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