New Environmental Evolutionary Graph Theorypuleo/mighty.pdf · 2014. 8. 21. · Puleo (RIT/UIUC)...

Environmental Evolutionary Graph Theory

Gregory [email protected]

Rochester Institute of TechnologyREU Host: University of Illinois at Urbana-Champaign

MIGHTY XLVIINovember 8, 2008

Puleo (RIT/UIUC) Environmental Graphs MIGHTY XLVII 1 / 14

What Is Evolutionary Graph Theory?

Lieberman, Hauert and Nowak,Nature 2005

Two species, red and blue, occupy thevertices of some graph G .

Each species has a fixed reproductivefitness.

Blue has unit fitness.Red has fitness r (r > 0).

In each update, an individual isselected to reproduce (replacing arandom neighbor) with pr.proportional to its fitness.

Fixation

This model is a stationary Markovchain.

The all-blue and all-red states areabsorbing.

When only one species remains, thatspecies has achieved fixation.

We will frequently be interested in thepr. that a randomly placed individualof one color achieves fixation:

ρR – the pr. that a single redindividual, placed uniformlyrandomly with everyone else blue,achieves fixationρB – the pr. that a lone blueindividual achieves fixation

Fixation

Adding the “Environmental”

We give the vertices backgroundcolors.

Fitness is no longer fixed, butenvironmental:

Individual gets unit fitness if its colormatches the background color.Individual gets fitness ε (0 < ε < 1)if it doesn’t match.

As before, an individual is picked forreproduction with pr. proportional toits fitness.

Cubes and Damaged Cubes

By symmetry, this cube graph must befair:

Whatever the value of ε,

ρR = ρB = 1/8

What about a damaged cube?

Damaged cubes are also fair:

ρR = ρB = 1/7

ρR = ρB = 1/8

ρR = ρB = 1/7

ρR = ρB = 1/8

ρR = ρB = 1/7

ρR = ρB = 1/8

ρR = ρB = 1/7

Hangers are Unfair

On the other hand, many graphs giveone color an advantage.

This “hanger” graph favors red:

ε ρR ρB

.9 .256 .243

.5 .284 .201

.1 .319 .080

Hangers are Unfair

ε ρR ρB

.9 .256 .243

.5 .284 .201

.1 .319 .080

Hangers are Unfair

ε ρR ρB

.9 .256 .243

.5 .284 .201

.1 .319 .080

A Star Graph

This star graph is fair.

ρR = ρB = 1/8

Question

Can we characterize fair graphs?

A Star Graph

ρR = ρB = 1/8

Question

A Star Graph

ρR = ρB = 1/8

Question

Properly Two-Colored Graphs

Definition

A graph is properly two-colored if no twovertices with the same background colorare adjacent.

Theorem

Let G be a properly two-colored graph, letS be any state of G, let v be any vertex ofG, and let w be any opposite-colorneighbor of v . Then φ(v) = φ(w).

Definition

Theorem

Definition

Theorem

Definition

Theorem

Notation for the Fixation Probability Vector

Let ~x be a 2n-vector where the entryxs is the pr. red fixates starting fromthe state S .

Identify each state S with the set ofvertices containing red individuals.

Define the quantity γ by

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

γ =1∑

v∈V (G)1

deg(v)

A Value for the Fixation Probability Vector

Theorem

For a properly two-colored graph G,

xS = γ∑v∈S

deg(v)

Corollary

All properly two-colored graphs are fair.

For the current state of the graph atleft, this yields

xS =10

))≈ .426

Theorem

xS = γ∑v∈S

deg(v)

Corollary

xS =10

))≈ .426

Theorem

xS = γ∑v∈S

deg(v)

Corollary

xS =10

))≈ .426

Proving Two-Colored Graphs are Fair

Proof.

For each vertex v , let yv = x{v}.

yv =γ

deg(v)

1deg(v)∑

w∈V (G)1

deg(w)

Now, consider this formula for ρR:

ρR =∑

v∈V (G)

Proof.

yv =γ

deg(v)

1deg(v)∑

w∈V (G)1

deg(w)

ρR =∑

v∈V (G)

Proof.

yv =γ

deg(v)=

1deg(v)∑

w∈V (G)1

deg(w)

ρR =∑

v∈V (G)

Proof.

yv =γ

deg(v)=

1deg(v)∑

w∈V (G)1

deg(w)

ρR =∑

v∈V (G)

Proof.

yv =γ

deg(v)=

1deg(v)∑

w∈V (G)1

deg(w)

ρR =1

∑v∈V (G)

Proof.

yv =γ

deg(v)=

1deg(v)∑

w∈V (G)1

deg(w)

ρR =1

∑v∈V (G)

Proof.

yv =γ

deg(v)=

1deg(v)∑

w∈V (G)1

deg(w)

ρR =1

∑v∈V (G)

ρR =1

Proof.

yv =γ

deg(v)=

1deg(v)∑

w∈V (G)1

deg(w)

ρR =1

∑v∈V (G)

ρB = ρR =1

Does the Converse Hold?

Proposition

All fair graphs are properly two-colored.

Open Question

What is the general rule characterizing fairgraphs?

Proposition

Open Question

Proposition

Open Question

Proposition

Open Question

Future Work

Open Question

Is there a nice closed-form expression for the entries of ~x on arbitrarygraphs?

Open Question

Is there an efficient algorithm for numerically calculating fixationprobabilities?

Generalizations and Extensions

Different types of graph

Directed graphsWeighted edges

Different “ecology”

Species-specific fitnessMore than two foreground/background colors

Future Work

Open Question

Future Work

Open Question

New Environmental Evolutionary Graph Theorypuleo/mighty.pdf · 2014. 8. 21. · Puleo (RIT/UIUC)...

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