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动态演化经典论文Evolutionary dynamics on graphs

动态演化经典论文Evolutionary dynamics on graphs
动态演化经典论文Evolutionary dynamics on graphs

Received 24September;accepted 16November 2004;doi:10.1038/nature03211.

1.MacArthur,R.H.&Wilson,E.O.The Theory of Island Biogeography (Princeton Univ.Press,Princeton,1969).

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7.Caswell,https://www.wendangku.net/doc/6715435392.html,munity structure:a neutral model analysis.Ecol.Monogr.46,327–354(1976).8.Bell,G.Neutral macroecology.Science 293,2413–2418(2001).9.Elton,C.S.Animal Ecology (Sidgwick and Jackson,London,1927).10.Gause,G.F.The Struggle for Existence (Hafner,New York,1934).

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Acknowledgements I thank the Makah Tribal Council for providing access to Tatoosh Island;J.Sheridan,J.Salamunovitch,F.Stevens,https://www.wendangku.net/doc/6715435392.html,ler,B.Scott,J.Chase,J.Shurin,K.Rose,L.Weis,R.Kordas,K.Edwards,M.Novak,J.Duke,J.Orcutt,K.Barnes,C.Neufeld and L.Weintraub for ?eld assistance;and NSF,EPA (CISES)and the Andrew W.Mellon foundation for partial ?nancial support.

Competing interests statement The author declares that he has no competing ?nancial interests.Correspondence and requests for materials should be addressed to J.T.W.(twootton@https://www.wendangku.net/doc/6715435392.html,).

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Evolutionary dynamics on graphs

Erez Lieberman 1,2,Christoph Hauert 1,3&Martin A.Nowak 1

1

Program for Evolutionary Dynamics,Departments of Organismic and Evolutionary Biology,Mathematics,and Applied Mathematics,Harvard University,Cambridge,Massachusetts 02138,USA 2

Harvard-MIT Division of Health Sciences and Technology,Massachusetts Institute of Technology,Cambridge,Massachusetts,USA 3

Department of Zoology,University of British Columbia,Vancouver,British Columbia V6T 1Z4,Canada

.............................................................................................................................................................................

Evolutionary dynamics have been traditionally studied in the context of homogeneous or spatially extended populations 1–4.Here we generalize population structure by arranging individ-uals on a graph.Each vertex represents an individual.The weighted edges denote reproductive rates which govern how

often individuals place offspring into adjacent vertices.The homogeneous population,described by the Moran process 3,is the special case of a fully connected graph with evenly weighted edges.Spatial structures are described by graphs where vertices are connected with their nearest neighbours.We also explore evolution on random and scale-free networks 5–7.We determine the ?xation probability of mutants,and characterize those graphs for which ?xation behaviour is identical to that of a homogeneous population 7.Furthermore,some graphs act as suppressors and others as ampli?ers of selection.It is even possible to ?nd graphs that guarantee the ?xation of any advantageous mutant.We also study frequency-dependent selec-tion and show that the outcome of evolutionary games can depend entirely on the structure of the underlying graph.Evolu-tionary graph theory has many fascinating applications ranging from ecology to multi-cellular organization and economics.Evolutionary dynamics act on populations.Neither genes,nor cells,nor individuals evolve;only populations evolve.In small populations,random drift dominates,whereas large

populations

Figure 1Models of evolution.a ,The Moran process describes stochastic evolution of a ?nite population of constant size.In each time step,an individual is chosen for

reproduction with a probability proportional to its ?tness;a second individual is chosen for death.The offspring of the ?rst individual replaces the second.b ,In the setting of evolutionary graph theory,individuals occupy the vertices of a graph.In each time step,an individual is selected with a probability proportional to its ?tness;the weights of the outgoing edges determine the probabilities that the corresponding neighbour will be replaced by the offspring.The process is described by a stochastic matrix W ,where w ij denotes the probability that an offspring of individual i will replace individual j .In a more general setting,at each time step,an edge ij is selected with a probability proportional to its weight and the ?tness of the individual at its tail.The Moran process is the special case of a complete graph with identical weights.

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are sensitive to subtle differences in selective values.The tension between selection and drift lies at the heart of the famous dispute between Fisher and Wright8–10.There is evidence that population structure affects the interplay of these forces11–15.But the celebrated results of Maruyama16and Slatkin17indicate that spatial structures are irrelevant for evolution under constant selection.

Here we introduce evolutionary graph theory,which suggests a promising new lead in the effort to provide a general account of how population structure affects evolutionary dynamics.We study the simplest possible question:what is the probability that a newly introduced mutant generates a lineage that takes over the whole population?This?xation probability determines the rate of evolu-tion,which is the product of population size,mutation rate and ?xation probability.The higher the correlation between the mutant’s?tness and its probability of?xation,r,the stronger the effect of natural selection;if?xation is largely independent of ?tness,drift dominates.We will show that some graphs are governed entirely by random drift,whereas others are immune to drift and are guided exclusively by natural selection.

Consider a homogeneous population of size N.At each time step an individual is chosen for reproduction with a probability pro-portional to its?tness.The offspring replaces a randomly chosen individual.In this so-called Moran process(Fig.1a),the population size remains constant.Suppose all the resident individuals are identical and one new mutant is introduced.The new mutant has relative?tness r,as compared to the residents,whose?tness is1.The ?xation probability of the new mutant is:

r1?

121=r

121=r N

e1T

This represents a speci?c balance between selection and drift: advantageous mutations have a certain chance—but no guaran-tee—of?xation,whereas disadvantageous mutants are likely—but again,no guarantee—to become extinct.

We introduce population structure as follows.Individuals are labelled i?1,2,…N.The probability that individual i places its offspring into position j is given by w ij.

Thus the individuals can be thought of as occupying the vertices of a graph.The matrix W?[w ij]determines the structure of the graph(Fig.1b).If w ij?0and w ji?0then the vertices i and j are not connected.In each iteration,an individual i is chosen for reproduction with a probability proportional to its?tness.The resulting offspring will occupy vertex j with probability w ij.Note that W is a stochastic matrix,which means that all its rows sum to one.We want to calculate the?xation probability r of a randomly placed mutant.

Imagine that the individuals are arranged on a spatial lattice that can be triangular,square,hexagonal or any similar tiling.For all such lattices r remains unchanged:it is equal to the r1obtained for the homogeneous population.In fact,it can be shown that if W is symmetric,w ij?w ji,then the?xation probability is always r1.The graphs in Fig.2a–c,and all other symmetric,spatially extended models,have the same?xation probability as a homogeneous population17,18.

There is an even wider class of graphs whose?xation probability is r1.Let T i?S j w ji be the temperature of vertex i.Avertex is‘hot’if it is replaced often and‘cold’if it is replaced rarely.The‘isothermal theorem’states that an evolutionary graph has?xation probability r1if and only if all vertices have the same temperature.Figure2d gives an example of an isothermal graph where W is not symmetric. Isothermality is equivalent to the requirement that W is doubly stochastic,which means that each row and each column sums to one.

If a graph is not isothermal,the?xation probability is not

given

Figure2Isothermal graphs,and,more generally,circulations,have?xation behaviour identical to the Moran process.Examples of such graphs include:a,the square lattice; b,hexagonal lattice;c,complete graph;d,directed cycle;and e,a more irregular circulation.Whenever the weights of edges are not shown,a weight of one is distributed evenly across all those edges emerging from a given vertex.Graphs like f,the‘burst’,and g,the‘path’,suppress natural selection.The‘cold’upstream vertex is represented in blue.The‘hot’downstream vertices,which change often,are coloured in orange.The type of the upstream root determines the fate of the entire graph.h,Small upstream populations with large downstream populations yield suppressors.i,In multirooted graphs,the roots compete inde?nitely for the population.If a mutant arises in a root then neither?xation nor extinction is possible.

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by r1.Instead,the balance between selection and drift tilts;now to one side,now to the other.Suppose N individuals are arranged in a linear array.Each individual places its offspring into the position immediately to its right.The leftmost individual is never replaced. What is the?xation probability of a randomly placed mutant with ?tness r?Clearly,it is1/N,irrespective of r.The mutant can only reach?xation if it arises in the leftmost position,which happens with probability1/N.This array is an example of a simple popu-lation structure whose behaviour is dominated by random drift. More generally,an evolutionary graph has?xation probability 1/N for all r if and only if it is one-rooted(Fig.2f,g).A one-rooted graph has a unique global source without incoming edges.If a graph has more than one root,then the probability of?xation is always zero:a mutant originating in one of the roots will generate a lineage which will never die out,but also never?xate(Fig.2i).Small upstream populations feeding into large downstream populations are also suppressors of selection(Fig.2h).Thus,it is easy to construct graphs that foster drift and suppress selection.Is it possible to suppress drift and amplify selection?Can we?nd structures where the?xation probability of advantageous mutants exceeds r1?

The star structure(Fig.3a)consists of a centre that is connected with each vertex on the periphery.All the peripheral vertices are connected only with the centre.For large N,the?xation probability of a randomly placed mutant on the star is r2?e121=r2T=e12 1=r2NT:Thus,any selective difference r is ampli?ed to r2.The star acts as evolutionary ampli?er,favouring advantageous mutants and inhibiting disadvantageous mutants.The balance tilts towards selection,and against drift.

The super-star,funnel and metafunnel(Fig.3)have the amazing property that for large N,the?xation probability of any advan-tageous mutant converges to one,while the?xation probability of any disadvantageous mutant converges to zero.Hence,these popu-lation structures guarantee?xation of advantageous mutants how-ever small their selective advantage.In general,we can prove that for suf?ciently large population size N,a super-star of parameter K satis?es:

r K?

121=r K

e2T

Numerical simulations illustrating equation(2)are shown in Fig.4a. Similar results hold for the funnel and metafunnel.Just as one-rooted structures entirely suppress the effects of selection,super-star structures function as arbitrarily strong ampli?ers of selection and suppressors of random drift.

Scale-free networks,like the ampli?er structures in Fig.3,have most of their connectivity clustered in a few vertices.Such networks are potent selection ampli?ers for mildly advantageous mutants(

r

Figure3Selection

‘leaves’and the number of vertices in each leaf grows large,these ampli?ers dramatically

increase the apparent?tness of advantageous mutants:a mutant with?tness r on an

ampli?er of parameter K will fare as well as a mutant of?tness r K in the Moran process.

a,The star structure is a K?2ampli?er.b–d,The super-star(b),the funnel(c)and the

guaranteeing the?xation

of any advantageous mutant.The latter three structures are shown here for K?3.The

funnel has edges wrapping around from bottom to top.The metafunnel has outermost

edges arising from the central vertex(only partially shown).The colours red,orange and

blue indicate hot,warm and cold vertices.

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close to 1),and relax to r 1for very advantageous mutants (r ..1)(Fig.4b).

Further generalizations of evolutionary graphs are possible.Suppose in each iteration an edge ij is chosen with a probability proportional to the product of its weight,w ij ,and the ?tness of the individual i at its tail.In this case,the matrix W need not be stochastic;the weights can be any collection of non-negative real numbers.

Here the results have a particularly elegant form.In the absence of upstream populations,if the sum of the weights of all edges leaving the vertex is the same for all vertices —meaning the fertility is independent of position —then the graph never suppresses selection.If the sum of the weights of all edges entering a vertex is the same for all vertices —meaning the mortality is independent of position —then the graph never suppresses drift.If both these conditions hold then the graph is called a circulation,and the structure favours neither selection nor drift.An evolutionary graph has ?xation probability r 1if and only if it is a circulation (see Fig.2e).It is striking that the notion of a circulation,so common in deterministic contexts such as the study of ?ows,arises naturally in this stochastic evolutionary setting.The circulation criterion com-pletely classi?es all graph structures whose ?xation behaviour is identical to that of the homogeneous population,and includes the subset of isothermal graphs (the mathematical details of these results are discussed in the Supplementary Information).

Let us now turn to evolutionary games on graphs 18,19.Consider,as before,two types A and B ,but instead of having constant ?tness,their relative ?tness depends on the outcome of a game with payoff matrix:

A B A B

a b c d

!In traditional evolutionary game dynamics,a mutant strategy A can invade a resident B if b .d .For games on graphs,the crucial condition for A invading B ,and hence the very notion of evol-utionary stability,can be quite different.

As an illustration,imagine N players arranged on a directed

cycle

Figure 4Simulation results showing the likelihood of mutant ?xation.a ,Fixation

probabilities for an r ?1.1mutant on a circulation (black),a star (blue),a K ?3super-star (red),and a K ?4super-star (yellow)for varying population sizes N .Simulation results are indicated by points.As expected,for large population sizes,the simulation results converge to the theoretical predictions (broken lines)obtained using equation (2).b ,The ampli?cation factor K of scale-free graphs with 100vertices and an average connectivity of 2m with m ?1(violet),m ?2(purple),or m ?3(navy)is compared to that for the star (blue line)and for circulations (black line).Increasing m increases the number of highly connected hubs.Scale-free graphs do not behave uniformly across the mutant spectrum:as the ?tness r increases,the ampli?cation factor relaxes from nearly 2(the value for the star)to circulation-like values of unity.All simulations are based on 104–106runs.Simulations can be explored online at http://www.univie.ac.at/

virtuallabs/.

Figure 5Evolutionary games on directed cycles for four different orientations.a ,Positive symmetric.The invading mutant (red)is favoured over the resident (blue)if b .c .b ,Negative symmetric.Invasion is favoured if a .d .For the Prisoner’s Dilemma,the implication is that unconditional cooperators can invade and replace defectors starting from a single individual.c ,Positive anti-symmetric.Invasion is favoured if a .c .The tables are turned:the invader behaves like a resident in a traditional setting.d ,Negative anti-symmetric.Invasion is favoured if b .d .We recover the traditional invasion of evolutionary game theory.

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(Fig.5)with player i placing its offspring into i t1.In the simplest case,the payoff of any individual comes from an interaction with one of its neighbours.There are four natural orientations.We discuss the ?xation probability of a single A mutant for large N .(1)Positive symmetric:i interacts with i t1.The ?xation prob-ability is given by equation (1)with r ?b /c .Selection favours the mutant if b .c .

(2)Negative symmetric:i interacts with i 21.Selection favours the mutant if a .d .In the classical Prisoner’s Dilemma,these dynamics favour unconditional cooperators invading defectors.

(3)Positive anti-symmetric:mutants at i interact with i 21,but residents with i t1.The mutant is favoured if a .c ,behaving like a resident in the classical setting.

(4)Negative anti-symmetric:Mutants at i interact with i t1,but residents with i 21.The mutant is favoured if b .d ,recovering the traditional invasion criterion.

Remarkably,games on directed cycles yield the complete range of pairwise conditions in determining whether selection favours the mutant or the resident.

Circulations no longer behave identically with respect to games.Outcomes depend on the graph,the game and the orientation.The vast array of cases constitutes a rich ?eld for future study.Further-more,we can prove that the general question of whether a population on a graph is vulnerable to invasion under frequency-dependent selection is NP (nondeterministic polynomial time)-hard.

The super-star possesses powerful amplifying properties in the case of games as well.For instance,in the positive symmetric orientation,the ?xation probability for large N of a single A mutant is given by equation (1)with r ?eb =d Teb =c TK 21:For a super-star with large K ,this r value diverges as long as b .c .Thus,even a dominated strategy (a ,c and b ,d )satisfying b .c will expand from a single mutant to conquer the entire super-star with a probability that can be made arbitrarily close to 1.The guaranteed ?xation of this broad class of dominated strategies is a unique feature of evolutionary game theory on graphs:without structure,all dominated strategies die out.Similar results hold for the super-star in other orientations.

Evolutionary graph theory has many fascinating applications.Ecological habitats of species are neither regular spatial lattices nor simple two-dimensional surfaces,as is usually assumed 20,21,but contain locations that differ in their connectivity.In this respect,our results for scale-free graphs are very suggestive.Source and sink populations have the effect of suppressing selection,like one-rooted graphs 22,23.

Another application is somatic evolution within multicellular organisms.For example,the hematopoietic system constitutes an evolutionary graph with a suppressive hierarchical organization;stem cells produce precursors which generate differentiated cells 24.We expect tissues of long-lived multicellular organisms to be organized so as to suppress the somatic evolution that leads to cancer.Star structures can also be instantiated by populations of differentiating cells.For example,a stem cell in the centre generates differentiated cells,whose offspring either differentiate further,or revert back to stem cells.Such ampli?ers of selection could be used in various developmental processes and also in the af?nity matu-ration of the immune response.

Human organizations have complicated network structures 25–27.Evolutionary graph theory offers an appropriate tool to study selection on such networks.We can ask,for example,which net-works are well suited to ensure the spread of favourable concepts.If a company is strictly one-rooted,then only those ideas that originate from the root will prevail (the CEO).A selection ampli?er,like a star structure or a scale-free network,will enhance the spread of favourable ideas arising from any one individual.Notably,scienti?c collaboration graphs tend to be scale-free 28.

We have sketched the very beginnings of evolutionary graph

theory by studying the ?xation probability of newly arising mutants.For constant selection,graphs can dramatically affect the balance between drift and selection.For frequency-dependent selection,graphs can redirect the process of selection itself.

Many more questions lie ahead.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 ?xation,and how do they lead to coexistence in ecological settings 29,30?Furthermore,how does the graph itself change as a consequence of evolutionary dynamics 31?Coupled with the present work,such studies will make increasingly clear the extent to which population structure affects the dynamics of evolution.A

Received 10September;accepted 16November 2004;doi:10.1038/nature03204.

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Supplementary Information accompanies the paper on https://www.wendangku.net/doc/6715435392.html,/nature .

Acknowledgements The Program for Evolutionary Dynamics is sponsored by J.Epstein.E.L.is supported by a National Defense Science and Engineering Graduate Fellowship.C.H.is grateful to the Swiss National Science Foundation.We are indebted to M.Brenner for many https://www.wendangku.net/doc/6715435392.html,peting interests statement The authors declare that they have no competing ?nancial interests.

Correspondence and requests for materials should be addressed to E.L.(erez@https://www.wendangku.net/doc/6715435392.html,).

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Supplementary Notes

Here we sketch the derivations of eq(1)for circulations and eq(2)for superstars.We give a brief discussion of complexity results for frequency-dependent selection and the computation underlying our results for directed cycles.We close with a discussion of our assumptions about mutation rate and the interpretations of?tness which these results can accommodate.

Evolution on graphs is a Markov process.

Let G be a graph whose adjacency matrix is given by W.Let P?V be the set of vertices occupied by a mutant at some iteration.P represents a state of the typical Markov chain E G which arises on an evolutionary graph.Analogously,the states P={1,2,...N}are the typical states of the Moran process M.

(For two types of individuals,the states of the explicit Markov chain E G are the2n possible arrangements of mutants on the graph.The transition probability between two states P,P is0unless|P\P |=1or vice versa.Otherwise,if P\P =v?,then the probability of a transition from P to P is

w(v,v?)

v∈G\P

N+|P|(r?1)

where the numerator is the sum of the weights of edges entering v*from vertices outside P.Similarly,the probability of a transition from P to P is

w(v,v?)

v∈P

N+|P |(r?1)

In practice,the resulting matrix is large and not very sparse.Consequently,it can be di?cult to work with directly,and we will not revisit it in the course of these notes.) We now de?ne the notion ofρ-equivalency.

De?nition1.A graph G isρ-equivalent to the Moran process if the cardinality map f(P)=|P|from the states of E G to the states of M preserves the ultimate?xation probabilities of the states.Equivalently,we need

ρ(r,G,P,N)=1?1/r P 1?1/r N

whereρ(r,G,P,N)is the probability that a mutant of?tness r on a graph G,given any initial population of size P,eventually reaches the?xation population of N.(Note that this function is often unde?ned:on most graphs,di?erent initial conditions with the same number of mutants have di?erent?xation probabilities.)

Note that eq(1)is obtained in the case P=1.

This shows that the requirement of preserving?xation probabilities leads inevitably to the preservation of transition probabilities between all the states.In particular,it means that the population size on G,|P|,performs a random walk with a forward bias of r,e.g.,where the probability of a forward step is r/(r+1).

Evolution on circulations is equivalent to the Moran process.

We now provide a necessary and su?cient condition forρ-equivalence to the Moran process for the case of an arbitrary weighted digraph G.The isothermal theorem for stochastic matrices is obtained as a corollary.First we state the de?nition of a circulation.

De?nition2.The matrix W de?nes a circulation?

?i,

j w ij=

j

w ji

This is precisely the statement that the graph G W satis?es

?v∈G,w o(v)=w i(v)

where w o and w i represent the sum of the weights entering and leaving v.

It is now possible to state and prove our?rst result.

Theorem1.(Circulation Theorem.)The following are equivalent:

(1)G is a circulation.

(2)|P|performs a random walk with forward bias r and absorbing states at{0,N}.

(3)G isρ-equivalent to the Moran process

(4)

ρ(r,G,P,P )=1?1/r P 1?1/r P

whereρ(r,G,P,P )is the probability that a mutant of?tness r on a graph G given any initial condition with P mutants eventually reaches a mutant population of P .

Proof.We show that(1)→(2)→(3)→(4)→(1).

To see that(1)→(2),letδ+(P)(resp.δ?(P))be the probability that the mutant population in a given state increases(resp.decreases),where P?V is just the set of vertices occupied by a mutant,corresponding to the present state.The mutant population size will only change if the edge selected in the next round is a member of an edge cut of P,e.g.,the head is in P and the tail is not,or vice-versa.

The probability of a population increase in the next round,δ+(P),is therefore just the weight of all the edges leaving P,adjusted by the?tness of the mutant r.Thus

δ+(P)=

w o(P)r

w o(P)r+w i(P)

where w o and w i represent the sum of the weights entering and leaving a vertex set P.Similarly,

δ?(P)=

w i(P)

w o(P)r+w i(P)

Dividing,we easily obtain

δ+(P)δ?(P)=r

w o(P)

w i(P)

We may also observe that

w o(P)?w i(P)=(

v∈P w o(v)?

e|e1,e2∈P

w(e))?(

v∈P

w i(v)?

e|e1,e2∈P

w(e)) =(

v∈P

w o(v)?

v∈P

w i(v))

where the second and fourth sums in the latter equality are over edges whose two endpoints are in P.Since this vanishes when G is a circulation,we?nd that on a circulation

?P?V,w o(P)=w i(P)

and therefore

δ+(P)

δ?(P)

=r

for all P.

Thus the population is simply performing a random walk with forward bias r as de-sired,yielding(1)→(2).

(2)→(3)follows immediately from the theory of random walks.

It is easy to see that(3)→(4)by conditional probabilities.We know that

?P ≥P,ρ(r,G,P,N)=ρ(r,G,P,P )?ρ(r,G,P ,N)

Therefore

?P ≥P,ρ(r,G,P,P )=ρ(r,G,P,N)ρ(r,G,P ,N)

=1?1/r P

1?1/r N

(

1?1/r P

1?1/r N

)?1 =

1?1/r P

1?1/r P

which is the desired result.

To complete the proof,we show that(4)→(1).By(4),we know

ρ(1,2,G,r)=1?1

r

1?1

r2

=

r

r+1

But this is only satis?ed for all populations of size1if we have

?v,δ+(v)

δ?(v

=r

As we saw above,this implies that

?v,w o(v)=w i(v)

which demonstrates that G must be a circulation and completes the proof.

The isothermal result is just a corollary.

Theorem2.(Isothermal Theorem.)G isρ-equivalent to the Moran process?G is isothermal,e.g.,W is doubly-stochastic.

Superstars are arbitrarily strong ampli?ers of natural selection.

We now sketch the derivation of the ampli?er theorem for superstars,denoted S K

L,M

, where K is the ampli?cation factor,L the number of leaves,and M the number of vertices in the reservoir of each leaf.First we must precisely de?ne these objects.

De?nition3.The Super-star S K

L,M

consists of a central vertex v center surrounded by L leaves.Leaf contains M reservoir vertices,r ,m and K-2ordered chain vertices c ,1 through c ,K?2.All directed edges of the form(r ,m,c ,1),(c ,w,c ,w+1),(c ,K+2,v center), and(v center,r ,m)exist and no others.In the case K=2,the edges are of the form (r ,m,v center),and(v center,r ,m).Illustrations for K=2and K=3are given in Fig 3.The weight of an edge(i,j)is given by1/d o(i),where d o is the out-degree.

Now we may move on to the theorem.

Theorem 3.(Super-star Theorem.)As the number and size of the leaves grows large,the?xation probability of a mutant of?tness r on a super-star of parameter K converges toward the behavior of a mutant of?tness r K on a circulation:

lim L,M→∞ρ(S K

L,M

)→

1?1/r K

1?1/r KN

Proof.(Sketch)The proof has several steps.

First we observe that for large M,the mutant is overwhelmingly likely to appear outside the center or the chain vertices.

Now we show that if the density of mutants in an upstream population is d,then the probability that an individual in a population immediately downstream will be a mutant at any given time is dr

1+d(r?1)

.In general,if we haveηpopulations,one upstream of the other,the?rst of which has mutant probability density d=d(1),we obtain the following probability density for theνth population

d(ν)=

drν

1+d(rν?1)

The result follows inductively from the observation that

d(j+1)=

dr j

1+d(r j?1)

r

1+dr j

1+d(r j?1)

(r?1)

=

dr j+1

1+d(r j+1?1)

For the super-star,this result is precise as we move inward from the leaf vertices along the chain leading into the central vertex,where derivation of an analogous result is necessary.Here we require careful bounding of error terms,and allowing L to go o?to the in?nite limit.This is in order to ensure that‘feedback’is su?ciently attenu-ated:otherwise,during the time required for information about upstream density to propagate to the central vertex,the upstream population will have already changed too signi?cantly.In this latter regime,‘memory’e?ects can give the resident a very

signi?cant advantage:the initial mutant has died before the central vertex is fully a?ected by its presence.For su?ciently many leaves feedback is irrelevant to?xation. In the relevant regime we establish that the central vertex is a mutant with probability

d(K?1)=

dr K?1

1+d(r K?1?1)

Our result follows by noting that the probability of an increase in the number of mutant leaf vertices during a given round is very nearly

r

N+P(r?1)

dr K?1

1+d(r K?1?1)

(1?d)

and the probability of a decrease is

1

N+P(r?1)

1?d

1+d(r K?1?1)

d

Dividing,all the terms cancel but an r K in the numerator.Thus the mutant popula-tion in the leaves performs a random walk with a forward bias of r K until?xation is guaranteed or the strain dies out.

In the spirit of this result,we may de?ne an ampli?cation factor for any graph G with N vertices as the value of K for whichρ(G)=1?1/r K

1?1/r KN

.We have seen above that a superstar of parameter K has an ampli?cation factor of K as N grows large.

The?xation problem for frequency-dependent evolution on graphs is at least as hard as NP.

NP-hard problems arise naturally in the study of frequency-dependent selection on graphs.Let us consider the general case of some?nite number of types;a state of the graph is a partition of its vertices among the types,or a coloring.Given a graph G and an initial state I,let VULNERABILITY be the decision problem of whether, given a graph G,an initial state I,a small constant ,and a desired winning type T, fewer than w individuals can mutate to T so as to ensure?xation of the graph by

T with probability at least1- .By reduction from the Boolean Circuit Satis?ability problem,it can be shown the VULNERABILITY is NP-hard.We omit the details of the proof here.

Frequency-dependent evolution on graphs leads to a multiplicity of inva-sion criteria.

The following computation establishes our observations about directed cycles.

Proposition4.(Fixation on Directed Cycles.)For large N,the directed cycle favors mutants where b>c(resp.a>d,a>c,b>d)in the positive symmetric(resp.negative symmetric,positive antisymmetric,negative antisymmetric)orientations.

Proof.(Sketch)For the positive symmetric case,we obtain eq(1)with r=b/c as a straightforward instance of gambler’s ruin with bias b/c.In the other three orientations,a bit more work is required to account for the case where the patch is of size1or size of N-1.In the negative symmetric and positive antisymmetric orientations,the mutant has an aberrant?tness of b for patch sizes of exactly1(near extinction).In both negative orientations,the resident has an aberrant?tness of c when the mutant patch is of size N-1(near?xation).Thus we must do some work to ensure that these aberrations do not ultimately a?ect which types of mutants are favored on large cycles.

We must evaluate the following expression to obtain the?xation probability of the biased random walk:

ρ=

1

1+ΣN?1

i=1

Πi

j=1

q i

i

The values of p i and q i represent probabilities of increase and decrease when the population is of size i.We obtain

ρ?s=

b(d?a)

bd?ab?ad+(d/a)N?2(ad+cd?ac)

ρ+a=

b(c?a)

bc?ab?ac+(c/a)N?2(c2)

ρ?a=

b(d?b)

?b2+(d/b)N?2(bd+cd?bc)

for the negative symmetric,positive antisymmetric,and negative antisymmetric cases. For large N,these expressions are smaller than the neutral?xation probability1/N if d/a(resp.c/a,d/b)is greater than one;if it is less than1,the?xation probabilities converge to

ρ?s=

b(a?d)

b(a?d)+ad

ρ+a=

b(a?c)

b(a?c)+ac

ρ?a=b(b?d)

b2

and the mutant is strongly favored over the neutral case.

Results hold if fertility and mortality are independent Poisson processes.

Finally,we will make some remarks about our assumptions regarding mutation rate and the meaning of our?tness values.

It is generally the case that suppressing either selection or drift,and in particular the latter,is time intensive.Good ampli?ers get arbitrarily large asρ→1or0,and have increasingly signi?cant bottlenecks.Thus,?xation times get extremely long the more e?ectively drift is suppressed.However,since we are working in the limit where mutations are very rare,this timescale can be ignored.The rate of evolution reduces to the product of population size,mutation rate,and?xation probability.

In our discussions,we have treated?tness as a measure of reproductive fertility.But a range of frequency-independent interpretations of?tness obtain identical results.If

instead of choosing an individual to reproduce in each round with probability propor-tional to?tness,we choose an individual to die with probability inversely proportional to?tness,and then replace it with a randomly-chosen upstream neighbor,theρval-ues obtained are identical.Put another way,as long as reproduction(leading to death of a neighbor by overcrowding)and mortality(leading to the reproduction of a neighbor that?lls the void)are independent Poisson processes,our results will hold.

中外动画赏析

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