scholarly journals Martingales and the characteristic functions of absorption time on bipartite graphs

2021 ◽  
Vol 8 (10) ◽  
Author(s):  
Travis Monk ◽  
André van Schaik
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Two Dimensions ◽  
Characteristic Functions ◽  
Spatial Constraints ◽  
Absorption Time ◽  
Moran Process ◽  
Conditional Time ◽  
Evolutionary Graph Theory

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.

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

2020 ◽  
Vol 476 (2241) ◽  
pp. 20200135
Author(s):  
Travis Monk ◽  
André van Schaik
Keyword(s):  
Stochastic Processes ◽  
Death Process ◽  
Characteristic Functions ◽  
Evolutionary Models ◽  
Conditional Distributions ◽  
Absorption Time ◽  
Moran Process ◽  
Conditional Time ◽  
Fixation Probabilities ◽  
Absorption Probabilities

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.

scholarly journals Wald’s martingale and the Moran process

2020 ◽  
Author(s):  
Travis Monk ◽  
André van Schaik
Keyword(s):  
Stochastic Processes ◽  
Population Size ◽  
Death Process ◽  
Characteristic Functions ◽  
Conditional Distributions ◽  
Mutant Population ◽  
Absorption Time ◽  
Moran Process ◽  
Conditional Time ◽  
Birth Death

AbstractMany models of evolution are stochastic processes, where some quantity of interest fluctuates randomly in time. One classic example is the Moran birth-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. The parameter dependence of the characteristic functions is explicit, so it is easy to explore their properties in parameter space. We also use them 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 we can quickly and tractably obtain absorption probabilities and times of evolutionary stochastic processes.Author summaryThe Moran process is a probabilistic birth-death model of evolution. A mutant is introduced to an indigenous population, and we randomly choose organisms to live or die on subsequent time steps. Our goals are to calculate the probabilities that the mutant eventually dominates the population or goes extinct, and the distribution of time it requires to do so. The conditional distributions of time are difficult to obtain for the Moran process, so we consider a slightly different but related problem. We instead calculate the conditional distributions of the number of times that the mutant population size changes before it dominates the population or goes extinct. We use a martingale identified by Abraham Wald to obtain elegant and exact expressions for those distributions. We then use them to approximate conditional time distributions, and we show when that approximation is accurate. Our analysis outlines the basic concepts martingales and demonstrates why they are a formidable tool for studying probabilistic evolutionary models such as the Moran process.

6. Graphs

2016 ◽  
pp. 86-108
Author(s):  
Robin Wilson
Keyword(s):  
Graph Theory ◽  
Bipartite Graph ◽  
Complete Graph ◽  
Planar Graphs ◽  
Complete Bipartite Graph ◽  
Travelling Salesman Problem ◽  
Chemical Bonds ◽  
Hamiltonian Graphs ◽  
Road Map ◽  
Cycle Graph

Graph theory is about collections of points that are joined in pairs, such as a road map with towns connected by roads or a molecule with atoms joined by chemical bonds. ‘Graphs’ revisits the Königsberg bridges problem, the knight’s tour problem, the Gas–Water–Electricity problem, the map-colour problem, the minimum connector problem, and the travelling salesman problem and explains how they can all be considered as problems in graph theory. It begins with an explanation of a graph and describes the complete graph, the complete bipartite graph, and the cycle graph, which are all simple graphs. It goes on to describe trees in graph theory, Eulerian and Hamiltonian graphs, and planar graphs.

scholarly journals Evolutionary graph theory revisited: when is an evolutionary process equivalent to the Moran process?

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150334 ◽  
Author(s):  
Karan Pattni ◽  
Mark Broom ◽  
Jan Rychtář ◽  
Lara J. Silvers
Keyword(s):  
Graph Theory ◽  
Evolutionary Dynamics ◽  
Evolutionary Process ◽  
Structured Populations ◽  
Fixation Probability ◽  
Finite Populations ◽  
Moran Model ◽  
Moran Process ◽  
Evolutionary Graph Theory ◽  
Associated Matrix

Evolution in finite populations is often modelled using the classical Moran process. Over the last 10 years, this methodology has been extended to structured populations using evolutionary graph theory. An important question in any such population is whether a rare mutant has a higher or lower chance of fixating (the fixation probability) than the Moran probability, i.e. that from the original Moran model, which represents an unstructured population. As evolutionary graph theory has developed, different ways of considering the interactions between individuals through a graph and an associated matrix of weights have been considered, as have a number of important dynamics. In this paper, we revisit the original paper on evolutionary graph theory in light of these extensions to consider these developments in an integrated way. In particular, we find general criteria for when an evolutionary graph with general weights satisfies the Moran probability for the set of six common evolutionary dynamics.

Graceful Labeling for Some Complete Bipartite Graph

2018 ◽  
Vol 9 (12) ◽  
pp. 2147-2152
Author(s):  
V. Raju ◽  
M. Paruvatha vathana
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Graceful Labeling

On the Crossing Number of $K_{m,n}$

10.37236/1748 ◽  
2003 ◽  
Vol 10 (1) ◽  
Author(s):  
Nagi H. Nahas
Keyword(s):  
Lower Bound ◽  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number

The best lower bound known on the crossing number of the complete bipartite graph is : $$cr(K_{m,n}) \geq (1/5)(m)(m-1)\lfloor n/2 \rfloor \lfloor(n-1)/2\rfloor$$ In this paper we prove that: $$cr(K_{m,n}) \geq (1/5)m(m-1)\lfloor n/2 \rfloor \lfloor (n-1)/2 \rfloor + 9.9 \times 10^{-6} m^2n^2$$ for sufficiently large $m$ and $n$.

scholarly journals Join Products K2,3 + Cn

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 925
Author(s):  
Michal Staš
Keyword(s):  
Bipartite Graph ◽  
Complete Bipartite Graph ◽  
Crossing Number ◽  
Arithmetic Means ◽  
Minimum Number ◽  
Edge Crossings

The crossing number cr ( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. The main goal of the paper is to state the crossing number of the join product K 2 , 3 + C n for the complete bipartite graph K 2 , 3 , where C n is the cycle on n vertices. In the proofs, the idea of a minimum number of crossings between two distinct configurations in the various forms of arithmetic means will be extended. Finally, adding one more edge to the graph K 2 , 3 , we also offer the crossing number of the join product of one other graph with the cycle C n .

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