EVOLUTIONARY DYNAMICS ON GRAPHS

Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7 EVOLUTIONARY DYNAMICS ON GRAPHS FERNANDO ALCALDE CUESTA, PABLO GONZÁLEZ SEQUEIROS, AND ÁLVARO LOZANO ROJO Abstrct. In this pper, we summrize some ides nd results bout evolutionry dynmics on grphs. This theory is illustrted with concrete exmple, the so-clled str grph, for which we clculte the verge fixtion probbility.. Introduction nd motivtion Popultion genetics studies the genetic composition of biologicl popultions, nd the chnges in this composition tht result from the ction of four different process: nturl selection, rndom drift, muttion nd migrtion. The modern evolutionry synthesis combines Drwin s thesis on the nturl selection nd Mendel s theory of inheritnce. According to this synthesis, the centrl object of study in evolutionry dynmics is the frequency distribution of the lterntive forms (llele) tht hereditry unit (gene) cn tke in popultion evolving under these four forces. Mny mthemticl models hve been proposed to understnd the evolutionry biologicl processes. For exmple, the Wright-Fisher model (stted explicitly by S. Wright [8], but present in the work of nd R. A. Fisher [4]) describes the chnge of gene frequency by rndom drift on popultion of finite fixed size N. For simplicity, the involved orgnisms re ssumed to be hploids (contining only one set of chromosomes) with only two possible lleles nd A for given locus, lthough the Wright-Fisher model cn be extended to multiple lleles in diploids orgnisms. Then there re only N + possible gene frequencies i/n for 0 i N. Assume tht in some popultion there re exctly i copies of the llele A (nd therefore N i copies of ). If ech of the N offspring contins copy of rndomly chosen llele from the present genertion, then the gene frequency in the next genertion could ssume ny of the N + possible vlues, except when i = 0 or i = N. The Morn model (introduced by P. A. P. Morn in [6]) shows prticulr equilibrium between nturl selection nd rndom drift. This model hs mny vrints, but we will consider the vrint which is closest to the Wright-Fisher model. We hve hploid popultion of N individuls hving only two possible 00 Mthemtics Subject Clssifiction. 05C8, 60J0 9D5. Key words nd phrses. Evolutionry dynmics, Morn process, fixtion probbility, str grph. Prtilly supported by the Ministry of Science nd Innovtion - Government of Spin (Grnt MTM00-547). 3

4 lleles nd A for given locus. In the Morn model, insted of ll individuls dying simultneously upon the birth of the next genertion, t ech unit of time, one individul is chosen t rndom for reproduction nd its clonl offspring replces nother individul chosen t rndom to die. To model nturl selection, it suffices to ssume tht the prent individuls with llele A hve reltive fitness r >, s compred to those with llele whose fitness is. In this cse, individuls with the dvntgeous llele A hve certin chnce of fixtion generting linege tht tkes over the whole popultion, wheres individuls with the disdvntgeous llele re likely to become extinct, lthough it will be never gurnteed. As in the previous models, evolutionry dynmics hs been usully studied for homogeneous popultions. But it is nturl question to sk how non-homogeneous structures ffects the dynmics. The study of evolutionry dynmics on directed grphs ws initited by E. Libermn, C. Huert nd M. A. Novk [5] (see lso [7]). Now ech vertex represents n individul in the popultion, nd the offspring of ech individul only replce direct successors, i.e. end-points of edges with origin in this vertex. The fitness of n individul represents gin its reproductive rte which determines how often offspring tkes over its neighbor vertices, lthough these vertices do not hve to be replced in equiprobble wy. In other words, the evolutionry process is given by the choice of stochstic mtrix W = (w ij ) where w ij denotes the probbility tht individul i plces its offspring into vertex j. In fct, further generliztions of evolutionry grphs re considered in [5] ssuming simply tht the probbility bove is proportionl to the product of weight w ij nd the fitness of the individul i. In this cse, the mtrix W does not need to be stochstic, but non-negtive. In this context, severl interesting nd importnt results hve been shown by Libermn, Huert nd Novk: Different grph structures support different dynmicl behviors mplifying or suppressing the reproductive dvntge of mutnt individuls (hving the dvntgeous llele A) over to the resident individuls (hving the disdvntgeous llele ). An isotherml theorem which sttes tht n evolutionry process on grph is equivlent to Morn process (in the sense tht there is well-defined fixtion probbility which coincides with the fixtion probbility for n homogeneous popultion) if nd only if it is defined by doubly stochstic mtrix W. More generlly, non-negtive mtrix W defines n evolutionry process equivlent to Morn process if nd only if it is circultion, i.e. ech vertex i hs the sme entering nd leving weight w (i) = N j= w ji = N j= w ij = w + (i), which is equl to in the stochstic cse. However, for evolutionry processes on grphs, the fixtion probbility depends usully on the strting position of the mutnt. The effect of the initil plcement on the mutnt spred hs been discussed by M. Broom, J. Rychtář nd B. Stdler in the cse of undirected grphs, see [] nd [3]. Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo 5 The im of this pper is summrize some fundmentl ides nd results on evolutionry dynmics on grphs. This theory will be illustrted with concrete exmple, the so-clled str grph, for which we clculte the (verge) fixtion probbility outlined in [5] (see lso []).. Morn process The Morn process ws introduced by Morn [6] to model rndom drift nd nturl selection for finite homogeneous popultions. As indicted in the introduction, we consider hploid popultion of N individuls hving only two possible lleles nd A for given locus. At the beginning, ll individuls hve the llele, then one resident individul is chosen t rndom nd replced by mutnt hving the neutrl or dvntgeous llele A. At successive steps, one rndomly chosen individul replictes with probbility proportionl to the reltive fitness nd its offspring replces nother individul rndomly chosen to be eliminted. Since the future sttes of the process depend only on the present stte, nd not on the sequence of events tht preceded it, the Morn process is defined by the Mrkov chin X n = number of mutnt individuls with the llele A t the step n with stte spce S = {0,..., N}. Moreover, this process is sttionry becuse the probbility to pss from i to j mutnt individuls P i,j = P[X n+ = j X n = i] = P[X n+ = j X 0 = i 0,..., X n = i n, X n = i] does not depend on the time. But the number of mutnt individuls cn chnge t most by one t ech time step nd therefore non-trivil trnsition exists only between stte i nd stte i, i or i +. Then, the trnsition mtrix of the stochstic process is tridigonl mtrix 0 0... 0 P 0,0 P 0,... P 0,N P,0 P,... P,N δ δ δ+ δ +... 0 P =...... =....... 0 0 0... δ + P N,0 P N,... P N N,N 0 0 0... where P i,i = δ i, P i,i+ = δ + i nd P i,i = δ i δ + i. The sttes i = 0 nd i = N re bsorbing while the other sttes re trnsient. A Figure. Morn process on homogeneous popultion Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

6 For generl birth-deth process, defined by tridigonl mtrix P, the chnce tht ny set of i mutnt individuls spred tking over the whole popultion is denoted by x i = P[ n : X n = N X 0 = i]. In prticulr, the probbility of one mutnt individul to rech fixtion x = P[ n : X n = N X 0 = ] is clled fixtion probbility nd lso denoted by Φ A. Now we consider the following system of liner equtions x 0 = 0 x i = δ i x i + ( δ i δ + i )x i + δ + i x i+ x N = (.) To find solution x = (x 0, x,..., x N ) of the liner eqution P x = x with the conditions x 0 = 0 nd x N =, it is useful to define y i = x i x i verifying N i= y i = x N x 0 =. Then, dividing ech side of (.) by δ + i, we hve: y i+ = γ i y i (.) where γ i = δ i /δ+ i is the deth-birth rte (which is reciprocl to the reproductive dvntge of ny set of i mutnt individuls). It follows: i y i = x γ j. j= Therefore, the fixtion probbility is given by x = + N i i= j= γ j (.3) Rndom drift. If none of lleles nd A is reproductive dvntgeous, the rndom drift phenomenon cn be modeled by the Morn process with reltive fitness r =. In this cse, the trnsition probbilities re given by P i,i = N i N. i N P i,i+ = i N. N i N P i,i = i N. i N + N i N.N i N (.4) Since γ i =, the fixtion probbility Φ A = /N. As for every birth-deth process, if the popultion reches one of the bsorbing sttes, then it stys there forever. In the other sttes, the popultion of mutnt individuls rndomly evolves, but eventully these individuls will either become extinct or tke over the whole popultion. Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo 7 Nturl selection. The effect of fitness on the evolutionry dynmics of popultion is described by the Morn process provided mutnt individuls with the llele A hve reltive fitness r >. Now, the trnsition probbilities re given by P i,i = P i,i+ = P i,i = N i ri + N i. i N ri ri + N i. N i N ri ri + N i. i N + N i ri + N i.n i N (.5) Since the deth-birth rte γ i = /r, the fixtion probbility Φ A = + r = N j= r i r N r. Thus, n dvntgeous muttion with r > reches fixtion with positive probbility but this is not lwys gurnteed, becuse this probbility is strictly less thn. 3. Evolutionry processes on grphs Evolutionry grph theory ws introduced by Libermn, Huert nd Novk [5]. Like for homogenous popultions, the first nturl question is to determine the chnce tht the offspring of mutnt individul hving n dvntgeous llele spreds through the grph reching ny vertex. But this chnce depends obviously on the initil position of the individul (see [] nd [3]) nd the globl grph structure my significntly modify the equilibrium between rndom drift nd nturl selection observed in homogeneous popultions (s proved in [5]; see lso [] nd [3]). Let G = (V, E) be directed grph, where V is the set of vertices nd E is the set of edges. We ssume G is finite, connected nd simple grph (without loop or multiple edges). Thus, E identifies to subset of V V which does not meet the digonl. Any grph structure on the vertex set V = {,..., N} is completely determined by the djcency mtrix ( ij ) where ij = E (i, j) for ech pir (i, j) V V. An evolutionry process on G is lso given by Mrkov chin, but ech stte is now described by set of vertices S S = P(V ) inhbited by mutnt individuls hving n dvntgeous llele A. This reproductive dvntge is mesured by the fitness r. The trnsition probbilities of this Mrkov chin re defined from non-negtive mtrix W = (w ij ) whose entries re edge weights stisfying w ij = 0 ij = 0. So evolutionry process on G cn be identified with the elements of the set W of such mtrices. The trnsition probbility between two sttes S, S S = P(V ) (which is still time-independent) Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

8 is given by r i S w ij r i S j V w ij + i V \S i V \S w ij j V w ij if S \S = {j} P S,S = r i S j V w ij + i V \S j V w if S\S = {j} ij r i,j S w ij + (3.6) i,j V \S w ij r i S j V w ij + i V \S j V w if S = S ij 0 otherwise where r i S j V w ij+ i V \S j V w ij is the sum of the reproductive weight of the mutnt nd resident individuls (equl to r#s +N #S = N +(r )#S when the mtrix W is stochstic). In other words, the process is defined by N N stochstic mtrix P = (P S,S ). As for the Morn process, S = nd S = V re bsorbing sttes, but there my exist other bsorbing sttes, s well s recurrent sttes, so the probbility tht resident or mutnt individuls become extinct cn be strictly less thn. Anywy, the fixtion probbility of ny other set S inhbited by mutnt individuls Φ S = P[ n : X n = V X 0 = S] cn be obtined s the solution of liner eqution, which is nlogous to (.) for the clssicl Morn process. Assuming the bsorbing sttes S = nd S = V re connected with other sttes nd using tht P is stochstic, it is possible to prove this eqution hs lwys unique solution. Detils will be reported elsewhere. By simplifying Φ S terms, this eqution reduces to the following eqution: Φ S = i S j V \S i S ( rwij Φ S {j} + w ji Φ S\{i} ) j V \S ( rwij + w ji ) (with P = 0 nd P V = ) used in [], [] nd [3]. In prticulr, for S = {i}, we hve the eqution: Φ {i} = j i rw ijφ {i,j} j i ( rwij + w ji ). Contrry to the cse of homogeneous popultions, the fixtion probbility depends on the strting position of the mutnt in the grph. This fct justifies the following definition: Definition 3.. For ny mtrix W W, we define the verge fixtion probbility on G s the verge Φ A = N Φ {i}. N The definitions nd results bove cn be illustrted by some exmples: i= Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo 9 Morn process. The clssicl Morn process coincides with the evolutionry process on the complete grph G = K N (where V = {,..., N} nd E = V V \ ) defined by the stochstic mtrix W = (w ij ) where w ij = N if i j. Since G is symmetric (i.e. its utomorphism group cts trnsitively on the vertex nd edge sets) nd W is preserved by the ction of the utomorphism group of G, the fixtion probbility Φ {i} = Φ {j} for ll i j. Therefore the verge probbility fixtion Φ A = Φ {i} for ll i. A Figure. Evolutionry process on complete grph Directed line grph. This grph is described in the figure below, nd the process is given by the djcency mtrix 0 0... 0 0 0... 0 W =........ 0 0 0... 0 0 0... 0 If the strting position of the mutnt individul coincide with the root, then this mutnt genertes with probbility linege tht will tke the whole popultion. But this will never be possible in other positions. In other words, Φ {i} = { if i = 0 if i According to [5], such grph structure is be sid to be suppressor of selection since the verge fixtion probbility Φ A = /N is the sme tht of the rndom drift for homogeneous popultion independently of the mutnt fitness. A Figure 3. The line grph Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

0 Cycle grph. As before, this grph is described in the figure below, but now the process is given by the stochstic mtrix 0 0... 0... 0 W =........ 0 0 0... 0 0 0 Since the grph is still symmetric nd W is preserved by the ction of its utomorphism group, even if the popultion is not yet homogeneous, the strting position of mutnt individul does not hve ny effect on the process. Thus, we cn ssume the strting stte is S = {}, which only my evolve to S =, S = {, } or S = {N, }. Arguing by recurrence, we see tht ny ccessible stte is connected subset, nd the non-trivil trnsition probbilities depend only on its crdinl number i. In more precise wy, these probbilities re given by P i,i = ri + N i + ri + N i P i,i+ = r ri + N i + r ri + N i P i,i = r(i ) ri + N i + ri N + N i i for i < N. = ri + N i = r ri + N i = P i,i P i,i+ A Figure 4. Cycle grph 4. Circultion theorem Complete grphs nd cycle grphs show the sme equilibrium between rndom drift nd nturl selection tht homogeneous popultion. Now, ccording to [5], it is nturl to dopt the following definition: Definition 4. ([5]). An evolutionry process on grph G defined by mtrix W = (w ij ) W is sid to be equivlent to the Morn process if the fixtion probbility of single copy of mutnt llele A hving fitness r > is well defined (tht is, it does not depend on the initil plcement of the mutnt llele) Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo nd equl to the fixtion probbility Φ A = r r N of the Morn process, where N is the number of vertices of G. Our next im is to recll the circultion theorem proved by Libermn, Huert nd Novk [5], where they give some necessry nd sufficient conditions for this equivlence. We strt by reclling the circultion condition: Definition 4. ([5]). A mtrix W = (w ij ) W defines circultion on G if for ny vertex i V the entering weight N w (i) = nd the leving weight w + (i) = re equl. The weighted grph (G, W ) is lso sid to be weight-blnced. In the cse where W is stochstic, the entering weight w (i) = N j= w ji is lso clled the temperture of the vertex i nd denote by T i, while the leving weight w + (i) = N j= w ij is lwys equl to. Circultion Theorem [5]. For ny mtrix W = (w ij ) W, the following conditions re equivlent: () W defines n evolutionry process equivlent to the Morn process. () The probbility tht initil popultion of n mutnt individuls hving fitness r > reches mutnt popultion of m individuls is given by Φ A (r, W, n, m) = r n r m. (3) W defines circultion on G. (4) The number of elements of stte S performs bised rndom wlk on the integer intervl [0, N] with forwrd bis r > nd bsorbing sttes 0 nd N. Proof. We prove cycle of implictions () () (3) (4) (). To verify () (), it suffices to remrk: j= N j= w ji w ij Φ A (r, W, n, N) = Φ A (r, W, n, m)φ A (r, W, m, N), m n since the probbility of reching one stte from nother stte depends only on their number of vertices. Now we prove () (3). First, for ech stte S, V, we define entering nd leving weights w (S) = w (i) = N w ji i S i S j= Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

nd w + (S) = w + (i) = N w ij. i S i S j= The probbility tht the mutnt popultion S increses or decreses of one individul is given by δ + (S) = rw + (S) rw + (S) + w (S) Then the birth-deth rte is equl to By hypothesis, we know: or δ (S) = w (S) rw + (S) + w (S). δ + (S) δ (S) = rw +(S) w (S). (4.7) Φ A (r, W,, ) = r r = r r +. In prticulr, this mens tht the evolutionry process does not depend on the initil plcement of the mutnt individul. Then, writing δ ± = δ ± ({i}) for ny i V, we hve lso: Φ A (r, W,, ) = δ + ( δ + δ ) k δ + = δ + + δ. k=0 We deduce: δ + δ = r. Combining this equlity with (4.7) for S = {i}, we obtin w + (i) = w (i) for ll i V, tht is, W defines circultion on G To show (3) (4), we need to prove tht the number of individuls k = #S of mutnt popultion S defines Mrkov chin X n verifying P[X n+ = k + X n = k] = δ + (S) nd P[X n+ = k X n = k] = δ (S) with forwrd bis δ + (S) δ (S) = r nd bsorbing sttes 0 nd N. Since W defines circultion, we hve: w + (S) w (S) = i S w + (i) w (i) = 0 for every stte S, V in S = P(V ). Using (4.7), we deduce tht the Mrkov chin X n verifies: δ + (S) δ (S) = rw +(S) w (S) = r for ll S, V. Finlly, we prove (4) (). By hypothesis, the fixtion probbility of single mutnt individul hving fitness r > does not depend on its initil plcement. More generlly, the probbility of reching the whole popultion V from one Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo 3 stte S depends only on its number of vertices k = #S. Thus, W defines Mrkov chin with trnsition mtrix 0 0... 0 δ δ δ+ δ +... 0 P =....... 0 0 0... δ + N 0 0 0... where δ + k δ k = r, k =,..., N by hypothesis. Arguing s for the cse of Morn process, we obtin: This completes the proof. Φ A (r, W,, N) = r r N. As corollry of this theorem, we hve: Isotherml Theorem [5]. For ny stochstic mtrix W = (w ij ) W, the following conditions re equivlent: () W defines n evolutionry process equivlent to the Morn process. () W defines n isotherml process on G, i.e. ll vertices i V hve the sme temperture T i = N j= w ji = T. (3) W is doubly stochstic, i.e. T i = N j= w ji = for ll i V. Proof. Firstly, we hve: n N N N N T i = w ji = w ji = N i= i= j= j= i= if W is stochstic. To see () (), it suffices to pply the circultion theorem, so the temperture T i = w (i) of ech vertex i is equl to its leving weight w + (i) =. The impliction () (3) follows from the eqution bove becuse n i= T i = NT = N nd hence T = when W defines n isotherml process. Finlly, ccording to the circultion theorem, ny doubly stochstic mtrix W defines circultion (nd hence n isotherml process) equivlent to the Morn process. 5. Str grph In [5], Libermn, Huert nd Novk showed tht there re some grph structures, clled str structures, which ct s evolutionry mplifiers fvoring dvntgeous lleles in non-homogeneous popultions. These structures hve lso been studied in []. We will explicitly describe the symptotic behvior of the verge fixtion probbility. A str grph consists of N = m + vertices lbelled 0,,..., m where only the center 0 is connected with the peripherl vertices,..., m, see the figure below. Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

4 A Figure 5. Str grph Since the utomorphism group is isomorphic to the symmetric group cting on the peripherl vertices, the stte spce reduces to the subsets nd {,..., i}, {0} nd {0,,..., i}, i m, which cn be described using ordered pirs. In the first entry, we write the number i of peripherl vertices inhbited by mutnt individuls. In the second one, we use or 0 to indicte whether or not there is mutnt individul t the center. Thus, the fixtion probbilities will be denoted by nd Φ i, = P[ n : X n = (m, ) X 0 = (i, )] Φ i,0 = P[ n : X n = (m, ) X 0 = (i, 0)]. As for the Morn process, the evolutionry dynmics of the str structure is described by the system of liner equtions Φ 0,0 = 0 Φ i, = δ + i, Φ i+, + δ i, Φ i,0 + ( δ + i, δ i, )Φ i, (5.8) Φ i,0 = δ + i,0 Φ i, + δ i,0 Φ i,0 + ( δ + i,0 δ i,0 )Φ i,0 (5.9) Φ m, = since non-trivil trnsition exists only between stte (i, ) (resp. (i, 0)) nd sttes (i +, ), (i, 0) nd (i, ) (resp. (i, 0), (i, ) nd (i, 0)), see the figure below. The non-trivil entries in the trnsition mtrix re given by δ + i, = P[X n+ = (i +, ) X n = (i, )] = δ i, = P[X n+ = (i, 0) X n = (i, )] = δ + i,0 = P[X n+ = (i, ) X n = (i, 0)] = δ i,0 = P[X n+ = (i, 0) X n = (i, 0)] = r r(i + ) + m i.m i m m i r(i + ) + m i ri ri + m + i ri + m + i. i m (5.0) (5.) (5.) (5.3) Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo 5 (0, ) (, ) (i, ) (i, ) (i +, ) (m, ) (m, ) (0, 0) (, 0) (i, 0) (i, 0) (i +, 0) (m, 0) (m, 0) Figure 6. Stte spce of str grph nd δ + i, δ i, = m + m. ri r(i + ) + m i δ + i,0 δ i,0 = m + m. m i ri + m + i In prticulr, we hve: Φ 0, = r r + m Φ, nd Φ,0 = rm rm + Φ,. (5.4) Thus, the deth-birth rtes re given by γ i, = δ i, δ + i, γ i,0 = δ i,0 δ + i,0 = m r = rm (5.5) (5.6) Arguing s for (.) nd dividing ech side of Equtions (5.8) nd (5.9) by δ + i, nd δ + i,0 respectively, we obtin equtions Φ i+, Φ i, = γ i, (Φ i, Φ i,0 ) = m r (Φ i, Φ i,0 ) (5.7) Φ i, Φ i,0 = γ i,0 (Φ i,0 Φ i,0 ) = rm (Φ i,0 Φ i,0 ) (5.8) nlogous to (.). From (5.8), we prove inductively the following lemm: Lemm 5.. For ech i =,..., m, the fixtion probbility i Φ i,0 = ( rm rm )i j ( rm + )i j+ Φ j,. j= Proof. For i =, the identity reduces to the second identity in (5.4). Assuming it is true for i, from (5.8), we deduce: (+ rm )Φ i,0 = Φ i, + rm Φ i,0 = Φ i, + i ( rm rm rm )i j ( rm + )i j+ Φ j,. This implies the formul. j= Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

6 Now, using (5.7), we obtin the following eqution: where Φ i+, Φ i, = m [ Φi, rm r rm + Φ i, rm.( rm rm + ) Φ i, i ( rm ] rm )i j ( rm + )i j+ Φ j, lim = i m + j= j= m r(rm + ) Φ m i, ( rm + ) Φ i, i j= rm rm )i j ( rm + )i j+ Φ j, m r ( rm rm )i j ( rm + )i j+ Φ j, = 0. m r ( Thus, when the number of peripherl vertices m tends to +, the peripherl process whose fixtion probbilities re equl to Φ i, becomes more nd more close to the Morn process determined by the system of liner equtions Φ i+, Φ i, = r (Φ i, Φ i, ) (5.9) Even though there is no limit process, we sy tht the peripherl process is symptoticlly equivlent to the Morn process determined by (5.9) nd whose reltive fitness is qudrticlly mplified. On the other hnd, ccording to (5.4), the verge fixtion probbility is equl to Φ A = m + Φ 0, + m m + Φ,0 ( m +. r r + m + m m +. rm rm + )Φ, nd therefore Φ A lso becomes more nd more close to the fixtion probbility of the Morn process determined by (5.9), which hs reltive fitness r >. We cn resume this discussion in the following sttement: Str Theorem [5]. The str structure is qudrticlly mplifier of selection in the sense tht the verge fixtion probbility of mutnt individul with reltive fitness r > is symptoticlly equivlent to the fixtion probbility Φ A = r r m for the Morn process with reltive fitness r > Actully, there re super-str structures which ct s mplifiers of selection of rbitrry polynomil degree [5]. Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7

F. Alclde Cuest, P. González Sequeiros, A. Lozno Rojo 7 6. Conclusion In this pper, we hve described some bsic ides of evolutionry grph theory sketched by Libermn, Huert nd Novk [5] by focusing in the study of the verge fixtion probbility of rndomly rising muttion. The effect of the strting position of the mutnt individul hs been discussed by Broom, Rychtář nd Stdler in [] nd [3] for smll-world networks nd other smll-order grphs. For exmple, like they observed, regulr grphs hve the worst structure for the mutnt spred. These uthors were lso interested in the time to fixtion of dvntgeous llele for strongly connected directed grphs nd undirected grphs, nd how it depends on the grph structure nd the initil plcement of the mutnt. In forthcoming pper, we will study the verge fixtion probbility nd the expected fixtion time in different types of grphs nd complex networks. References [] M. Broom, J. Rychtář. An nlysis of the fixtion probbility of mutnt on specil clsses of non-directed grphs. Proc. R. Soc. A, 464 (008), 609 67. [] M. Broom, J. Rychtář, B. Stdler. Evolutionry dynmics on smll-order grphs. J. Intesdiscip. Mth., (009), 9 40. [3] M. Broom, J. Rychtář, B. Stdler. Evolutionry dynmics on grphs - the effect of grph structure nd initil plcement on mutnt spred. J. Stt. Theory Prct., 5 (0), 369 38. [4] R.A. Fisher. The geneticl theory of nturl selection. Clrendon Press, Oxford, 930. [5] E. Liebermn, C. Huert, M. A. Novk. Evolutionry dynmics on grphs. Nture, 433 (005), 3 36. [6] P. A. P. Morn. Rndom processes in genetics. Proc. Cmb. Phil. Soc., 54 (958), 60 7. [7] M. A. Novk. Evolutionry dynmics: exploring the equtions of life. Hrvrd University Press, Cmbridge, 006. [8] S. Wright. Evolution in Mendelin popultions. Genetics, 6 (93), 97 59. Deprtmento de Xeometrí e Topoloxí, Fcultde de Mtemátics, Universidde de Sntigo de Compostel, Rú Lope Gómez de Mrzo s/n, E-578 Sntigo de Compostel (Spin) E-mil ddress: fernndo.lclde@usc.es Deprtmento de Didáctic ds Ciencis Experimentis, Fcultde de Formción do Profesordo, Universidde de Sntigo de Compostel, Avd. Rmón Ferreiro, 0, E-700 Lugo (Spin) E-mil ddress: pblo.gonzlez.sequeiros@usc.es Centro Universitrio de l Defens - IUMA Universidd de Zrgoz, Acdemi Generl Militr, Ctr. Huesc s/n, E-50090 Zrgoz (Spin) E-mil ddress: lvrolozno@unizr.es Rev. Semin. Iberom. Mt. 4 fsc. II (03) 3 7