An analysis of the fixation probability of a mutant on special classes of non-directed graphs

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1 An analysis of the fixation pobability of a mutant on special classes of non-diected gaphs By M. Boom and J. Rychtář Depatment of Mathematics, Univesity of Sussex, Bighton BN 9RF, UK Depatment of Mathematics and Statistics, Univesity of Noth Caolina Geensboo, NC74, USA Thee is a gowing inteest in the study of evolutionay dynamics on populations with some non-homogeneous stuctue. In this pape we follow the model of Liebeman et al. [Liebeman, E., Hauet, C., Nowak, M.A. 5 Evolutionay dynamics on gaphs. Natue 43 (3), 3 36] of evolutionay dynamics on a gaph. We investigate the case of non-diected equally weighted gaphs and find solutions fo the fixation pobability of a single mutant in two classes of simple gaphs. We futhe demonstate that finding simila solutions on gaphs outside these classes is fa moe complex. Finally we investigate ou chosen classes numeically, and discuss a numbe of featues of the gaphs; fo example we find the fixation pobabilities fo diffeent initial stating positions and obseve that aveage fixation pobabilities ae always inceased fo advantageous mutants as compaed against those of unstuctued populations. Keywods: Evolutionay dynamics, sta, linea gaph, andom walk, Makov chain. Intoduction Evolutionay dynamics models ae widespead, but have geneally assumed homogeneous populations. The study of evolutionay dynamics on gaphs was investigated in the pape Liebeman et al. (5), othe impotant wok on this subject being in Edös & Renyi (96); Nagylaki & Lucie (98); Baabasi & Albet (999). Each vetex o node epesents an individual in the population, and individuals can epoduce into neighbouing vetices, i.e. those connected by an edge. In Liebeman et al. (5) at each stage an individual was selected andomly, with pobability popotional to its fitness, which then copied itself into one of the vetices it was connected to. (It should be noted that thee ae othe possible dynamics. An example is the biased vote model, e.g. see Bamson & Giffeath (98), whee an individual is chosen at andom to be emoved and is eplaced by a copy of one of its neighbous). Liebeman et al. (5) consideed diected gaphs whee connections between vetices can be one-way only (e.g. it is possible fo an individual at to epoduce into, but not fo one at to epoduce into ), with geneal weightings indicating the pobability that any paticula vetex would be eplaced, given the chosen eplacing vetex. They showed seveal inteesting and impotant The eseach was suppoted by EPSRC gant EP/E434/ and by NSF Aticle submitted to Royal Society TEX Pape

2 M. Boom and J. Rychtář esults; fo instance, diffeent gaph stuctues could yield diffeent pobabilities fo fixation of a single mutant. In homogeneous populations the pobability of fixation in a population with N individuals (and so N vetices) is given by the Moan pobability P Moan = / / N (.) whee esident individuals have baseline fitness and mutants have fitness (each individual being chosen as the epoducing individual with pobability popotional to its fitness). It was shown in Liebeman et al. (5) that this pobability holds unde a condition on the weightings on the gaph, any gaph satisfying this condition being efeed to as an isothemal gaph. Howeve, othe gaph stuctues allow the pobability of fixation of an advantageous mutant ( > ) to convege to eithe o as N tends to infinity. Most of the inteesting esults fom Liebeman et al. (5) elied on gaphs being diected and the weights of connections fom a given vetex to be diffeent fom each othe. In this pape we look at non-diected gaphs with equal weights. We show that in this setting, the fomula (.) holds fo egula gaphs, gaphs whee evey vetex has the same degee, and only fo them. We then show that evolutionay dynamics on a gaph with N vetices leads to a system of N equations; with the exception of a cicle (a egula gaph case) and a line (a non-egula gaph). We use symmeties to educe the numbe of equations to n + fo a sta with N = n + vetices. In Liebeman et al. (5), the appoximation of the fixation pobability fo stas fo lage n was given by P = / n (.) Hee we find the exact fixation pobabilities fo any and n. We then analyze the dynamics on the line. The analysis is quite had to pefom even in this simple case, although we make substantial pogess. We also make suggestions about how to attack the moe geneal poblem without simply esoting to numeical methods and simulation.. Evolutionay dynamics on gaphs Let G = (V, E) be a finite, undiected and connected gaph, whee V is the set of vetices and E is the set of edges. We assume that the gaph is simple, i.e. no vetex is connected to itself and thee ae no paallel edges. We study evolutionay dynamics as descibed in Liebeman et al. (5), see also Nowak (6). We teat the dynamics as a discete time Makov chain. At the beginning, a vetex is chosen unifomly at andom and eplaced by a mutant with fitness, all emaining vetices having fitness. If the mutants aleady inhabit pecisely the vetices in the set C V, then in the next step the mutants will inhabit vetices in eithe ) a set C {j}, j C, povided a) a vetex i C was chosen fo epoduction and b) it placed its offsping into vetex j; o ) a set C \ {i}, i C, povided a) a vetex j C was selected fo epoduction and b) it placed its offsping into i; o Aticle submitted to Royal Society

3 An analysis of the fixation pobability ) a set C, povided an individual fom C (V \ C) eplaces anothe individual fom C (V \ C). The states and V ae the absobing points of the dynamics. The tansition pobabilities of the above Makov chain ae detemined by a) the pobability that a given vetex will be selected fo epoduction and b) the pobability that, once selected, it places its offsping into anothe given vetex. We set the fitness of an individual at vetex i as f i {, }, whee f i = means that the individual is a mutant. An individual at i is selected fo epoduction with pobability s i = f i j V f j. (.) The gaph stuctue is epesented by a matix W = (w ij ), whee w ij is the pobability of eplacing a vetex j by a copy of a vetex i, povided vetex i was selected fo epoduction, {, if i and j ae connected, ei w ij =, othewise, whee e i is the numbe of edges incident to the vetex i, so that edges have equal weights. Let P C denote the pobability of mutant fixation given mutants cuently inhabit a set C. The ules of the dynamics yield, see Liebeman et al. (5), ) (w ij P C {j} + w ji P C\{i} P C = i C j C ( ) (.) wij + w ji i C j C with P = and P V =. This system has a unique solution following fom the uniqueness of a Makov chain given a known initial distibution. Thee is a unique distibution ove the states at time (a single mutant is intoduced to the population at a andomly chosen vetex). The Kolmogoov equations then give a unique distibution at step s + conditional on uniqueness at step s. As s tends to infinity, thee is convegence to the set of absobing states (eithe all mutants o all esidents). This yields a unique limiting distibution, so a unique fixation pobability fom the initial distibution. The system (.) of linea equations is vey lage (typically of the ode of V equations, see 4) and vey spase (fom any state C, one can go to at most V othe states). 3. Regula gaphs A gaph is called isothemal if j w ji is constant as a function of i. A gaph is isothemal if and only if the matix W = (w ij ) is double stochastic (Liebeman et al., 5), i.e. w ji = j Aticle submitted to Royal Society

4 4 M. Boom and J. Rychtář It is poved in Liebeman et al. (5) that if a gaph is isothemal then P C = C V (3.) Hee we give a diffeent poof of this statement and one moe equivalent condition to being isothemal. Theoem 3.. A simple connected undiected gaph G = (V, E) is isothemal if and only if it is egula. Poof. Clealy, if G is egula, e i is constant, and thus G is isothemal. Now suppose that the elation in the othe diection is not tue. Conside a set C = {i, e i = min{e v, v V }}. Since, by ou assumption, C V, thee must be a vetex i C that is connected to a vetex j V \ C. Then, a contadiction. w vi = w ji + w vi < + e v i v j v j w vi e i + e i e i =, In ode to solve (.) fo an isothemal gaph, let us assume that P C depends upon the size of C, so that only P C = x C. (3.) By theoem 3., w ij attains only one nonzeo value (/k, whee k is the degee of any vetex in G) and thus (.) educes to x C = + x C x C. (3.3) This is a standad diffeence equation that gives the equied Moan pobabilities. Consequently, ou assumption (3.) leads to a solution of (.) and by the uniqueness of the solution, the solution must satisfy the popety (3.). 4. Complexity of the dynamics Since at evey vetex of a gaph G = (V, E) thee can be eithe a esident o a mutant, thee ae up to V potential mutant fomations and thus up to V equations in (.). Some fomations of mutants on a given gaph ae identical because of symmeties (automophisms) of the gaph. Cetain gaphs (like a complete gaph, o a sta gaph - see 5) thus have only a few possible mutant-esident pattens since thei automophisms goup is vey ich. Fo othe gaphs, like a line, the gaph stuctue itself yields only symbolic eduction of the numbe of pattens because the automophism goup consists of only a few nontivial elements. Taking the automophism goup of a gaph G, Aut(G), into account, we can use Bunside s obit counting theoem (Tucke, 994) to find the exact numbe Aticle submitted to Royal Society

5 An analysis of the fixation pobability... 5 of possible fomations. Conside the set X consisting of all possible V mutantesident pattens. Fo f Aut(G), let Fix(f) = {v V, f(v) = v}. If Fix(f) V, then f fixes Fix(f) + elements of X (one has feedom to put a mutant o a esident in any vetex v Fix(f), plus one can place eithe mutants o esidents into all vetices of V \ Fix(f)). Clealy, identity on G is the only f Aut(G) that fixes all elements of G. Bunside s theoem then yields the total numbe of Mutant-Resident Fomations (MRF) of G as MRF(G) = V + (4.) Aut(G) f Aut(G),f id G Fix(f) + The above consideed the gaph stuctue only, not consideing the ules of the dynamics at all. Clealy, any non-initial state of the dynamics contains at least one paent-offsping pai of connected vetices. Consequently, altenating pattens, i.e. pattens whee any pai of connected vetices is inhabited by a mutant at one vetex and a esident an the othe vetex, cannot be attained as a esult of the dynamics. Altenating pattes ae possible if and only if the gaph does not contains an odd cycle (i.e. if the gaph is bipatite). Thee ae at most altenating pattens. On a cicle o on a line, any mutant fomation esulting fom the dynamics consists of a connected segment. Hence, thee ae of the ode of V pattens on a cicle ( V possibilities whee the segment stats and V possibilities whee it ends, plus the pattens with all o no mutants) and V / pattens on a line ( V possibilities to stat, and on aveage V / possibilities whee the segment can end). Moeove, the otations on the cicle help us to educe the numbe of equations to V ; the symmety of a line also educes the numbe of equations by a facto to appoximately V /4. The next theoem shows that fo the vast majoity of gaphs, the system (.) consists of oughly MRF(G) equations. Theoem 4.. If a gaph contains a vetex of degee at least 3 (i.e. the gaph is neithe a line no a cicle), and the dynamics is in any non-absobing state, then thee is a nonzeo pobability that the dynamics will evolve to any of the possible MRF(G) states (MRF(G) states if the gaph is bipatite). Befoe poving theoem 4., we pove a esult equied fo the poof. Lemma 4.. Let vetices v and v be connected to a vetex t. Futhemoe, assume that thee is a mutant in v and a esident in v. Then, we can fill any patten to any subtee stuctue connected to t and not containing v and v. Poof of lemma 4.. The poof goes by the induction on the height of the subtee stuctue. If the height is (i.e. the stuctue is only the vetex t), we can clealy fill it by a mutant o a esident. Now assume that we can fill any patten to the subtee of height n and that ou stuctue has a height n. Fist, spead the esidents (fom v ) to get the stuctue which contains only esidents, see figue a). Next, spead the mutants (fom v ) to evey vetex but the leaves whee thee ae esidents in the taget patten. Afte this step, all of the leaves have the inhabitants of the taget patten, see figue b). Cut the leaves and what emains is the stuctue of height n. This can be filled by any patten by the induction hypothesis. This concludes the poof of lemma 4.. Aticle submitted to Royal Society

6 6 M. Boom and J. Rychtář v v v v v v v v v v t t t t t a) b) c) d) e) Figue. Filling a given patten (figue e)) to a subtee connected to t. The oiginal configuation is a) - a mutant in v and a esident in v. Poof of theoem 4.. We now pove theoem 4. using multiple applications of lemma 4.. To do this, thee ae some technical difficulties that have to be ovecome. Fistly the oiginal gaph has to be timmed to become a tee so we can apply the lemma. Secondly we need space to manoeuve since lemma 4. does not allow us to fill pattens behind the vetices v and v, and thus we need to abitaily flip the oles of vetices connected to t. We obtain this space by collapsing the taget patten by a single vetex, which allows us to have the cental vetex completely fee fo ou use. Finally, the timming could cause some pattens to become inaccessible (altenating) on the tee, although they wee not altenating on the oiginal gaph, so we have to deal with these pattens in one moe step. Fistly we tim the gaph to get a tee. Denote one of the vetices with degee at least 3 by t and label thee of its neighbous by t, t, t 3. Next, tim the gaph by cutting a sufficient numbe of edges to get a connected tee (a gaph with no cycle) consisting of all of the oiginal vetices and yet keeping all of the edges t t j, i =,, 3 intact. Label the emaining neighbous (if any) of t by t 4,..., t k. We will show that we can each any state that is non-altenating (on this tee) fom any non-absobing state even if only the edges of this timmed gaph ae used. Secondly we collapse a patten to get the cental vetex t fo fee usage. Since the taget state is not altenating, thee ae two connected vetices with the same type of inhabitants. We may assume that the vetices ae on the banch (linea set) B = b b b b 3 b l with b = t and b = t ; let b i and b i+ be the two vetices with the same inhabitants and with the lowest index i possible. Let S denote the taget state and S the state that is the same as the taget state except that at vetices b j, j =,, i it has inhabitants fom the vetices b j of the taget state. Note that S and S ae each attainable fom the othe by a shift of the patten along the line. See figue a) and b) fo an illustation of this. We now move the mutants into a position whee lemma 4. can be applied. Fom above it is enough to each the state S. Also, we may assume that in S, t is inhabited by a mutant (i.e. in the oiginal taget state, t is inhabited by a mutant). If the contay is tue, we would just intechange the ole of esidents and mutants in the following aguments. Clealy, any non-absobing state can evolve into a state with one mutant only. If the mutant is not at t aleady, we can elabel t and t 3 such that the shotest path fom the mutant s position to t goes though t. Now the mutants can spead to t by this shotest path, leaving the tail of mutants behind. The tail can be cleaed by speading esidents fom eithe t o t 3. See figue c)-e) fo illustation. Aticle submitted to Royal Society

7 An analysis of the fixation pobability... 7 III II. I. I. II. III. t t t t 3 t t t t 3 t t t t 3 a) b) c) d) e) Figue. a) and b) Shifting between pattens on the line. c), d) and e) Moving a single mutant fom a geneal position to t. Applying lemma 4. fo the fist time, we can fill any patten to subtees stating at t 3, t 4,, t k. We now otate the mutant-esident patten aound t and fill the banch behind vetex t ; then otate again and fill behind t. We use lemma 4. to place a esident at t 3 (thus having a mutant at t and esident at t 3 ) and use lemma 4. again to fill the patten S to the subtee stating at t. At the end, thee will be a mutant in t. Since now we have a mutant in t and a esident in t 3, we can fill the equied patten to the subtee stating at t. If thee has to be a mutant at t 3, place it thee by speading fom t though t. In any case, finish by shifting the patten fom S to S. So fa, we wee able to each any state that is not altenating on the timmed gaph. It is possible that an altenating state on the timmed gaph is not an altenating state on the oiginal gaph. If we want to each this patten, let us pick vetices w, w witnessing that the patten is not altenating on the oiginal gaph. We may assume that they ae both inhabited by mutants. If we change the mutant in w into a esident, we get a patten that is not altenating on the timmed gaph. In paticula, we can each it as shown above. To each the equied patten, it only emains to spead the mutant fom w into w ; we can do that because vetices w and w ae connected in the oiginal gaph. 5. Dynamics on stas In this section we conside a sta - a non-diected gaph with N = n + vetices labeled,,..., n whee the only edges ae between vetices and i, i =,..., n. The vetex is called a cente and the vetices,..., n can be called the leaves. The automophism goup is isomophic to the goup of pemutations on leaves and the state of the dynamics can be descibed by the numbe of mutants at the leaves and by an indicato of whethe o not thee is a mutant at the cente. See figue 3 fo a scheme of the dynamics. Let Pi, espectively) denote the pobability of fixation given thee ae i mutants at the leaves and thee is a (thee is no, espectively) mutant at the cente. The ules of the dynamics yield the following system of n + equations. (P i Aticle submitted to Royal Society

8 8 M. Boom and J. Rychtář Figue 3. States of a dynamics on a sta with n = 5 leaves. P i = P i = + n P i+ + n + n P i (5.) n n + P i + n + P i (5.) fo i =,..., n with bounday conditions P = and P n =. The equation (5.) can be eaanged to Pi = Pi + n (P i Pi ) (5.3) We can use (5.3) and (5.) to inductively calculate P, P, P,... as a function of P to get Pi = P + n i ( ) j n + n + (n + ) Since Pn =, we get Since, by (5.) and (5.), P = + n n+ P = P = j= n j= ( + n P n n + P ) j n+ (n) we get that the aveage fixation pobability fo a mutant is ϱ = Notice that fo lage n we get ( (n + ) + n n+ ϱ n n n + +n n j= + n j= ( n+ (n) = j n which means that we ae ecoveing the fomula (.) fom Liebeman et al. (5). Figue 4 contains illustations of the esults fo fixation pobability and a compaison with the fomula (.) and with the fixation fo a Moan pocess (.). ) j ) Aticle submitted to Royal Society

9 An analysis of the fixation pobability a) n b) Figue 4. The mean fixation pobability fo a sta (middle cuve) and compaison to P Moan given in (.) (lowest cuve) and fomula (.) (uppe cuve). This compaison is shown in a) fo a ange of values of n and =.5, and in b) fo a ange of values of and n =. 6. Dynamics on lines In this section we conside a line with N = n + vetices labeled,,..., n, which is a non-diected gaph whee vetices i and j ae connected if and only if i j =. Hence the vetices and n ae connected to just a single vetex, and will be efeed to as the end vetices, whilst all othe vetices ae connected to exactly two othes. In this section, we code the admissible mutant configuation by a pai of numbes (i, j), i j n +, tanslate the dynamics (.) into this coding, identify the evolution unde this dynamics with a andom walk on a tiangle in a -d squae lattice and then educe the numbe of equations fom the ode of n /4 to n. (a) States and tansition pobabilities The mutant population stats with a single individual; new mutants can only aise in vetices on neighbouing points on the line, and mutants can only be lost fom vetices connected to a esident individual. Thus the population of mutants foms a line segment i, i +,..., j fo some pai (i, j) whee i j, and the state of the system can be descibed by this pai of numbes only. The evolution of the population can thus be seen as a -d andom walk on a tiangula set T = {(i, j), i j n + }. Let P i,j denote the pobability of mutant fixation given that we ae at the state (i, j). Once the population eaches the diagonal state (i, i) fo i n +, the mutants ae extinct, i.e. P i,i =. (6.) Once a mutant eaches an end vetex, i.e. the population is in the state (, j) o (j, n + ) fo some j n +, it will neve be emoved fom the end vetex unless though extinction, since it can only be emoved by a esident on its sole neighbouing vetex, which in tun can only be pesent if the end vetex individual is the sole mutant in the population. Hence, the population stays on these bounday lines once it eaches them. In 6b we calculate that P,j = P n+ j,n+ = n+ n n+ j n+ n +. (6.) Aticle submitted to Royal Society

10 M. Boom and J. Rychtář We thus have to solve the -dimensional andom walk on a set T given the bounday conditions (6.) and (6.). We poceed to investigate the tansitions in the inteio of T. Fist, it should be noted that fo most choices of a vetex fo epoduction, the population does not change. The only change occus when we choose a vetex on a bounday between mutants and non-mutants. When i < j n, thee ae two boundaies and a change of state may occu if any of the vetices i, i, j o j ae chosen. None of these vetices is an end vetex and thus P i,j = ( + ) P i,j + ( + ) P i,j+ + ( + ) P i+,j + ( + ) P i,j. (6.3) When a mutant occupies one of the end vetices, thee is just one mutantesident bounday, and only two choices of vetices allow a change of state. P, = + P, (6.4) P,j = + P,j+ + + P,j, j n (6.5) P,n = + P,n + + (6.6) P j,n+ = P,n+ j, j n + (6.7) When a mutant occupies a vetex next to an end vetex ( o n ), with a esident at the coesponding end vetex we have, fo j n, P,j = + 3 P,j P,j P,j P,j, (6.8) P,n = + P,n + + P,n+ + + P,n + + P,n, (6.9) P j,n = P,n+ j. (6.) The system (6.3)-(6.) consists of the ode of n / equations. By symmety (equations (6.7) and (6.)), the system educes to the ode of n /4 equations. (b) Bounday conditions The equations (6.4), (6.5), and (6.6) ae diffeence equations fo P,j. Using standad methods (see e.g. Nois (997, Chapte )), by (6.5), we have to find the oots of ( + )x + x + =. Since the oots ae x = and x =, we get P,j = A + B ( ) j. (6.) The values A, B ae detemined afte technical calculation using equations (6.4) and (6.6). This yields, fo j n +, P,j = P n+ j,n+ = n+ j + n n+ + n n+ Aticle submitted to Royal Society

11 An analysis of the fixation pobability... = n+ j j + j n + n. (6.) (c) The inne bounday Using the equations (6.8), (6.9) and (6.) we obtain ( P,j ) P,j P,j = P,j + P,j, (6.3) ( + )P,n P,n = P,n. (6.4) Since we can calculate P,j, j =,..., n, we just need to calculate P,j in tems of P,k, k =,..., n to get a system of n equations fo n unknowns P,k, k =,..., n. (d) Inteio points In Mille (994) a -d andom walk on a squae lattice S = {(i, j); i, j n + } with abitay bounday conditions at states (, j), (n +, j), (i, ), (i, n + ), i, j n + was solved. In this pape we elabel the bounday coodinates of the squae to be appopiate to the application fom ou poblem, giving with bounday conditions at states S = {(i, j); i, j n} (, j), (n, j), (i, ), (i, n), i, j n. Ou goal is to extend the andom walk fom T onto S while giving fictional bounday conditions on S such that the estiction of the extended walk will give us exactly the oiginal walk on T with the bounday conditions (6.) and (6.). This is illustated in Figue 5. In the notation of Mille (994), we have P i,j = ( + ) n { a= P a, T (a,)ij + P,a T (,a)ij + P a,n T (a,n )ij + } P n,a T (n,a)ij (6.5) whee T (a,b)ij is the expected numbe of times the state (a, b) is visited given that the initial state is (i, j). By Mille (994, equation 4.3), T (a,b)ij = i a j+b f(a, b, i, j ), (6.6) Aticle submitted to Royal Society

12 M. Boom and J. Rychtář / / / / / / / / / / / / / / / / / / / / / /+ /+ Figue 5. States of a dynamics on a line with n + = 6 vetices (below the stais ). The lage ectangle epesents whee the dynamics will be extended in 6d. The dotted aows epesent fictional tansitions that will be used fo the extension. The states on the bounday of the lage ectangle ae absobing states of the fictional extension. The smalle ectangle shows the states whee the tansitions depend on the diection only and not on the actual state. The black cicles epesent mutants. The lines with gey cicles epesent the fictional states [i, j) fo j > i. whee f(a, b, i, j) = 4 (n ) n k,s= ( sin ikπ n ) ( sin [ cos ) ( sin ( akπ n kπ n bsπ n ) + cos ) ( ( sin jsπ sπ n n ) )]. Notice that f(a, b, j, j) = f(b, a, j, j). (6.7) (e) Fictitious bounday conditions It should be noted that in this section we ae not dealing with the fixation pobabilities as such, but athe finding methods of solving an abitay set of equations. Thus we will have expessions in tems that esemble pobabilities, but which ae not (e.g. P i,j whee i > j), which have negative solutions. These solutions howeve obey the coect tansition equations and have the coect bounday conditions fo the egion of inteest. In this section we give the fictitious bounday conditions fo the andom walk on the whole squae. So fa, in 6c we have equations fo P,a and P a,n fo all a n. We need to calculate P a, and P n,a. By (6.), (6.5), (6.6), and (6.7) = P j,j = ( + ) n { ] f(a,, j, j) [P a, a + P,a a a= +f(a, n, j, j) [P a,n n+ a + P n,a a+ n ]}. Aticle submitted to Royal Society

13 An analysis of the fixation pobability... 3 The above can be tue if This implies Denote P a, a + P,a a =, P a,n n+ a + P n,a a+ n =. P a, = a P,a, P n,a = n a P a,n. (f ) Reduction to n equations dx,y i,j = f(x, y, i, j) f(y, x, i, j). Consequently, by (6.), (6.5) and (6.6), P i,j = = i j ( + ) i j ( + ) n { a= n { a= P,a a d i,j,a + P a,n n+ a d i,j a,n P,a a d i,j,a + P,n a+ n+ a } d i,j a,n }. (6.8) Refeing back to (6.3) and (6.4), this gives us the following linea simultaneous equations in the pobabilities P,j, j n, and P,n P,j = j n { ( + ) a= P,a a d,j,a + P,n a+ n+ a P,j+ + P,n = ( + )P,n P,n. d,j a,n ( + 3 ) P,j P,j Afte solving the above system fo P,k, we can econstuct P i,j fo all i, j by (6.8). The fixation pobability fo a line is then given by P [fix] = n + n P i,i+. (6.9) 7. A compaison between line and cicle with a numeical example Using the above equations, we can find the fixation pobabilities fo any given position, and thus the oveall fixation pobability fo a line fo a paticula case. A cicle is a egula gaph and hence by theoem 3. has the Moan fixation pobability. In geneal the fixation pobability of a andom mutant is geate on a line than on a cicle fo mutants which ae fitte than the esidents > and is less on a line fo mutants which ae less fit. In figue 6, we can see how the aveage fixation pobability fo a line deceases as the numbe of vetices inceases. The decease is much i= } Aticle submitted to Royal Society

14 4 M. Boom and J. Rychtář a) Aveage Fixation Pobability Numbe of Vetices b) Aveage Fixation Pobability Numbe of Vetices Figue 6. Dependence of the aveage fixation pobability fo a line on the numbe of vetices (n + ) in the line; a) =., b) =.9. a) Diffeence between Line and Cicle Numbe of Vetices b) Diffeence between Line and Cicle Numbe of Vetices c) Relative Diffeence between Line and Cicle Numbe of Vetices d) Relative Diffeence between Line and Cicle Numbe of Vetices Figue 7. The diffeence between the aveage fixation pobability fo a line and fo a cicle (Moan pocess); a) =., absolute diffeence (line-cicle), b) =.9, absolute diffeence, c) =. elative diffeence (line-cicle)/line, d) =.9 elative diffeence. steepe fo <. Fo any, the aveage fixation pobability fo a line appoaches the Moan pobability (demonstated in figue 7). If >, the fixation pobability fo a line is geate than the fixation pobability fo the Moan pocess and it is smalle othewise. Figue 8 shows how the absolute and elative diffeence changes as changes fom to lage numbes. The absolute diffeence is at its lagest fo mutants which ae advantageous, but not ovewhelmingly so. This is easonable as these have an intemediate pobability of fixation and so stuctual changes have the geatest possibility of alteing this pobability. Vey advantageous mutants ae likely to achieve fixation whateve the stuctue, and non-advantageous ones ae unlikely to do so (note that the lage elative diffeence fo small in figue 8b coesponds to a vey small fixation pobability in each case). The dependence of the diffeence between a line and a cicle on is moe o less the same fo othe n as the one shown fo a line with n + = vetices. Why is the mutant fitte on a line than a cicle if and only if >? The key eason fo this is elated to the behaviou at the end vetices. The fixation Aticle submitted to Royal Society

15 An analysis of the fixation pobability... 5 a) Diffeence between a Line and a Cicle b) Relative diffeence between a Line and a Cicle Figue 8. The diffeence between the aveage fixation pobability fo a line and fo a cicle (Moan pocess) when thee ae vetices; a) absolute diffeence (line-cicle), b) elative diffeence ((line-cicle)/cicle). Fixation Pobability Fixation Pobability a) Vetex Numbe b) Vetex Numbe c) Fixation Pobability Vetex Numbe d) Fixation Pobability Vetex Numbe Figue 9. Fixation pobabilities at vetices in a line with n+ vetices. a) =, n+ =, b) =.9, n + = c) =., n + =, d) =.5, n + =. The line coesponds to the level of the Moan pocess fixation pobability. pobabilities fo a mutant placed into a specific vetex ae given in figue 9. The end vetices and n have the highest fixation pobability - because the only way the mutant can go extinct is by being eplaced by a esident fom vetex o n, espectively. But even if a esident at o n is selected fo epoduction, it has only a 5% chance (if n + > 3) that it will place its offsping in the cone. Thee is a steep dop in fixation pobabilities fo the vetex adjacent to the cone, o n, since a mutant placed at has a vey high chance of being eplaced by a esident fom (which, if selected has to place its offsping at ). As a vetex gets close to the middle of the line, then the fixation pobability inceases if >, and deceases if <. If >, then when the line is sufficiently long, the vetices close to the middle have appoximately the same fixation pobability as given by the fomula fo the Moan pocess. Geneally, the highe, the shote the line can be to have the cental vetices equivalent to the coesponding Aticle submitted to Royal Society

16 6 M. Boom and J. Rychtář Fixation Pobability Fixation Pobability a) Vetex Numbe b) Vetex Numbe Figue. Fixation pobabilities at vetices in a line with n + vetices. Cicles - single mutant pesent, diamonds - two mutants pesent in neighbouing vetices, boxes - thee mutants pesent in neighbouing vetices. a) =.9, n + =, b) =., n + =. The hoizontal lines coespond to the level of the Moan pocess fixation pobabilities when, o 3 mutants ae pesent in the population. Moan pocess. In othe wods, the highe, the shote is the ange of the effect of the cone end point. Being nea the cente can be thought of as equivalent to being in a cicle; fo an advantageous mutant once it has spead to be next to the cone (the fist time it is influenced by the cone) it is likely that thee will be many mutants and fixation will be almost assued. The most common way advantageous mutants ae eliminated is ealy, due to bad luck, and so the cones do not affect the fixation pobability of such mutants much (and hence why the lage, the stonge this effect). This is not tue, howeve, fo the case <. This is an inteesting qualitative distinction between < and >, and occus because non-advantageous mutants ae unlikely to each fixation unless by chance they each a lage popotion of the population, and so the cone will influence thei fixation pobability no matte whee they stat. In fact secuing a cone position seems impotant fo thei eventual suvival, so being nea a cone is bette than being in the cente, even though this means that vey ealy emoval is moe likely (the non-advantageous mutant needs to be lucky to each fixation). A simila patten holds fo lage goups of mutants on the line. Figue shows the situation once a small goup of mutants has been established, compaing the fixation pobabilities fo such a configuation of seveal mutants in thei diffeent possible positions. Any middle configuation of k mutants has appoximately the same fixation pobability. The cone configuation has a pobability equivalent to having one moe vetex in a non-cone position; the eason fo this elates to the fact that the fixation pobability of a cone mutant is appoximately twice that of its neighbou fo mutants whose fitness is close to that of the esidents. This is tue fo vaious line lengths and numbes of mutants (note that this appoximation becomes less good as the numbe of mutants inceases, and the fixation pobability becomes high). 8. Discussion In this pape we have consideed the use of evolutionay dynamics on gaphs populaised by Liebeman et al. (5). We have found an analytic way, using the Aticle submitted to Royal Society

17 An analysis of the fixation pobability... 7 wok of Mille (994), to obtain the fixation pobability of mutant populations fo one paticula type of gaph, a line. We cannot find explicit functional foms, but athe a set of N simultaneous linea equations, whee N is the population size, which need to be solved and then yield the pobability of fixation in any allowable situation. This is a significant saving on the ode of N equations deived diectly by consideing the tansition pobabilities between the states of ou system. We have used ou solutions to conside vaious examples and exploe the elationship between the fixation pobability of a mutant on the cicle, given by the Moan pobability, and the fixation pobability on the line; both the aveage such pobability and its value fo given stating positions. We see that fo mutants that ae fitte than the esident population the fixation pobability on the line is lage than on the cicle. Thee is also an inteesting patten in the fixation pobability fo the diffeent stating positions on the line. The best place fo a mutant to stat is always in the cone. Fo advantageous mutants, the place next to the cone is the wost and fixation pobabilities incease towads the cental positions. Fo mutants that ae not advantageous, the futhe fom the cone they ae, the wose the position they ae in. It should be noted that the pobability of fixation fo non-advantageous mutants fo gaphs with moe than a small numbe of vetices is geneally low, so the esults fo advantageous mutations ae the moe inteesting. In 4 we show that fo moe complex gaphs (which ae the vast majoity of gaphs not of ou linea type) almost all system states ae eachable fom almost all othes, and so the numbe of equations geneated by consideing the tansition pobabilities is not of the ode N but much lage. Fo some gaphs with a lot of symmety the numbe of equations can be educed consideably, and in 5 we analyse one well known such case, the sta, to poduce an exact solution fo the fixation pobability of a mutant. Howeve, gaphs which can be solved in this way ae special cases and the appoaches that we take hee fo a line o a sta will be had to implement elsewhee. Thus it is likely that we will have to esot to moe numeical methods, as in Rychtář & Stadle (8); Santos et al. (6); Paley et al. (7). Howeve, to gain an insight into deepe aspects of the poblem and the effect of vaious stuctues, analysis is useful and in futue wok we intend to use appoximation methods to investigate this. This has the benefit of extending to lage moe complex gaphical systems, such as the small wold netwoks of Bollobas & Chung (988) (see also Duett (7), Newman et al. (6), Newman & Watts (999) and Watts & Stogatz (998)). Small wold gaphs ae egula in fom with most vetices unconnected, but with a few added andom connections which geneally make the path length between any two vetices shot. In summay, in this pape we have found analytic solutions fo the mutant fixation pobabilities of impotant classes of gaphs, used these solutions to gain futhe undestanding of the undelying pocesses on gaphs in geneal and also demonstated the pactical limitations to extending ou methods. We have thus made a small step on the oad to undestanding the complex natue of evolutionay dynamics on gaphs. Aticle submitted to Royal Society

18 8 M. Boom and J. Rychtář Refeences Baabasi, A & Albet, R. 999 Emegence of scaling in andom netwoks. Science 86, Bollobas, B. & Chung, F.R.K. 988 The diamete of a cycle plus andom matching. SIAM J. Discete Math, Bamson, M. & Giffeat, D. 98 On the Williams-Bjeknes Tumou Gowth Model I. The Annals of Pobab. 9 (), Duett, R. 7 Random Gaph Dynamics, Cambidge Univesity Pess. Edös, P. & Renyi, A. 96 On the evolution of andom gaphs. Publ. Math. Inst. Hungaian Acad. Sci 5, 7 6. Liebeman, E., Hauet, C., Nowak, M.A. 5 Evolutionay Dynamics on Gaphs. Natue 433, Mille, J.W. 994 A matix equation appoach to solving ecuence elations in two-dimensional andom walks. J. Appl. Pobab. 3 (3), Nagylaki, T. & Lucie, B. 98 Numeical analysis of andom dift in a cline. Genetics 94, Newman, M.E.J., Baabasi, A.L., Watts, D.J. 6 The Stuctue and Dynamics of Netwoks, Pinceton Studies in Complexity, Pinceton Univesity Pess. Newman, M.E.J. & Watts, D.J. 999 Renomalization goup analysis of the smallwold netwok model Phys. Lett. A 63, Nois, J.R Makov Chains, Cambidge Univesity Pess. Nowak, M.A. 6 Evolutionay Dynamics: exploing the equations of life, Hawad Univesity Pess. Paley, C.S., Taashkin, S.N., Elliot, S.R. 7 Tempoal and dimensional Effects in Evolutionay Gaph Theoy. Physical Review Lettes 98 (9), 983. Rychtář, J. & Stadle, B. 8 Evolutionay Dynamics on Small-Wold Netwoks. Int. J. of Mathematics Sciences (), 4. Santos, P.C., Pacheco, J.M., Lenaets, T. 6 Evolutionay Dynamics of Social Dillemas in Stuctued Heteegenous Populations. PNAS 3 (9), Tucke, A.C. 994 Applied combinatoics (3d ed.), John Wiley & Sons, Inc. Watts, D.J. & Stogatz, S.H. 998 Collective dynamics of small-wold netwoks. Natue 393, Aticle submitted to Royal Society