Cmpe59e Evolutionary Dynamics
Evolutionary Graph Theory
Halil ERTAN
Bogazici University
Computer Engineering Department
April /12 /2011
CONTENT
 Introduction
 The Basic Idea
 First Observations
 The Isothermal Theorem
 Suppressing Selection
 Amplifying Selection
 Circulations
 Games on Graphs
 Research Questions
INTRODUCTION
 The effect of population structure on evolutionary
dynamics is our concern.
 Particular graphs can accelerate or decelerate the rate of
evolution by increasing or decreasing the fixation
probabilities of advantageous mutants.
 It is assumed that the graph does not change on time
scale under consideration.
INTRODUCTION
 Biology interpretation : How evolutionary dynamics
are affected by population structure.
 Social interpretation : ―We can ask, for example,
which networks are well suited to ensure the spread of
favorable concepts‖ (Lieberman et al., 2005)
INTRODUCTION
THE BASIC IDEA
 The probability that the offspring of i replaces j is given by
wij. If wij >0 there is an edge between from vertex i to j. If
wij = 0, there is no edge leading from vertex i to j.
THE BASIC IDEA
 Moran process is recovered as the special case of the
complete graph with identical weights, wij = 1/N for all
i and j. In each time step, an individual is chosen for
reproduction at random but proportional to its fitness.
FIRST OBSERVATIONS
What is the fixation probability of a new mutant that
arises at a random position on a graph?
 The Directed Cycle
 The Cycle
 The Line
 The Burst
The Directed Cycle
 Initially all individuals are of type A.
 After some time a mutant B is generated who gives rise
to a lineage :
- it will eventually dies
- or it takes over the whole population
The Directed Cycle
 Let m denote the number of B individuals.
 To reduce m by one, the A individual preceding the B
cluster must be chosen for reproduction.
 To increase m by one, B individual at the end of cluster
has to be chosen for reproduction.
Directed Cycle
 The ratio of these two probabilities,
 Finally, we get
 Thus, the fixation probability on a directed cycle is identical
to the fixation probability in the Moran process.
The Cycle
 Each individual can place its offspring into any one of the
two adjacent places.
The Line
 From vertex i the offspring can be placed into vertex
i+1.
 Vertex N places its offspring onto itself.
 No edge leads to vertex 1.
 With probability 1/N, the mutant arises in position i=1,
and its offspring lineage will take over the population.
The Burst
 There is one central vertex and N-1 peripheral vertices.
Edges lead from the center to the periphery. The mutant
must arise in the central position.
 The line and burst are suppressors of selection.
Balancing Drift and Selection
 If a graph G, has the same fixation probability as the
Moran process, then G is p-equivalent to the Moran
process.
 For an advantageous mutant, (r>1), if pG > pM, then G
favors selection over drift.
 For an advantageous mutant, (r>1), if pG < pM, then G
favors drift over selection.
Balancing Drift and Selection
 For a disadvantageous mutant, (r<1), if pG > pM, then G
favors drift over selection.
 For a disadvantageous mutant, (r<1), if pG < pM, then G
favors selection over drift.
 If pG = 1/N, then G is the strongest possible suppressor
of selection.
 Cycle and Directed Cycle are both p-equivalent to
Moran process, whereas line and burst completely
eliminate selection.
THE ISOTHERMAL THEOREM
The temperature of a vertex is defined as the sum of all
weights that lead into that vertex.
A vertex with a high temperature will change more
often than a vertex with a low temperature. If all the
vertices have the same temperature, then a
graph is isothermal.
THE ISOTHERMAL THEOREM
 Isothermal Theorem: A graph is p-equivalent to the
Moran process if and only if it is isothermal.
 For an isothermal graph, we have
= constant.
Since
it follows that
 The matrix W is doubly stochastic which means all rows
and all columns sum to one.
THE ISOTHERMAL THEOREM
 All symmetric graphs (W = WT) have the same fixation
probability as the Moran process, and they are
isothermal like the cycle.
 Many asymmetric graphs are also isothermal(the
directed cycle is asymmetric graph and isothermal,
however, the line is asymmetric but not isothermal)
 A symmetric graph:
SUPPRESSING SELECTION
 A root is a vertex that has no edge leading to it which
means zero temperature.
 If a graph is one-rooted, then it has fixation probability
1/N. Every one rooted graph completely eliminates
selection.
SUPPRESSING SELECTION
 If a graph has multiple roots, then any lineage arising
from a single mutant can never take over the whole
population.
 If a mutant arises in one of the roots, then it will give
rise to a lineage that will never become extinct. Thus,
graphs with multiple roots allow coexistence of different
lineages. A multiple root graph example:
AMPLIFYING SELECTION
 STAR : The relevant fitness r on a star is equivalent to a
relative fitness r2 in the Moran process.
 The star is an amplifier of selection. An advantageous
mutant r>1 behaves like an advantageous mutant with r2
in a standard Moran process, and vice a versa for
disadvantageous mutant.
AMPLIFYING SELECTION
 SUPERSTAR : The superstar amplifies a selective
difference r to rk, as the number of leaves and vertices
within each leaf grows.
 By increasing k, we can guarantee that p -> 1 if r>1, and
p ->0 if r<1.
k = length of each loop
l = number of leaves
m = number of loops in a leaf
AMPLIFYING SELECTION
 FUNNEL: There are k+1 layers. Layer 0 contains only a
single vertex. Layer j contains mj vertices. All edges that
originate from vertices in layer j lead into j-1. All edges
that originate from the single vertex in layer 0 lead into
layer k.
AMPLIFYING SELECTION
 METAFUNNEL :
CIRCULATIONS
 We can simply choose an edge instead of a vertex. Edge
ij is chosen with a probability proportional to wij
multiplied by fitness of its tail.
 A circulation is defined by
 A graph G is p-equivalent to the Moran process if and
only if it is a circulation in this framework.
 Note that every isothermal graph is a circulation, but not
every circulation is isothermal.
GAMES ON GRAPHS
 H (interaction graph) determines who plays with whom.
 G (replacement graph) specifies the reproductive events.
 For each neighbor, the cooperator pays a cost c, and the
neighbors receives a benefit b. Defectors do not provide
any help.
 Consider regular graphs of degree k.
GAMES ON GRAPHS
Three different update rules for the game dynamics:
 Birth-death process: An individual is chosen for
reproduction proportional to fitness, the offspring
replaces at random one of the neighbors on the
replacement graph.
for all b and c.
 In ―BD‖ process, selection always favors defectors.
GAMES ON GRAPHS
GAMES ON GRAPHS
 Death-birth process: A random individual is chosen
to die, the neighbors compete for the empty site
proportional to their fitness.
if b/c >k
 Imitation process: A random player is chosen for
updating his strategy, he either stay with his own strategy
or adapts a neighbor‘s strategy proportional to fitness.
if b/c > k+2
GAMES ON GRAPHS
The evolutionary outcomes are dependent on the
updating rules. Our three updating rules are only a small
subset of the possible dynamics on graphs. The ―BD‖,
―DB‖ and ―IM‖ updating rules assume ‗fertility-selection‘
where the payoffs of the game affect reproductive success
of players. However one can imagine ‗viability-selection‘
where the payoffs affect the survival of players.
RESEARCH QUESTIONS
 In the social interpretation of this model, r is how
strongly each individual favors the new concept.
 What if r is different for different people? If someone
especially favors the idea, their r will be large, and if they
are against it, their r will be small.
 Does the fixation probability depends more strongly on
some nodes‘ r than others? This would mean that some
people‘s opinions carry more weight than others,
because of their position in the network.
RESEARCH QUESTIONS
 If so, which ones? Does influence correlate with
centrality in the network? That is, does centralization
correspond to a concentration of power?
 If so, ―enhancing the spread of favorable ideas‖ would
come with a tradeoff — they would be the ideas
favorable to a select few, and what if those few aren‘t
representative?
 In the biological interpretation of this model,
Environmental Evolutionary Graph Theory.
RESEARCH QUESTIONS
 What is the maximum mutation rate compatible with
adaptation on graphs?
 How does sexual reproduction affect evolution on
graphs?
 What are the timescales associated with fixation, and
how do they lead to coexistence in ecological settings?
 Furthermore, how does the graph itself change as a
consequence of evolutionary dynamics?
REFERENCES
 Martin A. Nowak(2006). Evolutionary Dynamics, Harvard Univ.
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Press, Cambridge, MA.
Erez Lieberman, Christoph Hauert & Martin A. Nowak et.
al.(2005). Evolutionary dynamics on graphs,Nature, 433, 312316.
Lee Worden, Evolutionary Graph Theory and Structural Power,
Networks in Political Science, June 13–14, 2008, Cambridge,
Mass.
Gregory Puleo, Environmental Evolutionary Graph Theory,
Rochester Institute of Technology, November 8, 2008
Hisashi Ohtsuki, Jorge M. Pacheco, and Martin A. Nowak,
Evolutionary graph theory: breaking the symmetry between
interaction and replacement