Evolutionary graph theory
Mark Broom
City University London
Game Theoretical Modelling of Evolution in Structured Populations
NIMBioS
Knoxville
25-27 April 2016
Mark Broom (City University London)
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Origins
References I
This talk is based on work by various authors, which are cited throughout, but
in particular significantly follows Chapter 12 of the book:
1. Broom,M. and Rychtář,J. (2013) Game-Theoretical Models in Biology Chapman
and Hall/ CRC Press.
Other references are:
2. Moran,P. (1958) Random processes in genetics Mathematical Proceedings of the
Cambridge Philosophical Society 54 60-71.
3. Karlin,S. and Taylor,H. (1975). A First Course in Stochastic Processes. Academic
Press.
4. Antal,T. and Scheuring,I. (2006). Fixation of strategies for an evolutionary game
in finite populations. Bulletin of Mathematical Biology 68 1923-1944.
5. Kimura,M. and Crow,J. (1963). The measurement of effective population number.
Evolution 17 279-288.
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Origins
References II
6. Taylor,C., Fudenberg,D., Sasaki,A, and Nowak,M. (2004) Evolutionary game
dynamics in finite populations. Bulletin of Mathematical Biology 66 1621-1644.
7. Nowak,M. (2006). Evolutionary Dynamics, Exploring the Equations of Life.
Belknap Press.
8. Lieberman,E., Hauert,C. and Nowak,M. (2005). Evolutionary
games on graphs. Nature 433 312-316.
9. Broom,M. and Rychtář,J. (2008). An analysis of the fixation probability of a
mutant on special classes of non-directed graphs. Proceedings of the Royal Society of
London A 464 2609-2627.
10. Broom,M., Rychtář,J. and Stadler,B. (2011). Evolutionary dynamics on graphs:
The effect of graph structure and initial placement on mutant spread. Journal of
Statistical Theory and Practice 5 369-381.
11. Broom,M., Hadjichrysanthou,C. and Rychtář,J. (2010). Evolutionary games on
graphs and the speed of the evolutionary process Proceedings of the Royal Society of
London A 466 1327-1346.
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Origins
References III
12. Antal,T. Redner,S. and Sood,V. (2006). Evolutionary dynamics on
degree-heterogeneous graphs. Physical Review Letters 96 188104.
13. Hadjichrysanthou,C., Broom,M. and Rychtář,J. (2011). Evolutionary games on
star graphs under various updating rules. Dynamic Games and Applications 1
386-407.
14. Shakarain,P., Roos,P. and Johnson,A. (2012). A review of evolutionary graph
theory with applications to game theory. Biosystems 107 66-80.
15. Houchmandzadeh,B. and Vallade,M. (2010). Alternative to the diffusion equation
in population genetics. Physical Review E 82 051913.
16. Allen,B., Traulsen,A., Tarnita,C. and Nowak,M. (2012). How mutation affects
evolutionary games on graphs. Journal of Theoretical Biology 299 97-105.
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Finite populations and the Moran process
Outline
1
Finite populations and the Moran process
2
Games in finite populations
3
Evolution on graphs
4
Games on graphs
Mark Broom (City University London)
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Finite populations and the Moran process
Infinite populations
Traditional evolutionary models generally consider an infinite population
of individuals which is well-mixed.
Here the fitness of an individual depends upon its strategy and the
strategies adopted by others in the population.
The key static concept for such populations is the Evolutionarily Stable
Strategy, where a population playing one strategy can resist invasion by
all alternatives.
The dynamics of such populations, such as the Replicator Dynamics,
follow differential equations.
We consider finite populations, and subsequently structured populations,
and see that we need some new concepts distinct from those for infinite
population games.
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Finite populations and the Moran process
The Moran process
Here we consider a population of finite size N.
We shall start by assuming individuals have a fixed fitness ri for
individual Si , depending upon type, but independent of interactions with
others.
The standard dynamics applied to this population is as follows.
At each time step an individual is chosen for reproduction at random with
a probability proportional to its fitness
Its offspring replaces a randomly chosen individual (which could be its
parent).
This is called the Moran process, see Moran2 (1958).
2. Moran,P. (1958) Random processes in genetics Mathematical Proceedings of the
Cambridge Philosophical Society 54 60-71.
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Finite populations and the Moran process
Neutral fitness
We shall start by considering the neutral fitness case, where ri = 1 for all
Si , as in the original Moran process.
Suppose we have N individuals, made up of mi individuals of type i,
i = 1, . . . , n.
The population is thus described by a (row) vector m = (mi ) with
P
i mi = N.
At each time point a random individual is chosen to give birth, and
another to die, selected independently of each other.
Let ei be a (row) vector with 0’s everywhere except on the ith place
where there is a 1.
Mark Broom (City University London)
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Finite populations and the Moran process
A Markov process
The possible transitions in the population, together with the probabilities
of those transitions, are as follows:

m mj


 i
m∗ = m + ei − ej , i 6= j,


N
N


n
P(m → m∗ ) = X mi 2
(1)

m∗ = m,


N


i=1


0
otherwise.
Thus at any time t, m(t) only depends upon m(t − 1) and no earlier time
points are relevant; thus the Moran process is a Markov process.
Mark Broom (City University London)
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Finite populations and the Moran process
Non-neutral evolution
Now suppose that not all of the values of ri are equal.
We note that there are many ways we can incorporate this in the process.
As described above, we shall make the birth rate depend upon fitness and
the death rate not.
One way of thinking of the fitness of an individual is as the number of
offspring that it will have that will survive to adulthood.
We can thus perhaps think that at any given time step, and for every type
Si there are mi ri offspring of that type that may be born.
P
An offspring of type Si will thus be born with probability mi ri / ( l ml rl ).
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Finite populations and the Moran process
The revised transition probabilities
The transition probabilities now become

mi ri mj



m∗ = m + ei − ej , i 6= j.
 Pn m r N

l
l

 l=1
n
P(m → m∗ ) = X mi ri mi
Pn

m∗ = m.


N
m
r
l
l

l=1

i=1


0
otherwise.
(2)
Multiplying all ri by a constant leaves the probabilities unchanged, so
without loss of generality we set one of our fitnesses to be 1.
There is a non-zero probability that any given type will reach fixation,
since ri > 0 for all i, and it is certain that one type will eventually do so.
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Finite populations and the Moran process
The fixation probability
The long-term outcome of the process described above is a population
consisting of just a single type.
The important question is, which type is likely to dominate, i.e. how
likely is each such fixation to occur?
Thus the probability of fixation, the fixation probability, is the single
most important property of a finite evolutionary system.
This is usually considered as the probability of fixation of a single
mutant in a population otherwise entirely made up of a resident type.
It thus makes sense for us to consider the case where we have two types
of individuals only, type A with fitness r, and type B with fitness 1.
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Finite populations and the Moran process
The Chapman-Kolmogorov equations
The state of the population is described by a single number, NA , the
number of individuals of type A.
We can find an expression for the probability of the population
containing i mutants at time t + 1, πi (t + 1), in terms of the probabilities
of occupying the different population sizes at time t and these transition
probabilities using the equation
X
πi (t + 1) =
pj,i πj (t),
(3)
j
where pi,j = Prob (i, N − i) → (j, N − j) is the probability that NA = j
at time point t + 1 given that NA = i at time point t.
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Finite populations and the Moran process
Transition probabilities for two types of individual
By (2), we get
N−i
ir
,
ir + N − i N
N−i i
pi,i−1 =
,
ir + N − i N
i
N−i N−i
ir
+
pi,i =
ir + N − i N
ir + N − i N
ir
N−i
N−i i
=1−
−
.
ir + N − i N
ir + N − i N
pi,i+1 =
(4)
(5)
(6)
(7)
In the terminology of Markov processes, (3) are the
Chapman-Kolmogorov forward equations, see Karlin & Taylor3 (1975).
3. Karlin,S. and Taylor,H. (1975). A First Course in Stochastic
Processes. Academic Press.
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Finite populations and the Moran process
The fixation probability equation
Denoting Pi as the fixation probability of A given NA = i, we obtain the
following difference equation
Pi = Pi−1 pi,i−1 + Pi pi,i + Pi+1 pi,i+1
(8)
with the obvious boundary conditions, the fixation probabilities on the
absorbing states, P0 = 0, PN = 1.
We can solve the above equations to obtain

i
 1 − (1/r)
r 6= 1,
Pi = 1 − (1/r)N

i/N
r = 1.
Mark Broom (City University London)
(9)
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Finite populations and the Moran process
The Moran probability
This in turn gives the fixation probability of a single mutant of type A in
a population of type Bs as

 1 − (1/r)
r 6= 1,
PA = P1 = 1 − (1/r)N
(10)

1/N
r = 1.
This is the Moran probability, plotted on the next slide.
This is the benchmark against which fixation probabilities in more
complex structured populations are compared.
By symmetry, the fixation probability of a B mutant in an A population if
r 6= 1 is
r−1
PB = N
.
(11)
r −1
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Finite populations and the Moran process
A plot of the Moran probability
1
N=2
N=4
N=16
Fixation probability
0.75
0.5
0.25
0
0.001
0.01
0.1
1
10
100
1000
r
Figure: The Moran fixation probability for various N.
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Finite populations and the Moran process
General Birth-Death processes
The recurrence relation (8) is in fact equivalent to the Birth-Death
process, or equivalently the random walk, with the birth rate βi and death
rate δi when the population is at state i.
The general equations are given by
xi = βi−1 xi−1 + (1 − βi − δi )xi + δi+1 xi+1 ,
(12)
where the probability of fixation (i.e. reaching state N) starting from
state i is denoted by xi , and so x0 = 0, xN = 1.
We note that Karlin & Taylor3 (1975) discussed this process without
upper limit N, so that the summation in the denominator of (10) went to
infinity.
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Finite populations and the Moran process
General Birth-Death processes and the Moran process
A solution to (12) is
xi =
1+
1+
Pi−1 Qj
δk
j=1
k=1 βk
PN−1 Qj δk
j=1
k=1 βk
.
(13)
We recover the Moran probability by appropriate substitution in the
solution for x1 .
In general (13) can be directly applied to some cases of evolutionary
processes including games on graphs.
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Finite populations and the Moran process
Fixation and absorption times
Another important question in such a population is, how long does it take
for a mutant to fixate?
There is always some chance that the mutant will be eliminated even if
r > 1 and it is fitter than the resident, so we must distinguish between
two times.
We define Ti , the absorption time, as the expected time until the
population contains only one type of individual, either A or B, given
there are currently NA = i A individuals in the population.
The fixation time Fi is the expected time to mutant fixation, conditional
on fixation occurring.
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Finite populations and the Moran process
General absorption times
As seen above, at any time step, the population moves from state NA = i
(for 1 ≤ i ≤ N − 1) to state NA = i + 1 with probability pi,i+1 , to state
NA = i − 1 with probability pi,i−1 , and remains at state NA = i with
probability 1 − (pi,i+1 + pi,i−1 ).
This yields the following system of equations
Ti = 1 + pi,i+1 Ti+1 + pi,i−1 Ti−1 + (1 − pi,i+1 − pi,i−1 )Ti ,
(14)
for 1 ≤ i ≤ N − 1 with boundary conditions T0 = TN = 0.
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Finite populations and the Moran process
General fixation times I
The situation with fixation times is somewhat more difficult.
Following Antal & Scheuring4 (2006), define Ai (t) as the probability of
fixation occurring at time
P t, given that the population is in state i at time
0. Then clearly Pi = ∞
t=0 Ai (t) and
P∞
P∞
tAi (t)
tAi (t)
t=0
= t=0
.
(15)
Fi = P∞
Pi
t=0 Ai (t)
Similarly to (8) Ai (t) satisfies
Ai (t) = Ai−1 (t − 1)pi,i−1 + Ai (t − 1)pi,i + Ai+1 (t − 1)pi,i+1 .
(16)
4. Antal,T. and Scheuring,I. (2006). Fixation of strategies for an evolutionary game
in finite populations. Bulletin of Mathematical Biology 68 1923-1944.
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Finite populations and the Moran process
General fixation times II
Multiplying both sides of (16) by t, summing from 0 to ∞ and using the
fact that
∞
X
tAi (t − 1) =
t=0
∞
X
(t + 1)Ai (t) = Pi (Fi + 1),
(17)
t=0
we obtain
Pi Fi = Pi + pi,i+1 Pi+1 Fi+1 + pi,i−1 Pi−1 Fi−1 + (1 − pi,i+1 − pi,i−1 )Pi Fi ,
(18)
for 1 ≤ i ≤ N − 1 with boundary condition FN = 0.
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Finite populations and the Moran process
The diffusion approximation I
For a very large population, the Moran process can be approximated by a
continuous stochastic process.
The population follows a sequence of small changes (a proportion 1/N
of the population) in an equivalently short time interval.
As N tends to infinity this tends to a continuous stochastic process, the
diffusion approximation, originating with Kimura & Crow5 (1963).
The mathematical consequences of the process is that the
Chapman-Kolmogorov forward equations (3) become a partial
differential equation in their limit.
5. Kimura,M. and Crow,J. (1963). The measurement of effective population number.
Evolution 17 279-288.
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Finite populations and the Moran process
The diffusion approximation II
Denoting the fraction of mutants ρ = i/N, the probability c(ρ, t) that the
mutant density is ρ at time t satisfies
∂
∂2
∂c(ρ, t)
=−
v(ρ)c(ρ, t) + 2 D(ρ)c(ρ, t) ,
∂t
∂ρ
∂ρ
(19)
where v(ρ) and D(ρ), the drift and diffusion terms, are given by
δρ
R(ρ) − L(ρ) ,
δt
1 (δρ)2
D(ρ) =
R(ρ) + L(ρ) .
2 δt
v(ρ) =
(20)
(21)
This is Fokker-Planck equation.
In our well-mixed population L(ρ) = R(ρ) = ρ(1 − ρ).
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Games in finite populations
Outline
1
Finite populations and the Moran process
2
Games in finite populations
3
Evolution on graphs
4
Games on graphs
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Games in finite populations
The payoff matrix
Taylor et al.6 (2004) extended the Moran process approach above to
playing games in finite populations. The results in this section mainly
relate to those from this paper and Nowak7 (2006).
For two types of individuals, mutants M and residents R, we consider the
standard 2 × 2 payoff matrix
M
R
M
a
c
R
b
.
d
(22)
6. Taylor,C., Fudenberg,D., Sasaki,A, and Nowak,M. (2004) Evolutionary game
dynamics in finite populations. Bulletin of Mathematical Biology 66 1621-1644.
7. Nowak,M. (2006) Evolutionary Dynamics, Exploring the Equations of Life.
Belknap Press.
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Games in finite populations
Mean payoffs
The average payoffs to a mutant individual in a population where there
are mM = i mutants in total is thus
EM,i =
a(i − 1) + b(N − i)
,
N−1
(23)
Similarly the average payoff to a resident individual in such a population
is
ci + d(N − i − 1)
.
(24)
ER,i =
N−1
Note that we assume above that an individual cannot play a game with
itself and hence we have the factors (i − 1) and (N − i − 1).
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Games in finite populations
ESSs in finite populations I
If a resident strategy is an ESS in an infinite population, it is fitter than
any mutant in a population comprising a mixture of a sufficiently small
mutant group and the remainder playing the resident strategy.
A natural extension to finite populations is that a mutant should be less
fit than a resident in a population of one mutant and N − 1 residents
(selection opposes M invading R) i.e. EM,1 < ER,1 which is
b(N − 1) < c + d(N − 2).
(25)
However, this is insufficient for stability, since a chance increase in the
mutant population can lead to greater mutant fitness. Thus we have a
second condition, that selection opposes the replacement of R by M,
PM <
Mark Broom (City University London)
1
.
N
(26)
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Games in finite populations
ESSs in finite populations II
We thus have the definition:
For a finite population size N and a 2 × 2 matrix game (22), a pure
strategy R is called an evolutionarily stable strategy, ESSN , if (25) and
(26) hold.
Similarly as in (13) PM is given by
PM =
1+
1
PN−1 Qj
j=1
pk,k−1
k=1 pk,k+1
,
(27)
where pk,k+1 and pk,k−1 are the transition probabilities equivalent to (4)
and (5).
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Games in finite populations
The intensity of selection
Since the formulae (4) and (5) use fitness, we need to find a way to
translate the payoffs of the game EM,i and ER,i into the fitness of the
respective types rM,i and rR,i .
If we assume that the fitness is equal to the payoff, i.e. rM,i = EM,i and
rR,i = ER,i , the stability condition (26) is very complex.
Thus the idea of intensity of selection is often used.
Assume that the fitness is given by
rM,i = 1 − w + wEM,i ,
(28)
rR,i = 1 − w + wER,i ,
(29)
where 0 < w ≤ 1 is the intensity of selection.
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Games in finite populations
Weak selection
A small w represents weak selection and means that the game has a small
effect on the process of evolution. This gives the transition probabilities
(4) and (5) as
i(1 − w + wEM,i )
N−i
,
i(1 − w + wEM,i ) + (N − i)(1 − w + wER,i ) N
(N − i)(1 − w + wER,i )
i
=
.
i(1 − w + wEM,i ) + (N − i)(1 − w + wER,i ) N
pi,i+1 =
(30)
pi,i−1
(31)
For small w ≈ 0, pi,i−1 /pi,i+1 ≈ 1 + w(ER,i − EM,i ), and after
substituting into (27) we get that the stability condition (26) is equivalent
to
a(N − 2) + b(2N − 1) < c(N + 1) + d(2N − 4).
(32)
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Games in finite populations
The rule of 1/3
For a large population the conditions for ESSN (25) and (32) reduce to
b < d, and
(33)
a + 2b < c + 2d.
(34)
The first of these two conditions is the standard condition for an ESS in
an infinite population.
The addition of the second condition leads to the rule of 1/3. The rule
says that if a > c and b < d (so in an infinite population there are two
pure ESSs) selection favours M replacing R, if the unstable internal
equilibrium value is less than a third i.e.
d−b
1
< .
a−c+d−b
3
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(35)
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Evolution on graphs
Outline
1
Finite populations and the Moran process
2
Games in finite populations
3
Evolution on graphs
4
Games on graphs
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Evolution on graphs
Representing population structure by a graph
In this part of the talk, we assume that a population consists of N
individuals and that each individual occupies a vertex in a given graph
G = (V, E).
G is thus a finite and undirected graph, which we assume is connected
and simple, i.e. no vertex is connected to itself and there are no parallel
edges.
We moreover assume that every vertex is occupied, and by one
individual only.
Two individuals can interact only if they are connected by an edge of the
graph.
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Evolution on graphs
The graph weights
The graph structure is represented by a matrix W = (wij ), where
wij is the probability of replacing a vertex j by a copy of a vertex i, given
that vertex i was selected for reproduction.
wij = 0 if there is no edge between vertices i and j.
For connected vertices, we often assume equal weightings, i.e. wij = 1/ei
where ei is the degree of vertex i, see the figure on the next slide.
The well-mixed population that we have considered up until now is a
special case of this. It is represented by the complete graph, the graph
where every pair of vertices are connected, with all weights equal.
We note that this treatment with weights wij is sufficiently general to
allow us to consider directed graphs or graphs where edges carry a
different weight.
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Evolution on graphs
Graphs with four vertices
(a)
(b)
(c)
(d)
2
1
3
4
1
2
3
4
(e)
1
0
2
3
1
2
1
2
1
3
0
1
2
1
2
0
0
1
3
(f)
4
0
0
0 13
1 0
Figure: Connected undirected graphs with 4 vertices. For one of these, the graph and
its corresponding weighting matrix W, in the case of equal weights.
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Evolution on graphs
Evolutionary dynamics on the graph
We suppose that the population evolves according to an evolutionary
dynamics and the evolutionary process can be represented as a discrete
time Markov chain.
Supposing that C ⊆ V is the set of vertices occupied by mutants, then at
the next time step the set occupied by mutants will become either
1) C ∪ {j}, j 6∈ C, provided a) a vertex i ∈ C was chosen for reproduction and
b) it placed its offspring into vertex j; or
2) C \ {i}, i ∈ C, provided a) a vertex j 6∈ C was selected for reproduction
and b) it placed its offspring into i; or
3) C, provided an individual from C (V \ C) replaces another individual from
C (V \ C).
The states ∅ and V are the absorbing points of the dynamics.
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Evolution on graphs
The Invasion Process
It is generally assumed that at the beginning of the evolutionary process,
all vertices are occupied by residents and then one vertex is chosen
uniformly at random and replaced by a mutant.
We have outlined the possible transitions in the Markov chain, but not
the probabilities.
Whilst the possible transitions are generally the same for any of the
evolutionary dynamics commonly used, the transition probabilities are
not and depend upon a choice of evolutionary dynamics.
We shall initially assume the Invasion Process (IP) where an individual
is selected to give birth proportional to its fitness, and then copies itself
into one of its neighbours (usually at random, with equal probability).
An example of one step of the IP dynamics is shown in the following
figure.
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Evolution on graphs
An illustration of the Invasion Process
Selection for reproduction
Initial population
1
1
r
1
r
3+r
Neighbour replacement
1
1
r
1
3+r
1
3
1
1
3
1
3
1
1
1
3+r
r
1
2
1
1
3+r
1
2
1
1
r
1
1
1
1
1
r
1
1
2
1
2
Figure: One step of an Invasion Process.
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Evolution on graphs
The fixed fitness case
The individual at vertex i is selected for reproduction with probability
proportional to the corresponding fitness 1 or r.
Following Lieberman et al.8 , the invasion process yields
X X
rwij PC∪{j} + wji PC\{i}
PC =
i∈C j6∈C
XX
rwij + wji
(36)
i∈C j6∈C
with P∅ = 0 and PV = 1, where PC denotes the probability of mutant
fixation given C is the set currently occupied by mutants.
Note that the system (36) of linear equations is very large.
8. Lieberman,E., Hauert,C. and Nowak,M. (2005). Evolutionary
games on graphs. Nature 433 312-316.
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Evolution on graphs
Analytical solutions
The general analytical solution of (36) is only known in a few cases.
The graphs where analytical progress has been made have essentially
been of three types (see Broom & Rychtář9 (2008)):
(i) regular graphs, where the size of the system can be reduced by
symmetries,
(ii) graphs where the greatest degree of the vertices is two, which means that
mutants must always be in the form of a line segment, significantly
reducing the number of states that need to be considered,
(iii) graphs with a high degree of symmetry, such as the star, where many of
the states are isomorphic.
9. Broom,M. and Rychtář,J. (2008). An analysis of the fixation probability of a
mutant on special classes of non-directed graphs. Proceedings of the Royal Society of
London A 464 2609-2627.
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Evolution on graphs
Symmetries on the star graph
[a]
[b]
Figure: Symmetries allow us to reduce the size of the system (36) such as in the
dynamics on a star graph. (a) All distinct states, (b) states relevant to the system (36).
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Evolution on graphs
Regular graphs
A graph is regular if the number of edges ei is constant for all vertices.
P
It follows that Ti = j wji , the temperature of vertex i, is constant,
which is known as the isothermal property.
Thus, if we assume that PC depends only on the size of set C ⊆ V, we
can replace PC by the probability of fixation from |C| mutants, x|C| , and
we get that for regular graphs the system (36) reduces to
x|C| =
1
r
x|C|+1 +
x
,
r+1
1 + r |C|−1
(37)
with boundary conditions x0 = 0, xN = 1.
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Evolution on graphs
The fixation probability for regular graphs
The equation (37) is a special case of (12) and the solution is thus given
by (13) as

 1 − (1/r)
r 6= 1,
x1 = 1 − (1/r)N
(38)

1/N
r = 1,
which is the Moran fixation probability (10).
Substituting this solution into (36) shows that this does indeed solve the
system of equations.
Since we have found a solution to (36), and this solution must be unique,
it also follows that our assumption that we can replace PC by x|C| must
be correct.
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Evolution on graphs
Selection amplifiers and suppressors I
Lieberman et al.8 (2005) considered a variety of directed graphs,
generating some surprising results.
The burst completely suppresses selection, so that the fixation probability
of a randomly placed mutant is 1/N, irrespective of the fitness of the
individuals (the type of the individual at the centre always fixates).
We note that a burst is not a connected graph, as there is no way to reach
the central vertex from other vertices, and no connected graph can
completely suppress selection in this way.
At the other extreme there are a number of selection enhancers, where
the probability of selection of a random mutant tends to 1 as N → ∞ for
any advantageous mutant (r > 1).
Two selection enhancers are the superstar and the funnel.
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Evolution on graphs
Some directed graphs
Superstar
Funnel
Burst
Figure: Directed graphs can be selection suppressors (the burst and any one rooted
graph in general) and amplifiers (the superstar or the funnel).
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Evolution on graphs
Selection amplifiers and suppressors II
A superstar of level k has approximate fixation probability
1 − (1/r)k
,
1 − (1/r)kN
(39)
i.e. equivalent to an individual of fitness rk in a well-mixed population.
In general for equally weighted graphs, regular graphs tend to have the
lowest fixation probability for a randomly placed advantageous mutant,
while highly irregular graphs such as the star have the highest.
There is a high degree of correlation between the average fixation
probability of a single mutant and the variance of vertex degrees on the
graph, as shown in Broom et al.10 (2011).
10. Broom,M., Rychtář,J. and Stadler,B. (2011). Evolutionary dynamics on graphs:
The effect of graph structure and initial placement on mutant spread.
Journal of Statistical Theory and Practice 5 369-381.
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Games on graphs
Outline
1
Finite populations and the Moran process
2
Games in finite populations
3
Evolution on graphs
4
Games on graphs
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Games on graphs
Games on graphs
When considering games on graphs the fitness of each individual
depends upon the types of all of its neighbours.
Using the standard payoff matrix, the payoffs to an M individual at
vertex i and an R individual at vertex j are given by
aNM,i + bNR,i
,
NM,i + NR,i
cNM,j + dNR,j
,
fj =
NM,j + NR,j
fi =
(40)
(41)
where NM,i (NR,i ) is the number of neighbours of i of type M (R).
Here we consider the example of a classical game, the Hawk-Dove game.
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Games on graphs
An example Hawk-Dove game I
Broom et al.11 (2010) generated theoretical formulae for the exact
solutions of fixation probabilities, absorption times and fixation times for
the star and the circle.
In one example the payoffs matrix (22) of the Hawk-Dove game becomes
Hawk
Dove
Hawk
a = (15 − C)/2
c=5
Dove
b = 10
,
d = 15/2
(42)
which is equivalent to a reward V = 5, an arbitrary cost C, plus a
“background fitness” of 5.
11. Broom,M., Hadjichrysanthou,C. and Rychtář,J. (2010). Evolutionary games on
graphs and the speed of the evolutionary process Proceedings of the Royal Society of
London A 466 1327-1346.
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Games on graphs
Fixation on three graphs
Figure: The Fixation probability (left figure) and the mean time to absorption (right
figure) when a mutant Hawk invades into a resident population of Doves between a
star (crosses), a circle (circles) and a complete graph (boxes) when N = 100 and C
varies.
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Games on graphs
An example Hawk-Dove game II
Changing C has a gradual effect on the fixation probability on the circle,
a sudden and dramatic effect on the fixation probability on the complete
graph and almost no effect on the fixation probability on the star.
Comparing the complete graph and the circle, it is clear that the fixation
probability can be significantly different for different regular graphs (as
opposed to the fixed fitness case, where this is the Moran probability).
The time to absorption is hardly affected by the value of C on the circle
or star (except as C approaches 15 and so the payoff of a mutant against
another mutant approaches 0).
The larger times for intermediate C on the complete graph corresponds
to when the equivalent infinite population has an ESS corresponding of
roughly equal numbers of Hawks and Doves.
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Games on graphs
Dynamics and fitness I
Evolutionary game models consider a number of dynamics, not just the
Invasion Process, the most common being the four related ones below.
IP dynamics or BD-B - an individual is chosen for reproduction with
probability proportional to its fitness and its offspring replaces a randomly
chosen neighbour.
BD-D process - an individual is chosen for reproduction at random and its
offspring replaces a neighbouring individual which is chosen with
probability inversely proportional to its fitness.
Voter model or DB-D - an individual first dies with a probability inversely
proportional to its fitness and is then replaced by the offspring of a
randomly chosen neighbour.
DB-B process - an individual first dies at random and is then replaced by
an offspring of a neighbour that is chosen with probability proportional to
its fitness.
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Games on graphs
Dynamics and fitness II
In general, the dynamics are similar for regular graphs. In particular the
fixation probability for the IP and the VM are identical Antal et al12 .
The fixation probabilities for the BD-D and DB-B processes are different
for small population sizes, these differences disappear for sufficiently
large populations.
However, the choice of a dynamics can be very important for irregular
graphs as seen in the next slide, which considers evolution on the star for
the Hawk-Dove game, from Hadjichrysanthou et al13 (2011).
12. Antal,T. Redner,S. and Sood,V. (2006). Evolutionary dynamics on
degree-heterogeneous graphs. Physical Review Letters 96 188104.
13. Hadjichrysanthou,C., Broom,M. and Rychtář,J. (2011). Evolutionary games on
star graphs under various updating rules. Dynamic Games and Applications 1
386-407.
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Games on graphs
Dynamics and fitness III
In all of the processes except the IP, Hawks have lower fixation
probabilities on the star than on the complete graph for low costs,
whereas the reverse is true for the IP.
The differences between the fixation probabilities for the different
processes are very large for medium to large values of the population
size n + 1.
Thus the dynamics used can have a profound effect on the fixation
probability, and indeed it can have similar effects on other properties
such as the fixation time.
It should be noted that a similar figure can be obtained for the fixed
fitness case as well.
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Games on graphs
Fixtation probabilities for different dynamics
Dyn Games Appl (2011) 1:386–407
[a]
401
[b]
5 The average
fixation probability
probability of a single
Hawk
star graph
under IP
the IP
(crosses), theBD-D
Figure:Fig.Average
fixation
of amutant
Hawk
ononaa star
graph;
(crosses),
BD-D process (diamonds), the VM (circles) and the DB-B process (boxes) in the Hawk–Dove game described
(diamonds),
VM
(circles)
and
DB-B
(boxes)
in
the
Hawk-Dove
game
with
(a)
by the payoff matrix (4.15) in the case where (a) C/V = 1.5 and n varies, (b) n = 60 and C/V varies.
fb =
2
w1=and
1. Thenthick
lines represent
the 100,
respective
case1onand
the complete
graph Background
and the dashed-dotted
line is 2.
V = Cand
=
varies,
(b)
n
=
V
=
C
varies.
fitness
represents the fixation probability of a single mutant in the case of neutral drift, 1/(n + 1)
The thick lines represent the complete graph and the dashed-dotted line neutral drift.
Figure 5b suggests that when Hawks are favoured over Doves in the different update
rules, the complete graph promotes the fixation of Hawks compared to the star graph in
the BD-D, VM and DB-B process. Moreover, in the IP, favoured Hawks have much higher
Mark Broom
(Cityto
University
London)
NIMBioS2016
chance
fixate on
a star graph.
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Games on graphs
What is fitness on evolutionary graphs?
What do we mean by fitness on evolutionary graphs?
Fitness can be thought of as the expected number of offspring (raised to
adulthood) of an individual.
The simplistic definition of fitness used in the current chapter can be far
away from fitness in the above meaning.
In particular one might suppose that a mutant with 0 fitness should have
0 probability to fixate, and that an individual with infinite fitness would
be certain to fixate within a population of unit fitness.
We note that the IP and VM processes are consistent with this
interpretation, but that the BD-D and DB-B processes are not.
Thus it is perhaps more accurate to say that “true” fitness is an increasing
function (for the fixed fitness case) of r, rather than r itself.
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Games on graphs
Further developments on evolutionary graphs
Work on evolutionary graphs continues to develop at a very fast pace.
A very good review article which covers the developments up to 2012 is
Shakarian et al14 (2012).
A version of the diffusion approximation for large graphs has been
developed by Houchmandzadeh & Vallade15 (2010).
A model with mutation has been introduced by Allen et al16 (2012).
14. Shakarain,P., Roos,P. and Johnson,A. (2012). A review of evolutionary graph
theory with applications to game theory. Biosystems 107 66-80.
15. Houchmandzadeh,B. and Vallade,M. (2010). Alternative to the diffusion equation
in population genetics. Physical Review E 82 051913.
16. Allen,B., Traulsen,A., Tarnita,C. and Nowak,M. (2012). How mutation affects
evolutionary games on graphs. Journal of Theoretical Biology 299 97-105.
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