further diverge over time. Herein, we explore indirect reciprocity when information transmission is private and noisy. We find that in the presence of perception errors, most leading-eight strategies cease to be stable. Even if a leading-eight strategy evolves, cooperation rates may drop considerably when errors are common. Our research highlights the role of reliable information and synchronized reputations to maintain stable moral systems.
AU - Hilbe, Christian
AU - Schmid, Laura
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 2
IS - 48
JF - PNAS
TI - Indirect reciprocity with private, noisy, and incomplete information
VL - 115
ER -
TY - JOUR
AB - Because of the intrinsic randomness of the evolutionary process, a mutant with a fitness advantage has some chance to be selected but no certainty. Any experiment that searches for advantageous mutants will lose many of them due to random drift. It is therefore of great interest to find population structures that improve the odds of advantageous mutants. Such structures are called amplifiers of natural selection: they increase the probability that advantageous mutants are selected. Arbitrarily strong amplifiers guarantee the selection of advantageous mutants, even for very small fitness advantage. Despite intensive research over the past decade, arbitrarily strong amplifiers have remained rare. Here we show how to construct a large variety of them. Our amplifiers are so simple that they could be useful in biotechnology, when optimizing biological molecules, or as a diagnostic tool, when searching for faster dividing cells or viruses. They could also occur in natural population structures.
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin A.
ID - 5751
IS - 1
JF - Communications Biology
SN - 2399-3642
TI - Construction of arbitrarily strong amplifiers of natural selection using evolutionary graph theory
VL - 1
ER -
TY - JOUR
AB - The fixation probability is the probability that a new mutant introduced in a homogeneous population eventually takes over the entire population. The fixation probability is a fundamental quantity of natural selection, and known to depend on the population structure. Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. In this work we focus on symmetric population structures represented as undirected graphs. In the regime of undirected graphs, the strongest amplifier known has been the Star graph, and the existence of undirected graphs with stronger amplification properties has remained open for over a decade. In this work we present the Comet and Comet-swarm families of undirected graphs. We show that for a range of fitness values of the mutants, the Comet and Cometswarm graphs have fixation probability strictly larger than the fixation probability of the Star graph, for fixed population size and at the limit of large populations, respectively.
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 512
IS - 1
JF - Scientific Reports
SN - 20452322
TI - Amplification on undirected population structures: Comets beat stars
VL - 7
ER -
TY - DATA
AB - Strong amplifiers of natural selection
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak , Martin
ID - 5559
KW - natural selection
TI - Strong amplifiers of natural selection
ER -
TY - GEN
AB - The fixation probability is the probability that a new mutant introduced in a homogeneous population eventually takes over the entire population.
The fixation probability is a fundamental quantity of natural selection, and known to depend on the population structure.
Amplifiers of natural selection are population structures which increase the fixation probability of advantageous mutants, as compared to the baseline case of well-mixed populations. In this work we focus on symmetric population structures represented as undirected graphs. In the regime of undirected graphs, the strongest amplifier known has been the Star graph, and the existence of undirected graphs with stronger amplification properties has remained open for over a decade.
In this work we present the Comet and Comet-swarm families of undirected graphs. We show that for a range of fitness values of the mutants, the Comet and Comet-swarm graphs have fixation probability strictly larger than the fixation probability of the Star graph, for fixed population size and at the limit of large populations, respectively.
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 5449
SN - 2664-1690
TI - Amplification on undirected population structures: Comets beat stars
ER -
TY - GEN
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 5451
SN - 2664-1690
TI - Strong amplifiers of natural selection
ER -
TY - GEN
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 5452
SN - 2664-1690
TI - Arbitrarily strong amplifiers of natural selection
ER -
TY - GEN
AU - Pavlogiannis, Andreas
AU - Tkadlec, Josef
AU - Chatterjee, Krishnendu
AU - Nowak, Martin
ID - 5453
SN - 2664-1690
TI - Arbitrarily strong amplifiers of natural selection
ER -
TY - CONF
AB - Balanced knockout tournaments are ubiquitous in sports competitions and are also used in decisionmaking and elections. The traditional computational question, that asks to compute a draw (optimal draw) that maximizes the winning probability for a distinguished player, has received a lot of attention. Previous works consider the problem where the pairwise winning probabilities are known precisely, while we study how robust is the winning probability with respect to small errors in the pairwise winning probabilities. First, we present several illuminating examples to establish: (a) there exist deterministic tournaments (where the pairwise winning probabilities are 0 or 1) where one optimal draw is much more robust than the other; and (b) in general, there exist tournaments with slightly suboptimal draws that are more robust than all the optimal draws. The above examples motivate the study of the computational problem of robust draws that guarantee a specified winning probability. Second, we present a polynomial-time algorithm for approximating the robustness of a draw for sufficiently small errors in pairwise winning probabilities, and obtain that the stated computational problem is NP-complete. We also show that two natural cases of deterministic tournaments where the optimal draw could be computed in polynomial time also admit polynomial-time algorithms to compute robust optimal draws.
AU - Chatterjee, Krishnendu
AU - Ibsen-Jensen, Rasmus
AU - Tkadlec, Josef
ID - 1182
TI - Robust draws in balanced knockout tournaments
VL - 2016-January
ER -