39,012 research outputs found
Absorption Time of the Moran Process
The Moran process models the spread of mutations in populations on graphs. We
investigate the absorption time of the process, which is the time taken for a
mutation introduced at a randomly chosen vertex to either spread to the whole
population, or to become extinct. It is known that the expected absorption time
for an advantageous mutation is O(n^4) on an n-vertex undirected graph, which
allows the behaviour of the process on undirected graphs to be analysed using
the Markov chain Monte Carlo method. We show that this does not extend to
directed graphs by exhibiting an infinite family of directed graphs for which
the expected absorption time is exponential in the number of vertices. However,
for regular directed graphs, we show that the expected absorption time is
Omega(n log n) and O(n^2). We exhibit families of graphs matching these bounds
and give improved bounds for other families of graphs, based on isoperimetric
number. Our results are obtained via stochastic dominations which we
demonstrate by establishing a coupling in a related continuous-time model. The
coupling also implies several natural domination results regarding the fixation
probability of the original (discrete-time) process, resolving a conjecture of
Shakarian, Roos and Johnson.Comment: minor change
Absorption time of the Moran process
The Moran process models the spread of mutations in populations on graphs. We investigate the absorption time of the process, which is the time taken for a mutation introduced at a randomly chosen vertex to either spread to the whole population, or to become extinct. It is known that the expected absorption time for an advantageous mutation is O(nâ´) on an n-vertex undirected graph, which allows the behaviour of the process on undirected graphs to be analysed using the Markov chain Monte Carlo method. We show that this does not extend to directed graphs by exhibiting an infinite family of directed graphs for which the expected absorption time is exponential in the number of vertices. However, for regular directed graphs, we show that the expected absorption time is âŚ(n log n) and O(n²). We exhibit families of graphs matching these bounds and give improved bounds for other families of graphs, based on isoperimetric number. Our results are obtained via stochastic dominations which we demonstrate by establishing a coupling in a related continuous-time model. The coupling also implies several natural domination results regarding the fixation probability of the original (discrete-time) process, resolving a conjecture of Shakarian, Roos and Johnson
Martingales and the characteristic functions of absorption time on bipartite graphs
Evolutionary graph theory investigates how spatial constraints affect processes that model evolutionary selection, e.g. the Moran process. Its principal goals are to find the fixation probability and the conditional distributions of fixation time, and show how they are affected by different graphs that impose spatial constraints. Fixation probabilities have generated significant attention, but much less is known about the conditional time distributions, even for simple graphs. Those conditional time distributions are difficult to calculate, so we consider a close proxy to it: the number of times the mutant population size changes before absorption. We employ martingales to obtain the conditional characteristic functions (CCFs) of that proxy for the Moran process on the complete bipartite graph. We consider the Moran process on the complete bipartite graph as an absorbing random walk in two dimensions. We then extend Wald's martingale approach to sequential analysis from one dimension to two. Our expressions for the CCFs are novel, compact, exact, and their parameter dependence is explicit. We show that our CCFs closely approximate those of absorption time. Martingales provide an elegant framework to solve principal problems of evolutionary graph theory. It should be possible to extend our analysis to more complex graphs than we show here
Parameterised Approximation of the Fixation Probability of the Dominant Mutation in the Multi-Type Moran Process
The multi-type Moran process is an evolutionary process on a connected graph
in which each vertex has one of types and, in each step, a vertex
is chosen to reproduce its type to one of its neighbours. The probability of a
vertex being chosen for reproduction is proportional to the fitness of the
type of . So far, the literature was almost solely concerned with the
-type Moran process in which each vertex is either healthy (type ) or a
mutant (type ), and the main problem of interest has been the (approximate)
computation of the so-called fixation probability, i.e., the probability that
eventually all vertices are mutants.
In this work we initiate the study of approximating fixation probabilities in
the multi-type Moran process on general graphs. Our main result is an FPTRAS
(fixed-parameter tractable randomised approximation scheme) for computing the
fixation probability of the dominant mutation; the parameter is the number of
types and their fitnesses. In the course of our studies we also provide novel
upper bounds on the expected absorption time, i.e., the time that it takes the
multi-type Moran process to reach a state in which each vertex has the same
type.Comment: 14 page
Phase transitions of the Moran process and algorithmic consequences
The Moran process is a random process that models the spread of genetic mutations through graphs. On connected graphs, the process eventually reaches âfixationâ, where all vertices are mutants, or âextinctionâ, where none are. Our main result is an almost-tight upper bound on expected absorption time. For all Îľ > 0, we show that the expected absorption time on an n-vertex graph is o(n3+Îľ). Specifically, it is at most n3eO((log log n)3), and there is a family of graphs where it is âŚ(n3). In proving this, we establish a phase transition in the probability of fixation, depending on mutantsâ fitness r. We show that no similar phase transition occurs for digraphs, where it is already known that the expected absorption time can be exponential. Finally, we give an improved FPRAS for approximating the probability of fixation. On degree-bounded graphs where some basic properties are given, its running time is independent of the number of vertices
A probabilistic view on the deterministic mutation-selection equation: dynamics, equilibria, and ancestry via individual lines of descent
We reconsider the deterministic haploid mutation-selection equation with two
types. This is an ordinary differential equation that describes the type
distribution (forward in time) in a population of infinite size. This paper
establishes ancestral (random) structures inherent in this deterministic model.
In a first step, we obtain a representation of the deterministic equation's
solution (and, in particular, of its equilibrium) in terms of an ancestral
process called the killed ancestral selection graph. This representation allows
one to understand the bifurcations related to the error threshold phenomenon
from a genealogical point of view. Next, we characterise the ancestral type
distribution by means of the pruned lookdown ancestral selection graph and
study its properties at equilibrium. We also provide an alternative
characterisation in terms of a piecewise-deterministic Markov process.
Throughout, emphasis is on the underlying dualities as well as on explicit
results.Comment: J. Math. Biol., in pres
Fixation, transient landscape and diffusion's dilemma in stochastic evolutionary game dynamics
Agent-based stochastic models for finite populations have recently received
much attention in the game theory of evolutionary dynamics. Both the ultimate
fixation and the pre-fixation transient behavior are important to a full
understanding of the dynamics. In this paper, we study the transient dynamics
of the well-mixed Moran process through constructing a landscape function. It
is shown that the landscape playing a central theoretical "device" that
integrates several lines of inquiries: the stable behavior of the replicator
dynamics, the long-time fixation, and continuous diffusion approximation
associated with asymptotically large population. Several issues relating to the
transient dynamics are discussed: (i) multiple time scales phenomenon
associated with intra- and inter-attractoral dynamics; (ii) discontinuous
transition in stochastically stationary process akin to Maxwell construction in
equilibrium statistical physics; and (iii) the dilemma diffusion approximation
facing as a continuous approximation of the discrete evolutionary dynamics. It
is found that rare events with exponentially small probabilities, corresponding
to the uphill movements and barrier crossing in the landscape with multiple
wells that are made possible by strong nonlinear dynamics, plays an important
role in understanding the origin of the complexity in evolutionary, nonlinear
biological systems.Comment: 34 pages, 4 figure
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