FAST AND DETERMINISTIC COMPUTATION OF FIXATION PROBABILITY IN EVOLUTIONARY GRAPHS

Transcription

1 FAST AND DETERMINISTIC COMPUTATION OF FIXATION PROBABILITY IN EVOLUTIONARY GRAPHS Paulo Shakaran Network Scence Center and Dept. of Electrcal Engneerng and Computer Scence Unted States Mltary Academy West Pont, NY emal: Patrck Roos Dept. of Computer Scence Unversty of Maryland College Park, MD emal: ABSTRACT In evolutonary graph theory [] bologsts study the problem of determnng the probablty that a small number of mutants overtake a populaton that s structured on a weghted, possbly drected graph. Currently Monte Carlo smulatons are used for estmatng such fxaton probabltes on drected graphs, snce no good analytcal methods exst. In ths paper, we ntroduce a novel determnstc algorthm for computng fxaton probabltes for strongly connected drected, weghted evolutonary graphs under the case of neutral drft, whch we show to be a lower bound for the case where the mutant s more ft than the rest of the populaton (prevously, ths was only observed from smulaton). We also show that, n neutral drft, fxaton probablty s addtve under the weghted, drected case. We mplement our algorthm and show expermentally that t consstently outperforms Monte Carlo smulatons by several orders of magntude, whch can allow researchers to study fxaton probablty on much larger graphs. KEY WORDS Modellng of Evoluton, Network Scence, Stochastc Models, Network Dffuson Introducton Evolutonary graph theory, ntroduced n [] studes the problem of a mutant overtakng a populaton whose underlyng structure s represented as a drected, weghted graph. Ths model has been appled to problems n evolutonary bology [2], physcs [], and game theory []. Most work wth evolutonary graphs concerns computng the fxaton probablty - the probablty that a certan subset of mutants overtakes the populaton. Although good analytcal approxmatons are avalable for the undrected case [5, 6], these break down for drected, weghted graphs as shown n [7], even n the case of neutral drft. As a result, most work dealng wth evolutonary graphs rely on Monte Carlo smulatons to approxmate the fxaton probablty [8, 9, 0]. In ths paper we develop a novel determnstc algorthm to compute fxaton probablty n the case of neutral drft n Neutral drft s the case where mutants and resdents have the same ftness. drected, weghted evolutonary graphs based on the convergence of vertex probabltes to the fxaton probablty as tme approaches nfnty. We then prove that the fxaton probablty computed n neutral drft s a lower bound for the fxaton probablty when the mutant s more ft than the resdent, confrmng the smulaton observatons of []. We also show that fxaton probablty under neutral drft s addtve (even for weghted, drected graphs), whch extends the work of [6] whch proved ths for undrected graphs. Further, we mplemented our algorthm and show that t outperforms Monte Carlo smulatons by several orders of magntude. The paper s organzed as follows. In Secton 2 we revew the orgnal model of [] and ntroduce the dea of vertex probabltes. In Secton we show how vertex probabltes can be used to fnd the fxaton probablty. Ths s followed by our expermental evaluaton n Secton. Fnally, related work s dscussed n Secton 5. 2 Techncal Prelmnares The Moran Process [2] s a stochastc process used to model evoluton n a well-mxed populaton. All the ndvduals n the populaton are ether mutants or resdents. The am of such work was to determne f a set of mutants cold take over a populaton of resdents. In [], evolutonary graph theory (EGT) s ntroduced, whch generalzes the model of the Moran Process by specfyng relatonshps between the N ndvduals of the populaton n the form of a drected, weghted graph. We re-ntroduce ther model n ths secton. Frst, we defne an evolutonary graph s defned as follows. Defnton (Evolutonary Graph (EG) []). Gven natural number N, set of ndvduals V = {v,..., v,..., v N } and adacency matrx W = [w ] s.t., w = and w = 0, the tuple N, V, W s an evolutonary graph (EG). Intutvely, w s the weght of the edge from vertex v to v. In all lterature on evolutonary graph theory known to the authors, the evolutonary graph s assumed to be strongly connected as defned below. We make the same assumpton.

2 0.7 2 Fgure : Example strongly connected, drected, and weghted graph. Defnton 2 (Strongly Connected). An evolutonary graph N, V, W s strongly connected f for all dstnct vertces v, v V, there s a drected path from v to v such that for all edges n the path (v a, v b ), w ab > 0. As all ndvduals n the populaton are ether mutants or resdents, we defne a confguraton of mutants and resdents n the EG. Defnton (Confguraton). Gven evolutonary graph N, V, W, a confguraton C s a subset of V such that all ndvduals n C are mutants and all ndvduals n V C are resdents. The set 2 V s the set of all confguratons. In [], the authors specfy a stochastc process known as the Moran Process on Graphs whch we defne below. Defnton (Graphcal Moran Process (GMP) []). Gven evolutonary graph N, V, W and real number r > 0 (known as the relatve ftness), the Graphcal Moran Process (GMP) s specfed as follows. For any confguraton C V, some vertex v V s selected for brth. If v s n C t s selected wth a probablty 0. r r C + V C and r C + V C otherwse. Then vertex v s selected wth probablty w for death and replaced by a clone of v. Hence, f v s a mutant, the new confguraton s C {v } and C {v } otherwse. Note f r =, we say we are n the specal case of neutral drft. The followng s an example of ths process. Example. Consder the evolutonary graph specfed by Fgure, whch s drected, weghted, and strongly connected. Suppose vertex n the graph s a mutant. Then, one possble sequence that leads to fxaton s shown n Fgure 2 and has the probablty of under neutral drft (r = ). For nstance, to transton from the confguraton labeled A to the confguraton labeled B (where node 2 becomes a mutant as well) requres for node to be pcked for brth (wth a probablty 5) and node 2 selected to de (whch occurs wth a probablty of by the edge weght). Hence, the transton to confgurataon B from A s Based on the defnton of GMP, we wll use the notaton Pr(C (t) ) to refer to the probablty of beng n confguraton C after t tmesteps of the GMP. Perhaps the most wdely studed problem n evolutonary graph theory s to determne the fxaton probablty. We defne t formally below A C E B D F Fgure 2: A sequence of mutant-resdent confguratons leadng to fxaton that happens wth a probablty of under neutral drft. Mutant nodes are shaded. A s the ntal state and F s the fnal absorbng state. Defnton 5 (Fxaton Probablty). Gven an evolutonary graph N, V, W, real number r > 0, and confguraton C V the fxaton probablty, P C, s lm t Pr(V (t) C (0) ). Intutvely, ths s the probablty that an ntal set C of mutants takes over the entre populaton. Smlarly, we wll use the term the extncton probablty, P C, to be lm t Pr( (t) C (0) ). If the graph f strongly connected, we have the followng well-known result. Proposton. Gven EG N, V, W, real number r > 0, and confguraton C V, f the EG s strongly connected, then under GMP we have P C + P C = The above result essentally says that for a strongly connected graph, a mutant ether fxates or becomes extnct. As [] showed that an optmzaton problem related to computng fxaton probablty s NP-hard, much of the work on evolutonary graph theory reles on Monte Carlo smulatons to calculate P C. Algorthm shows pseudo-code to fnd the P C for the GMP usng Monte Carlo smulatons for the case of neutral drft. The tme complexty s smply O(RT sm ), where R s the number of smulaton runs executed (loop at lne 2) and T sm s the average tme t takes for the evolutonary process to reach mutant extncton or fxaton (the frst nested loop at lne )

3 Algorthm - Monte Carlo Smulaton to Compute Fxaton Probabltes Input: Evolutonary Graph N, V, W, confguraton C V, and natural number R > 0. Output: Estmate of fxaton probablty of mutant. : x 0 {x wll count number of tmes the mutant fxates} 2: for = 0 R do : Set current confguraton C = C : whle Whle (C V ) or (C = ) do 5: Select vertex v V wth probablty /N 6: Select vertex v V wth probablty w 7: If v C set C = C {v }. Otherwse set C = C {v } 8: end whle 9: If C = V then x + + 0: end for : return x/r {the estmated fxaton probablty} Further, based on the commonly-accepted defnton of estmated standard error from statstcs, we can obtan the estmated standard error for the soluton returned by Algorthm. Proposton 2. The estmated standard error of the soluton, S returned by Algorthm s S( S) R. Ths work uses the dea of a vertex probltes to create an alternatve to Algorthm. A vertex probablty s defned formally below. Defnton 6 (Vertex Probablty). Gven an evolutonary graph, N, V, W and tme t 0, the vertex probablty for v V s wrtten ) and defned as C 2 V s.t. v Pr(C(t) ). C Hence, the vertex probablty s the probablty that a gven vertex at a certan tme s a mutant. When we refer to the set vertex probabltes of each element of V at some tme t, we shall use the term vertex probablty vector. Vewng the probablty that a specfc vertex s a mutant at a gven tme has not, to our knowledge, been extensvely studed before wth respect to evolutonary graph theory (or the voter models n statstcal physcs). The key nsght of ths paper s that studyng these probabltes sheds new lght on the problem of calculatng fxaton probabltes. For example, t s easy to show the followng relatonshp. Proposton. Let C be a subset of V and t be an arbtrary tme pont. Iff for all v C, ) = and for all v / C, ) = 0, then Pr(C (t) ) = and for all C 2 V s.t. C C, Pr(C (t) ) = 0. Proof Sketch. Suppose, BWOC, there s some C C where Pr(C (t) ) > 0. Clearly, there must exst some element v C that s not n C. By Defnton 6, ths causes ) <, whch s a contradcton. Gong the other drecton, we need only to consder that Pr(C (t) ) s n the summand of each ) assocated only wth a v C. We wll also use random varables S (t) to denote the event that vertex v was selected for reproducton and R (t) to denote the event of v replacng v. We wll often use condtonal probabltes. For example, C (0) ) s the probablty that v s a mutant gven the ntal set C of mutants. Throughout ths paper, unless noted otherwse, all of our probabltes wll be condtoned on C (0). We wll drop t for ease of notaton wth the understandng that some set C of V were mutants at t = 0. Hence, ) = C (0) ). It s easy to verfy that P C > 0 ff v V, lm t ) > 0. Hence, n ths paper, we shall generally assume that lm t ) > 0 holds for all vertces v and specfcally state when t does not. As an asde, for a gven EG, ths assumpton can be easly checked n n polynomal tme. Smply ensure for v V C that exsts some v C s.t. there s a drected path from v to v. Drectly Calculatng Fxaton Probablty Now that we have ntroduced the model and the dea of vertex probabltes we wll show how to leverage ths nformaton n our algorthm to compute fxaton probablty. Frst, we show that as tme approaches nfnty, the vertex probabltes for all vertces converge to the fxaton probablty when the graph f strongly connected. Theorem. If the graph s strongly connected,, lm t ) = P C. Proof Sketch. As tme approaches nfnty, there are only two possble confguratons of mutants - sets and V correspondng wth extncton and fxaton respectvely. By Defnton 6, as tme approaches nfnty, the probablty of any vertex beng a mutant must be the equal to the fxaton probablty. Hence, the statement follows. Now let us consder how to calculate ) for some v and t. For t = 0, where we know that we are n the state where only vertces n a gven set are mutants, we need only appeal to Proposton - whch tells us that we assgn a probablty of to all elements n that set and 0 otherwse. For subsequent tmesteps, we have developed Theorem 2 shown next. Theorem 2. ( (S (t) (v,v ) E w Pr(M (t ) ) equals w Pr(M (t ) ) Pr(S (t) ) Pr(S (t) M(t ) ) ) M(t ) ) + Pr(M (t ) ) s true ff v s selected for reproducton at tme t.) Proof Sketch. Note we use the varable R (t) s true ff v replaces v at tme t. Frst we show that ) =

4 Pr(M (t ) M (t ) M(t ) (v,v ) E S(t) ) + (v,v ) E Pr(S(t) R (t) ) + (v,v ) E Pr(S(t) R (t) ) by the orgnal model. Then, by the defnton of condtonal probablty, we get Pr(M (t ) (v,v ) E S(t) ) = Pr(M(t ) ) ( (v,v ) E Pr(S(t) M(t ) )). Then, by further manpulatng the probablty axoms as well as Bayes Theorem, we obtan Pr(S (t) R (t) M (t ) ) = w Pr(M (t ) ) Pr(S (t) M(t ) ) as well as Pr(S (t) R (t) M (t ) ) = ( w ) Pr(M (t ) ) Pr(S (t) M(t ) ). After showng all of these tems, we obtan the statement of the theorem through algebrac manpulaton. Although fndng Pr(S (t) M (t ) ) may be computatonally ntractable n practce, the good news s that for neutral drft (r = ), these condtonal probabltes are trval - specfcally, we have Pr(S (t) ) = /N for all and ths s ndependent of the current set of mutants or resdents n the EG. Hence, we can smplfy ) as follows. Corollary. Under r =, N (v,v ) E ( w Pr(M (t ) ) equals ) ) Pr(M (t ) ) + Pr(M (t ) ) Studyng evolutonary graph theory under neutral drft was a central theme n work such as [6, 7, ] as t provdes an ntuton on the effects of network topology on mutant spread. We shall focus on neutral drft n ths paper as well. Ths specal case also allows us to strengthen the statement of Theorem to a necessary and suffcent condton - showng that when the probabltes of all nodes are equal, then we can determne the fxaton probablty. Theorem. When r =, f for some tme t,, the value ) s the same, then ) = P C. Proof Sketch. (v w,v ) E (Pr(M (t ) N t,, we have Pr(M (t ) Consder ) = Pr(M (t ) ) + ) Pr(M (t ) )) when for ) = Pr(M (t ) ). Clearly, ) = Pr(M (t ) ). As the n ths case, the value for probabltes of all vertces was the same at t, they reman so at t. Therefore, n ths case, lm t ) = ). By Theorem, ths equals P C. Therefore, under neutral drft, we can determne fxaton probablty when the equaton of Corollary causes all ) s to be equal. We can also use Corollary to fnd bounds on the fxaton probablty for some tme t by the followng result. Theorem. For any tme t, under neutral drft (r = ), P C [mn ), max )]. Proof Sketch. For any tme t, under neutral drft (r = ), P C max ). We show that for each tme step t, max Pr(M (t ) ) max ). Hence, by showng that, for any tme t, ) lm t max ) whch we have max Pr(M (t ) by allows us to apply Theorem and obtan the statement of ths theorem. Let v be the vertex that at tme t becomes acheves the greatest ncrease n vertex probablty. At tme t. The rest follows by Corollary, and the fact that max Pr(M (t ) ) Pr(M (t ) ) Whch gves us the upper bound. The second part of the proof, whch s used for the lower bound, mrrors ths part. Under neutral drft, we can show that fxaton probablty s addtve for dsont sets. A smlar result was proved for a specal case (that s, undrected evolutonary graphs) n [6]. However, our proof dffers from thers n that we leverage Corollary. Theorem 5. When r = for dsont sets C, D V, P C + P D = P C D. Proof Sketch. Consder some tme t and vertex v. Clearly, by Corollary, ) can be expressed as a lnear combnaton of the form v (C V Pr(M (0) )) where C s a coeffcent. We note that these coeffcents are the same regardless of the ntal confguraton of mutants that M (t) s condtoned on. Hence, C (0) ) s ths postve functon wth Pr(M (0) ) = f v C and 0 otherwse (see Proposton ). Hence, for dsont C, D, for any v V, we have C (0) ) + D(0)) = (C D) (0) ). The statement follows. In [], the author observes expermentally (through smulaton) that the fxaton probablty computed wth neutral drft appears to be a lower bound on the fxaton probablty calculated wth a mutant ftness r >. Here, we prove ths to be true mathematcally. Theorem 6. For a gven set C, let P () (C) be the fxaton probablty under neutral drft and P (r) (C) be the fxaton probablty calculated usng a mutant ftness r >. Then, P () (C) P (r) (C). Proof Sketch. After ntroducng some notaton, 2 we show frst, by the rules of dynamcs, we show that f a some tme perod t, the probablty dstrbuton over mutant confguratons s I, mutant ftness r, and the transton functons used to reach the next tme step are χ (r) +, χ (r), and all 2 Proof Setup. We defne an nterpretaton, I : 2 V [0, ] as probablty dstrbuton over mutant confguratons. Hence, for some I we have C 2 V I(C) =. Next, we defne a transton functon that maps confguratons of mutants to probabltes, χ : 2 V [0, ] where for any C 2 V, C 2 V χ(c, C ) =. We wll use χ + and χ to ndcate f the transton s made wth a mutant beng selected for brth (χ + ) or resdent (χ ). Hence, for some C V and v / C, χ (C, C {v}) = 0 and χ + (C {v}, C) = 0. Hence, for all C 2 V, C 2 V (χ +(C, C ) + χ (C, C )) =. If the transton functon s based on brth-death and computed wth some r >, then we wll wrte t as χ (r) +, χ(r) respectvely. If computed wth r =, then we wrte χ (nd) +, χ(nd) respectvely. For some C 2C, let nc(c) be the set of all elements D 2 V s.t. D C and χ + (C, D) > 0. For some C 2 C, let dec(c) be the set of all elements D 2 V s.t. D C and χ (C, D) > 0. Gven set C V, we wll use P (r) C to denote the probablty of fxaton gven ntal set of mutants C where the value r s used to calculate all transton probabltes.

5 Table : Example of Algorthm 2 for Example 2. T v v v v τ subsequent transtons are computed usng the same dynamcs wth neutral drft, then the fxaton probablty s: P(I, r) = C 2 V I(C) D dec(c) (χ(r) ( D nc(c) (χ(r) () + (C, D) P D )+ )) ). Then, by algebrac manp- () (C, D) P D ulaton, we show that for some r r, for all C, D 2 V, we have χ (r) + (C, D) χ (r ) + (C, D) and χ (r) (C, D) χ (r ) (C, D). From Theorem 5, we know that gven some C 2 V, for all pars D, D where D nc(c) and D dec(c), we have P () D P () D. We then prove that gven nterpretaton I, for some r >, P(I, r) P(I, 0), from whch the theorem follows. Algorthm 2 - Our Novel Soluton Method to Compute Fxaton Probabltes Input: Evolutonary Graph N, V, W, confguraton C V, natural number R > 0, and real number ɛ 0. Output: Estmate of fxaton probablty of mutant. : p s the th poston n vector p correspondng wth vertex v V. 2: Set p = f v C and p = 0 otherwse. : q p {q wll be p from the prevous tme step.} : τ 5: whle τ > ɛ do 6: for v V do 7: sum 0 8: m {v V w > 0} 9: for v m do 0: sum = sum + w (q q ) : end for 2: p q + /N sum : end for : q p 5: τ (/2) (max p mn p) 6: end whle 7: return (mn p) + τ. The Algorthm Now we have shown all the necessary propertes of vertex probabltes to create our algorthm. Algorthm 2 shows pseudo-code to compute the fxaton probablty usng our method. As descrbed above, our method has found the exact fxaton probablty when all the probabltes n ) (represented n the pseudo-code as the vector p) are equal. We use Theorem to provde a convergence crtera based on value ɛ, whch we can prove to be the tolerance for the fxaton probablty. Next, we show the tme complexty and approxmaton guarantee of the algorthm, whch follows from the results of ths secton. We also provde an example of how t works n Example 2. Proposton. Where T sol s the number of tme steps to convergence (based on ɛ) and K s the average n-degree of the vertces n the evolutuonary graph, O(T sol NK) s the tme complexty of Algorthm 2. Algorthm 2 returns the fxaton probablty P C wthn ±ɛ. Example 2. Consder the scenaro ntroduced n Example. Suppose we decde to use Algorthm 2 to compute the fxaton probablty wth ɛ = Table shows the vertex probablty vector at each tme step, along wth the value for τ from Algorthm 2. Hence, at 50 tmesteps, the algorthm returns a fxaton probablty of In the above result, Snce T depends on a desred tolerance, we cannot compare our algorthm s performance to monte-carlo smulaton wthout the rght termnaton condton. Thus n our later experments we frst fnd fxaton probabltes usng monte-calro smulaton and then fnd the number of tmesteps T that t takes our soluton method to fnd a fxaton probablty wthn standard error of the smulaton method. It s easy to show that the expected number of mutants as tme approaches nfnty s equal to P C N - as P C s the probablty of beng n the state where all the vertces are mutants and N s the populaton (as the only other possble state as tme approaches nfnty s extncton - whch has no mutants). For any tme step t, the expected number of mutants s v V Pr(M(t) ) whch follows drectly from Defnton 6. Hence, returnng the average vertex probablty for a suffcently large value of t may also provde a good approxmaton. We also note, that as the vertex probabltes converge, the standard devaton of the p vector n Algorthm 2 could be a potentally faster convergence crtera. Note that usng standard devaton of p and returnng the average vertex probablty would no longer provde us of the guarantee n Proposton, however t may provde good results n practce. The modfcatons to the algorthm would be as follows: lne 5 would be τ st.dev(p) and lne 7 would be return avg(p). We wll refer to ths as Algorthm 2 wth alternate convergence crtera or Algorthm 2-ACC for short.

6 MnP MaxP AvgP Fnal t Fgure : Convergence of the mnmum (MnP), maxmum (MaxP), and average (AvgP) of vertex probabltes towards the fnal fxaton probablty as a functon of our algorthm s teratons t for a graph of 00 nodes. Expermental Evaluaton Our novel method for computng fxaton probabltes on strongly connected drected graphs allows us to compute near-exact fxaton probabltes wthn a desred tolerance. The tme complexty of our method s hghly dependent on how fast the vertex probabltes converge. In ths secton we expermentally evaluate how the vertex probabltes n our algorthms converge. We also provde results from comparson experments to support the clam that Algorthm 2-ACC fnds adequate fxaton probabltes order of magntudes faster than Monte Carlo smulatons (Algorthm ). All algorthms were mplemented n Python and run on a 2.GHz Intel Xeon CPU t stdev n vertex probabltes0.0 Fgure : Standard devaton of vertex probabltes as a functon of our algorthm s teratons for the same 00 node graph of Fgure.. Convergence of Vertex Probabltes We ran our algorthms to compute fxaton probabltes on randomly weghted and strongly connected drected graphs n order to expermentally evaluate the convergence of the vertex probabltes. We generated the graphs to be scale free usng the standard preferental attachment growth model [] and randomly assgned an ntal mutant node. We replaced all edges n the graph gven by the growth model wth two drected edges and then randomly assgned weghts to all the edges. Fgure shows the convergence of the mnmum, maxmum, and the average of vertex probabltes towards the fnal fxaton probablty value for a small graph of 00 nodes. We can observe that the average converges to the fnal value at a logarthmc rate and much faster than the mnmum and maxmum vertex probablty values. Ths suggests that whle Algorthm 2-ACC does not gve the same theoretcal guarantees as Algorthm 2, t s much preferable for speed snce the mnmum and maxmum vertex probabltes take much longer to converge to the fnal soluton than the average. The fact that the average of the vertex probabltes s much preferable as a fast estmaton of fxaton probabltes s supported by the logarthmc decrease of the standard devaton of vertex probabltes (see Fgure ). Convergences for other and larger graphs are not shown here but are qualtatvely smlar to the relatve convergences shown n the provded graphs..2 Speed Comparson to Monte Carlo Smulaton In order to compare our method s speed compared to the standard Monte Carlo smulaton method, we must determne how many teratons our algorthm must be run to fnd a fxaton probablty estmate comparable to that of Algorthm. Thankfully, as we have seen, we can get a standard error on the fxaton probablty returned by Algorthm as per Proposton 2. Whle we dd not theoretcally prove anythng about how smoothly fxaton probabltes from our methods approach the fnal soluton, the convergences of the average and standard devaton as shown above strongly suggest that estmates from our method approach the fnal soluton qute gracefully. In fact, n the followng experments once our method has arrved at a fxaton probablty estmate wthn the standard error of smulatons, the estmate never agan fell outsde the wndow of standard error (although the estmate dd not always approach the fnal estmate monotoncally). Ths s n stark contrast to Monte Carlo smulatons, from whch estmatons can vary greatly before the method has completed enough sngle runs to acheve a good probablty estmate. We generated a number of randomly weghted and strongly connected drected graphs of varous szes on whch we compare our soluton method to Monte Carlo approxmaton of fxaton probabltes. The graphs were generated as n our convergence experments. For each graph of a dfferent sze, we generate a number of dfferent ntal mutant confguratons. We found fxaton probabltes both usng Monte Carlo estmaton wth 2000 smulaton runs and our drect soluton method, termnatng when we have reached wthn the standard error of the Monte Carlo estmaton. Snce the average vertex probablty proved to be such a good fast estmate of the true fxaton probablty, we used Algorthm 2-ACC.

7 Table 2: Experment Results. For each graph sze, shows average number of tmesteps needed for each sngle smulaton n the Monte Carlo estmaton to reach extncton or fxaton (Avg T sm ), and the average number of teratons our soluton algorthm must be run to get a fxaton probablty wthn the standard error of smulatons ( AVG T sol ). Table : Experment Results. For each graph sze, shows the average seconds taken for the Monte Carlo method (Avg Secs Sm) and our soluton method (Avg Secs Sol) to get a fxaton probablty wthn the standard error of smulatons. Also shows the average speedup our algorthm provdes. N Avg T sm T sol N Avg Secs Sm Avg Secs Sol Avg Speedup Tables 2 and show results from these experments, ncludng the average measured values for T sm (average tmesteps to extncton r fxaton for smulatons), T sol (average tmesteps for our soluton method to get wthn standard error of Monte Carlo estmaton), and actual seconds taken for each method mplemented. Fgure 5 shows the speedup our soluton provdes over Monte Carlo smulaton. Here speedup s defned as the rato of the tme t takes for smulatons to complete over the tme t takes our algorthm to fnd a fxaton probablty wthn the standard devaton. log(speedup) graph sze Fgure 5: Average speedup (on a log scale) for fndng fxaton probabltes acheved by our algorthm vs Monte Carlo smulaton for graphs of dfferent szes. We can observe from our experments that computng fxaton probabltes usng Monte Carlo smulatons showed to be a very tme-expensve process, hghlghtng the need for faster soluton methods as the one we have presented. Especally for larger graph szes, the tme complexty of our soluton to acheve smlar results to Monte Carlo smulaton has shown to be orders of magntude smaller than the standard method. 5 Related Work Whle most work dealng wth evolutonary graphs rely on Monte Carlo smulaton, there are some good analytcal approxmatons for the undrected cased based on the degree of the vertces n queston. In [5], the authors use the meanfeld approach to create these approxmatons for the undrected case. In [6], the authors derve an exact analytcal result for the undrected case n neutral drft, whch agrees wth the results of [5]. They also show that fxaton probablty s addtve n that case (a result whch we extend n ths paper usng a dfferent proof technque). However, the results of [7] demonstrate that mean-feld approxmatons break down n the case of weghted, drected graphs. That work s followed by [] whch also studes weghted, drected graphs, but does so by usng Monte Carlo smulaton. In [8] the authors derve exact computaton of fxaton probablty through means of lnear programmng. However, that approach requres an exponental number of both constrants and varables and s ntractable. In [0], the authors present a technque for speedng up Monte Carlo smulatons by early termnaton. However, our algorthm dffers n that t does not rely on smulaton at all and provdes a determnstc result. Our method totally avods smulaton and nstead leverages propertes of the model. Recently, n [], the authors study the related problem of determnng the probablty of fxaton gven a sngle, randomly placed mutant n the graph where the vertces are slands and there are many ndvduals resdng on each sland n a wellmxed populaton. They use quas-fxed ponts of ODE s to obtan an approxmaton of the fxaton probablty and performed experments wth a maxmum of 5 slands (vertces) contanng 50 ndvduals each. It s unclear f ther methods can be appled to the problem presented n ths paper. In [5, 6] the authors present a generalzed frameworks for socal network dffuson. These works prmarly focus on dffuson models that are monotonc - where the number of vertces wth a certan property ncreases at each tme step. Whle [5] does show how to adust ther framework for non-monotonc models, they only do so for a fnte number of tme steps (fxaton probablty s n the lmt of tme).

8 6 Concluson In ths paper, we presented a novel determnstc algorthm for quckly computng fxaton probablty n drected, weghted strongly connected evolutonary graphs under neutral drft, whch we prove to be a lower bound for the case where a mutant s more ft than the rest of the populaton. In our experments, we showed our approach to outperform Monte Carlo smulatons, whch are currently used n most evolutonary graph theory research, by several orders of magntude. We also show that under neutral drft, fxaton probablty s addtve - showng optmal substructure n ths case. Our algorthm reled on the convergence of the vertex probabltes - the probablty of an ndvdual beng a mutant at a certan tme. As our experments demonstrated that ths convergence generally occurs rapdly, t s a temptng conecture that the tme to convergence s polynomal n the sze of the graph. If ths s the case, we suspect that ftness-based selecton of the reproducng ndvdual n the network may be a source of complexty n ths problem. A more complete complexty analyss of evolutonary graph problems may stll be n order. An mportant lmtaton of our algorthm s that t s lmted to strongly connected graphs. However, we beleve that our algorthm can be used n solutons to general graphs by breakng the graph nto ts strongly connected components and consderng transton edges between these n the computaton of the overall fxaton probabltes. Such an extenson s an mmedate goal for future work. Acknowledgements P.S. s supported by ARO grant 602B7F. P.R. s supported by ONR grant W9NF080. The opnons n ths paper are those of the authors and do not necessarly reflect the opnons of the funders. References [] E. Leberman, C. Hauert, M. A. Nowak, Evolutonary dynamcs on graphs, Nature (702),2005, 2 6. [2] P. A. Zhang, P. Y. Ne, D. Q. Hu, F. Y. Zou, The analyss of b-level evolutonary graphs., Bosystems 90 () 2007, [] V. Sood, T. Antal, S. Redner, Voter models on heterogeneous networks, Physcal Revew E 77 () 2008, 02. [] J. M. Pacheco, A. Traulsen, M. A. Nowak, Actve lnkng n evolutonary games, Journal of Theoretcal Bology () 2006, 7. [5] T. Antal, S. Redner, V. Sood, Evolutonary dynamcs on degree-heterogeneous graphs, Physcal Revew Letters 96 (8) 2006, 880. [6] M. Broom, C. Hadchrysanthou, J. Rychtar, B. T. Stadler, Two results on evolutonary processes on general non-drected graphs, Proceedngs of the Royal Socety A 66 (22) 200, [7] N. Masuda, H. Ohtsuk, Evolutonary dynamcs and fxaton probabltes n drected networks, New Journal of Physcs () 2009, 002. [8] J. Rychtář, B. Stadler, Evolutonary dynamcs on small-world networks, Internatonal Journal of Computatonal and Mathematcal Scences 2 (). [9] M. Broom, J. Rychtář, An analyss of the fxaton probablty of a mutant on specal classes of nondrected graphs, Proceedngs of the Royal Socety A , [0] V. C. Barbosa, R. Donangelo, S. R. Souza, Early apprasal of the fxaton probablty n drected networks, Physcal Revew E 82 () 200, 06. [] N. Masuda, Drectonalty of contact networks suppresses selecton pressure n evolutonary dynamcs, Journal of Theoretcal Bology 258 (2) 2009, 2. [2] P. Moran, Random processes n genetcs, Mathematcal Proceedngs of the Cambrdge Phlosophcal Socety 5 (0) 958, [] H. Jeong, Z. Neda, A. L. Barabas, Measurng preferental attachment n evolvng networks, EPL (Europhyscs Letters) 6 () 200, 567. [] B. Houchmandzadeh, M. Vallade, The fxaton probablty of a benefcal mutaton n a geographcally structured populaton, New Journal of Physcs (7) 20, [5] D. Kempe, J. Klenberg, E. Tardos, Maxmzng the spread of nfluence through a socal network, n: KDD 0: Proceedngs of the nnth ACM SIGKDD nternatonal conference on Knowledge dscovery and data mnng, ACM, New York, NY, USA, 200, pp [6] P. Shakaran, V. Subrahmanan, M. L. Sapno, Usng generalzed annotated programs to solve socal network optmzaton problems, n: M. Hermenegldo, T. Schaub (Eds.), Techncal Communcatons of the 26th Internatonal Conference on Logc Programmng, Lebnz Internatonal Proceedngs n Informatcs (LIPIcs), Dagstuhl, Germany, 200, pp