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Evolutionary Graph Theory
Uses of graphical framework• Graph can represent relationships in a social
network of humans• E.g. “Six degrees of separation”
You Dad David:dad’s friend from US
George:David’s college mate
George’s wife
Michelle
Mr. Obama
• It can also analyze the effect of population structure on evolutionary dynamics…
Uses of graphical framework
The Basic Idea
• Vertices – individuals in a population
• Edge – competitive interaction
• wij = Probability that an offspring of i replaces j
• Edge can go both directions and so describes a digraph
i j
wij
i j
wij
The Basic Idea• Label all the individuals in the population with
i=1,2,…,N• Represent each with a vertex• At each time step, choose a random individual
for reproduction• Determine direction of edge
• Every edge has weight = wij – wij >0 → an edge from i to j
– Wij =0 → no edge from i to j
• Hence process determined by W=[wij]; 0< wij <1
• The matrix W defines a weighted digraph
Moran Process
• Consider a homogeneous population of size N consisting of residents (white) and mutants (black).
Moran Process
• At each time step, choose an individual for reproduction with a probability proportional to its fitness
• Here, a resident is selected for reproduction
Moran Process
• A randomly chosen individual is eliminated
• Here, a mutant is selected for death
Moran Process
• The offspring replaces the eliminated individual.
Why Moran Process?
• Represents the simplest possible stochastic model to study selection in a finite population
• Where 2 individuals are chosen at each time step
• One for reproduction & one for elimination
• Offspring of first replaces the second
• Total population size, N, is strictly constant
Moran Process
• Represented by a complete graph with identical weights
• An unstructured population is given by a complete graph: an edge btw any 2 vertices
• Evolutionary process is equivalent to the Moran process
Moran Process• Fixation probability: probability that a mutant invading
a population of N -1 residents will produce a lineage that takes over the whole population
• Fixation probability α evolution rate• Suppose
all resident individuals are identical and one new mutant is introduced
new mutant has relative fitness r, as compared to the residents, whose fitness is 1
• Fixation probability of the mutant is then given by: R = (1-1/r)/(1-1/rN)
Evolutionary Suppressors• Line
• Burst
The line• Suppose N individuals are arranged in a linear chain:
• Each individual places its offspring into the position immediately to its right.
• The leftmost individual is never replaced. • Mutant can only reach fixation if it arises in the
leftmost position, which happens with probability 1/N.
• Fixation probability = 1/N, independent of r
The Burst• A new mutant can only reach fixation if it arises
in the center:
• Probability that a randomly placed mutant originates in the center = 1/N
• Hence fixation probability is again independent of r, the relative fitness of the new mutant
• Represent suppressors of selection • All mutants – irresp of their fitness – have the
same fixation probability as a neutral mutant in the Moran process
Evolutionary Amplifiers
• Star • Superstar
The Star
• For a large N, a mutant with a relative fitness r has a fixation probability
• ρ = (1-1/r2)/(1-1/r2N). • So, a relative fitness r on a
star is equivalent to a relative fitness r2 in the Moran process
• Thus, a star is an amplifier of selection
The Superstar
• l= no. of leaves• m= no. of loops in a leaf• k= the length of each loop• for sufficiently large N, a
super-star of parameter k satisfies:
• Fixation probability• The superstar amplifies a
selective difference r to rk
• A powerful amplifier of selection!
l=5
k=3
m=5
References
• Nowak, Martin A. “Evolutionary Graph Theory.” Evolutionary Dynamics: exploring the equations of life. Cambridge, Massachusetts, and London : Belknap Press of Harvard University Press, 2006: 123-144.
• Graph images from: Lieberman, Erez and Hauert , Christoph and Nowak, Martin A. “Evolutionary dynamics on graphs.” Nature. 433 (2005): 312-316.
• Image in slide #3 from: http://en.wikipedia.org/wiki/6_degrees_of_separation