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one of the most basic models of evolution in finite populations is the moran process @xcite . in the moran process , a population consisting of two types , a mutant type and a wild type , is continually updated via a birth - death process until only one type remains . the mutant and wild types are distinguished by only their reproductive fitness , which is assumed to be an intrinsic property of a player . a mutant type has fitness @xmath0 relative to the wild type ( whose fitness relative to itself is @xmath1 ) , and in each step of the process an individual is selected for reproduction with probability proportional to fitness . reproduction is clonal , and the offspring of a reproducing individual replaces another member of the population who is chosen for death uniformly at random . eventually , this population will end up in one of the monomorphic absorbing states : all mutant type or all wild type . in this context , a fundamental metric of the success of the mutant type is its ability to invade and replace a population of wild - type individuals @xcite .    in a population of size @xmath2 , the probability that a single mutant in a wild - type population will fixate in the moran process is    @xmath3    in this version of a birth - death process , the members of the population are distinguished by only their types ; in particular , there is no notion of spatial arrangement , i.e. the population is _ well - mixed_. @xcite extend the classical moran process to graph - structured populations , which are populations with links between the players that indicate who is a neighbor of whom . in this structured version of the moran process , reproduction happens with probability proportional to fitness , but the offspring of a reproducing individual can replace only a _ neighbor _ of the parent . since individuals are now distinguished by both their types ( mutant or wild ) _ and _ locations within the population , a natural question is whether or not the fixation probability of a single mutant type depends on where this mutant appears in the population . @xcite show that this fixation probability is independent of the location of the mutant if everyone has the same number of neighbors , i.e. the graph is _ regular _ @xcite . in fact , remarkably , the fixation probability of a single mutant on a regular graph is the same as that of eq . ( [ fpmoran])an observation first made in a special case by @xcite . this result , known as the isothermal theorem , is independent of the number of neighbors the players have ( i.e. the _ degree _ of the graph ) . the moran process is _ frequency - independent _ in the sense that the fitness of an individual is determined by type and is not influenced by the rest of the population . however , the moran model can be easily extended to account for frequency-_dependent _ fitness . a standard way in which to model frequency - dependence is through evolutionary games @xcite . in the classical setup , each player in the population has one of two strategies , @xmath4 and @xmath5 , and receives an aggregate payoff from interacting with the rest of the population . this aggregate payoff is usually calculated from a sequence of pairwise interactions whose payoffs are described by a payoff matrix of the form    @xmath6    each player s aggregate payoff is then translated into fitness and the strategies in the population are updated based on these fitness values . since a player s payoff depends on the strategies of the other players in the population , so does that player s fitness . traditionally , this population is assumed to be infinite , in which case the dynamics of the evolutionary game are governed deterministically by the replicator equation of @xcite . more recently , evolutionary games have been considered in finite populations @xcite , where the dynamics are no longer deterministic but rather stochastic . in order to restrict who interacts with whom in the population , these populations can also be given structure . popular types of structured populations are graphs @xcite , sets @xcite , and demes @xcite . we focus here on evolutionary games in graph - structured populations that proceed in discrete time steps . such processes define discrete - time markov chains , either with or without absorbing states ( depending on mutation rates ) . typically , in evolutionary game theory , one starts with a population of players and repeatedly updates the population based on some update rule such as birth - death @xcite , death - birth @xcite , imitation @xcite , pairwise comparison @xcite , or wright - fisher @xcite . these update rules can be split into two classes : cultural and genetic ( see * ? ? ? cultural update rules involve strategy imitation while genetic update rules involve reproduction and inheritance . without mutations , an update rule may be seen as giving a probability distribution over a number of strategy - acquisition scenarios : a player inherits a new strategy through imitation ( cultural rules ) or is born with a strategy determined by the parent(s ) ( genetic rules ) . mutation rates disrupt these scenarios by placing a small probability of a player taking on a novel strategy . the way in which strategy mutation rates are incorporated into an evolutionary process depends on both the class of the update rule and the specifics of the update rule itself . in a general sense , we say that strategy mutations are _ homogeneous _ if they depend on neither the players themselves nor the locations of the players . this notion of homogeneous strategy mutations is analogous to that of a symmetric game , which is a game for which the payoffs depend on the strategies played but are independent of the identities and locations of the players . the isothermal theorem seems to indicate that populations structured by regular graphs possess a significant degree of homogeneity , meaning that different locations within the population appear to be equivalent for the purposes of evolutionary dynamics . however , it is important to note that ( i ) fixation probability is just one metric of evolutionary success and ( ii ) the moran process is only one example of an evolutionary process . for example , in addition to the probability of fixation , one could look at the _ absorption time _ , which is the average number of steps until one of the monomorphic absorbing states is reached . moreover , one could consider frequency - dependent processes , possibly with different update rules , in which fitness is no longer an intrinsic property of an individual but is also influenced by the other members of the population . we show that the isothermal theorem does not extend to arbitrary frequency - dependent processes such as evolutionary games . furthermore , we show that this theorem does not apply to fixation _ times _ ; that is , even for the moran process on a regular graph , the average number of updates until a monomorphic absorbing state is reached can depend on the initial placement of the mutant . given that the isothermal theorem does not extend to other processes defined on regular graphs , the next natural question is the following : what is the meaning of a spatially - homogeneous population in evolutionary game theory ? in fact , we argue using asymmetric games @xcite that the term  homogeneous " should apply to an evolutionary process _ as a whole _ rather than to just the population structure . even for populations that appear to be spatially homogeneous , such as populations on complete graphs , non - uniform distribution of resources within the population can result in heterogeneity of the overall process . similarly , for symmetric games , heterogeneity can be introduced into the dynamics of an evolutionary process through strategy mutations . therefore , a notion of homogeneity of an evolutionary game should take into account _ at least _ ( i ) population structure , ( ii ) payoffs , and ( iii ) strategy mutations . if the strategy - mutation rates are miniscule , then the population spends most of its time in monomorphic states . with small mutation rates , one can define an embedded markov chain on the monomorphic states and use this chain to study the success of each strategy @xcite . that is , when a mutation occurs , the population is assumed to return to a monomorphic state before another mutant arises . thus , the states of interest are the monomorphic states and the states consisting of a single mutant in an otherwise - monomorphic population . we say that an evolutionary game is _ homogeneous _ if any two states consisting of a single mutant ( @xmath4-player ) in wild - type population ( @xmath5-players ) are mathematically equivalent . we make precise what we mean by  mathematically equivalent " in  [ sec : markov ] , but , informally , this equivalence means that any two such states are the same up to relabeling . in particular , _ all _ metrics such as fixation probability , absorption time , etc . are the same for any two states consisting of a single @xmath4-mutant in a @xmath5-population . we show that an evolutionary game in a graph - structured population is homogeneous if the graph is vertex - transitive (  looks the same " from each vertex ) , the payoffs are symmetric , and the strategy mutations are homogeneous . this result holds for any update rule and selection intensity . finally , we explore the effects of population structure on asymmetric evolutionary games . in the weak selection limit , we show that asymmetric matrix games with homogeneous strategy mutations can be reduced to symmetric games if the population structure is arc - transitive (  looks the same " from each edge in the graph ) . this result is a finite - population analogue of the main result of @xcite , which states that a similar reduction to symmetric games is possible in sufficiently large populations . thus , we establish that this reduction applies to any population size if the graph possesses a sufficiently high degree of symmetry . our explorations , both for symmetric and asymmetric games , clearly demonstrate the effects of population structure , payoffs , and strategy mutations on symmetries in evolutionary games . the evolutionary processes we consider here define discrete - time markov chains on finite state spaces . the notions of symmetry and evolutionary equivalence that we aim to introduce for evolutionary processes can actually be stated quite succinctly at the level of the markov chain . we first work with general markov chains , and later we apply these ideas to evolutionary games . [ def : symmetry ] suppose that @xmath7 is a markov chain on a ( finite ) state space , @xmath8 , with transition matrix @xmath9 . an automorphism of @xmath10 is a bijection @xmath11 such that @xmath12 for each @xmath13 . two states @xmath13 are said to be _ symmetric _ if there exists @xmath14 such that @xmath15 .    definition [ def : symmetry ] says that the states of the chain can be relabeled in such a way that the transition probabilities are preserved . this relabeling may affect the long - run distribution of the chain since it need not fix absorbing states , so we make one further refinement in order to ensure that if two states are symmetric , then they behave in the same way :    [ def : evolutionaryequivalence ] states @xmath16 and @xmath17 are _ evolutionarily equivalent _ if there exists an automorphism of the markov chain , @xmath14 , such that    1 .   @xmath15 ; 2 . if @xmath18 is a stationary distribution of @xmath10 , then @xmath19 .    for a markov chain with absorbing states , the notions of symmetry and evolutionary equivalence of states need not coincide ( see example [ ex : swapstrategies ] of appendix b ) . however , if the markov chain has a unique stationary distribution ( as would be the case if it were irreducible ) , then symmetry implies evolutionary equivalence :    [ prop : ergodicsymmetry ] if @xmath10 has a unique stationary distribution , then two states are symmetric if and only if they are evolutionarily equivalent . we show in appendix b that a markov chain symmetry preserves the set of stationary distributions ( lemma [ lem : stationarydistribution ] ) , so if there is a unique stationary distribution , then condition ( ii ) of definition [ def : evolutionaryequivalence ] is satisfied automatically by any symmetry . proposition [ prop : ergodicsymmetry ] is then an immediate consequence of this result . if @xmath16 and @xmath17 are evolutionarily equivalent , then it is clear for absorbing processes that the probability that @xmath16 fixates in absorbing state @xmath20 is the same as the probability that @xmath17 fixates in state @xmath20 ( and similarly for fixation times ) . if the process has a unique stationary distribution , then the symmetry of @xmath16 and @xmath17 implies that this distribution puts the same mass on @xmath16 and @xmath17 . these properties follow at once from the fact that the states @xmath16 and @xmath17 are equivalent up to relabeling . our focus is on evolutionary games on fixed population structures . if @xmath21 is a finite set of strategies ( or  actions " ) available to each player , and if the population size is @xmath2 , then the state space of the markov chain defined by an evolutionary game in such a population is @xmath22 . for evolutionary games without random strategy mutations , the absorbing states of the chain are the monomorphic states , i.e. the strategy profiles consisting of just a single unique strategy . thus , states @xmath16 and @xmath17 are evolutionarily equivalent if they are symmetric relative to the monomorphic states . on the other hand , evolutionary processes with strategy mutations are typically irreducible ( and have unique stationary distributions ) ; in these processes , the notions of symmetry and evolutionary equivalence coincide by proposition [ prop : ergodicsymmetry ] .    in order to state the definition of a _ homogeneous _ evolutionary process , we first need some notation . for @xmath23 , denote by @xmath24 the state in @xmath25 whose @xmath26th coordinate is @xmath27 and whose @xmath28th coordinate for @xmath29 is @xmath30 ; that is , all players are using strategy @xmath30 except for player @xmath26 , who is using @xmath27 . [ def : homogeneousevolutionaryprocess ] an evolutionary process on @xmath25 is _ homogeneous _ if for each @xmath23 , the states @xmath24 and @xmath31 are evolutionarily equivalent for each @xmath32 . an evolutionary process is _ heterogeneous _ if it is not homogeneous .    in other words , an evolutionary process is homogeneous if , at the level of the markov chain it defines , any two states consisting of a single mutant in an otherwise - monomorphic population appear to be relabelings of one another . as noted in [ subsec : generalmarkov ] , _ all _ quantities with which one could measure evolutionary success are the same for these single - mutant states if the process is homogeneous . we consider evolutionary games in graph - structured populations . unless indicated otherwise , a  graph " means a directed , weighted graph on @xmath2 vertices . a directed graph is one in which the edges have orientations , meaning there may be an edge from @xmath26 to @xmath28 but not from @xmath28 to @xmath26 . moreover , the edges carry weights , which we assume are nonnegative real numbers . a directed , weighted graph is equivalent to a nonnegative @xmath33 matrix , @xmath34 , where there is an edge from @xmath26 to @xmath28 if and only if @xmath35 . if there is such an edge , then the weight of this edge is simply @xmath36 . since there is a one - to - one correspondence between directed , weighted graphs on @xmath2 vertices and @xmath33 real matrices , we refer to graphs and matrices using the same notation , describing @xmath34 as a graph but using the matrix notation @xmath36 to indicate the weight of the edge from vertex @xmath26 to vertex @xmath28 . every graph considered here is assumed to be connected ( strongly ) , which means that for any two vertices , @xmath26 and @xmath28 , there is a ( directed ) path from @xmath26 to @xmath28 . this assumption is not that restrictive in evolutionary game theory since one can always partition a graph into its strongly - connected components and study the behavior of an evolutionary process on each of these components . moreover , for evolutionary processes on graphs that are not strongly connected , it is possible to have both ( i ) recurrent non - monomorphic states in processes without mutations and ( ii ) multiple stationary distributions in processes with mutations . some processes ( such as the death - birth process ) may not even be defined on graphs that are not strongly connected . therefore , we focus on strongly - connected graphs and make no further mention of the term  connected . " since our goal is to discuss symmetry in the context of evolutionary processes , we first describe several notions of symmetry for graphs . the three types of graphs we treat here are _ regular _ , _ vertex - transitive _ , and _ symmetric_. informally speaking , a regular graph is one in which each vertex has the same number of neighboring vertices ( and this number is known as the _ degree _ of the graph ) . a vertex - transitive graph is one that looks the same from any two vertices ; based on the graph structure alone , a player can not tell if he or she has been moved from one location to another . a symmetric ( or _ arc - transitive _ ) graph is one that looks the same from any two edges . that is , if two players are neighbors and are both moved to another pair of neighboring vertices , then they can not tell that they have been moved based on the structure of the graph alone . we recall in detail the formal definitions of these terms in appendix b. the relationships between these three notions of symmetry , as well as some examples , are illustrated in fig . [ fig : graphdiagram ] .    ) and vertex - transitive but not symmetric ( fig . [ fig : transitivesymmetric](a)).[fig : graphdiagram ] ]    we turn now to evolutionary processes in graph - structured populations :      consider the moran process on a graph , @xmath34 . @xcite show that if @xmath34 is regular , then the fixation probability of a randomly placed mutant is given by eq . ( [ fpmoran ] ) , the fixation probability of a single mutant in the classical moran process . this result ( known as the isothermal theorem ) proves that , in particular , this fixation probability does not depend on the initial location of the mutant . ( we refer to this latter statement as the  weak " version of the isothermal theorem . ) our definition of homogeneity in the context of evolutionary processes ( definition [ def : homogeneousevolutionaryprocess ] ) is related to this independence of initial location and has nothing to do with fixation probabilities in the classical moran process . naturally , the isothermal theorem raises the question of whether or not this location independence extends to absorption times ( average number of steps until an absorbing state is reached ) when @xmath34 is regular . suppose that @xmath34 is the _ frucht graph _ of fig . [ fig : fruchtgraph ] . the frucht graph is an undirected , unweighted , regular ( but not vertex - transitive ) graph of size @xmath37 and degree @xmath38 @xcite . the fixation probabilities and absorption times of a single mutant in a wild - type population are given in fig . [ fig : moranbargraphs ] as a function of the initial location of the mutant . the fixation probabilities do not depend on the initial location of the mutant , as predicted by the isothermal theorem , but the absorption times _ do _ depend on where the mutant arises . in fact , the absorption time is distinct for each different initial location of the mutant . the details of these calculations are in appendix c. therefore , even the weak form of the isothermal theorem fails to hold for absorption times . in particular , the moran process on a regular graph need not define a homogeneous evolutionary process .     of the frucht graph . in the snowdrift game , the probability that cooperators fixate depends on the initial location of this mutant on the frucht graph ( even if the intensity of selection is weak).[fig : fruchtgraph ] ]    this setup involving two types of players , frequency - independent interactions , and a population structure defined by a single graph , can be generalized considerably :      a powerful version of evolutionary graph theory uses _ two _ graphs to define relationships between the players : an _ interaction graph _ , @xmath39 , and a _ dispersal graph _ , @xmath34 @xcite . these graphs both have nonnegative weights . as an example of how these two graphs are used to define an evolutionary process , we consider a birth - death process based on two - player , symmetric interactions :    [ interactiondispersalexample ] consider a symmetric matrix game with @xmath40 strategies , @xmath41 , and payoff matrix    @xmath42    if @xmath43 , then the total payoff to player @xmath26 is    @xmath44    if @xmath45 is the intensity of selection , then the fitness of player @xmath26 is    @xmath46    in each time step , a player ( say , player @xmath26 ) is chosen for reproduction with probability proportional to fitness . with probability @xmath47 , the offspring of this player adopts a novel strategy uniformly at random from @xmath48 ; with probability @xmath49 , the offspring inherits the strategy of the parent . next , another member of the population is chosen for death , with the probability of player @xmath28 dying proportional to @xmath36 . the offspring then fills the vacancy created by the deceased neighbor , and the process repeats . @xmath39 is called the  interaction " graph since it governs payoffs based on encounters , and @xmath34 is called the  dispersal " graph since it is involved in strategy propagation .      we now explore the ways in which population structure and strategy mutations can introduce heterogeneity into an evolutionary process . consider the snowdrift game with strategies @xmath50 ( cooperate ) and @xmath51 ( defect ) and payoff matrix    @xmath52    suppose that @xmath39 and @xmath34 are both the ( undirected , unweighted ) frucht graph ( see fig . [ fig : fruchtgraph ] ) . if the selection intensity is @xmath53 , then the fixation probability of a single cooperator in a population of defectors in a death - birth process depends on the initial location of the cooperator ( fig . [ fig : snowdriftbargraphs ] ) . since the frucht graph is regular ( but not vertex - transitive ) , this example demonstrates that the isothermal theorem does not extend to frequency - dependent games . in particular , symmetric games on regular graphs can be heterogeneous , and regularity of the graph does not imply that the  fixation probability of a randomly placed mutant " is well - defined . this dependence of the fixation probability on the initial location of the mutant is not specific to the snowdrift game or the death - birth update rule ; one can show that it also holds for the donation game in place of the snowdrift game or the birth - death rule in place of the death - birth rule , for instance .    with @xmath53 , the selection intensity is fairly strong , which raises the question of whether or not these fixation probabilities still differ if selection is weak . in fact , our observation for this value of @xmath54 is not an anomaly : suppose that @xmath16 and @xmath17 are states ( indicating some non - monomorphic initial configuration of strategies ) , and that @xmath55 and @xmath56 are monomorphic absorbing states ( indicating states in which each player uses the same strategy ) . let @xmath57 denote the probability that state @xmath26 is reached after starting in state @xmath16 , and let @xmath58 denote the average number of updates required for the process to reach an absorbing state after starting in state @xmath16 . each of @xmath57 and @xmath58 may be viewed as functions of @xmath54 , and we have the following result :    [ prop : functionofbeta ] each of the equalities    @xmath59    holds for either ( i ) _ every _ @xmath45 or ( ii ) at most finitely many @xmath45 . thus , if one of these equalities fails to hold for even a single value of @xmath54 , then it fails to hold for all sufficiently small @xmath60 .    for a proof of proposition [ prop : functionofbeta ] , see appendix a. this result allows one to conclude that if there are differences in fixation probabilities or times for large values of @xmath54 ( where these differences are more apparent ) , then there are corresponding differences in the limit of weak selection . even if a symmetric game is played in a well - mixed population , heterogeneous strategy mutations may result in heterogeneity of the evolutionary process . consider , for example , the pairwise comparison process @xcite based on the symmetric snowdrift game , ( [ snowdriftgame ] ) , in a well - mixed population with @xmath61 players . we model this well - mixed population using a complete , undirected , unweighted graph of size @xmath38 for each of @xmath39 and @xmath34 ( see fig . [ fig : wellmixedthree ] ) . for @xmath62 , let @xmath63 $ ] be the strategy - mutation (  exploration " ) rate for player @xmath26 . these strategy mutations are incorporated into the process as follows : at each time step , a focal player ( player @xmath26 ) is chosen uniformly at random to update his or her strategy . a neighbor ( one of the two remaining players ) is then chosen randomly as a model player . if @xmath54 is the selection intensity , @xmath64 is the payoff of the focal player , and @xmath65 is the payoff of the model player , then the focal player imitates the strategy of the model player with probability    @xmath66    and chooses to retain his or her strategy with probability    @xmath67    with probability @xmath68 , the focal player adopts a new strategy uniformly at random from the set @xmath69 , irrespective of the current strategy . provided at least one of @xmath70 , @xmath71 , and @xmath72 is positive , the markov chain on @xmath73 defined by this process is irreducible and has a unique stationary distribution , @xmath18 . let @xmath74 and @xmath75 . since the mutation rate depends on the location , @xmath26 , the strategy mutations are _ heterogeneous_. if the selection intensity is @xmath53 , then a direct calculation ( to four significant figures ) gives    @xmath76    where @xmath77 ( resp . @xmath78 ) is the mass @xmath18 places on the state @xmath79 ( resp . @xmath80 ) . therefore , by proposition [ prop : ergodicsymmetry ] and definition [ def : homogeneousevolutionaryprocess ] , this evolutionary process is _ not _ homogeneous , despite the fact that the population is well - mixed and the game is symmetric . this result is not particularly surprising , but it clearly illustrates the effects of hetergeneous strategy - mutation rates on symmetries of the overall process . the behavior of an evolutionary process sometimes depends heavily on the choice of update rule . as a result , a particular problem in evolutionary game theory is often stated ( such as the evolution of cooperation ) and subsequently explored separately for a number of different update rules . for example , consider the donation game ( an instance of the prisoner s dilemma ) in which cooperators pay a cost , @xmath81 , in order to provide the opponent with a benefit , @xmath82 . defectors pay no costs and provide no benefits . on a large regular graph of degree @xmath83 , @xcite show that selection favors cooperation in the death - birth process if @xmath84 , but selection _ never _ favors cooperation in the birth - death process . therefore , the approach of exploring a problem in evolutionary game theory separately for several update rules has its merits . on the other hand , one might expect that high degrees of symmetry in the population structure , payoffs , and strategy mutations induce symmetries in an evolutionary game for a variety of update rules .    before stating our main theorem for symmetric matrix games , we must first understand the basic components that make up an evolutionary game . evolutionary games generally have two timescales : interactions and updates . in each ( discrete ) time step , every player in the population has a strategy , and this strategy profile determines the _ state _ of the population . neighbors then interact ( quickly ) and receive payoffs based on these strategies and the game(s ) being played . the total payoff to a player determines his or her fitness . in the update step of the process , the strategies of the players are updated stochastically based on the fitness profile of the population , the population structure , and the strategy mutations . popular examples of evolutionary update rules are birth - death , death - birth , imitation , pairwise comparison , and wright - fisher . since interactions happen much more quickly than updates , there is a separation of timescales . the most difficult part of an evolutionary game to describe in generality is the update step . if @xmath21 is the strategy set of the game and @xmath2 is the population size , then a state of the population is simply an element of @xmath25 , i.e. a specification of a strategy for each member of the population . implicit in the state space of the population being @xmath25 is an enumeration of the players . that is , if @xmath85 is an @xmath2-tuple of strategies , then this profile indicates that player @xmath26 uses strategy @xmath55 . for our purposes , we need only one property to be satisfied by the update rule of the process , which we state here as an axiom of an evolutionary game :    the update rule of an evolutionary game is independent of the enumeration of the players . [ axiomremark ] as an example of what this axiom means , consider a death - birth process in which a player is selected uniformly at random for death and is replaced by the offspring of a neighbor . a neighbor is chosen for reproduction with probability proportional to fitness , and the offspring of this neighbor inherits the strategy of the parent with probability @xmath49 and takes on a novel strategy with probability @xmath86 for some @xmath87 . if all else is held constant ( fitness , mutations , etc . ) , the fact that a player is referred to as the player at location @xmath26 is irrelevant : let @xmath88 be the symmetric group on @xmath2 letters . if @xmath89 is a permutation that relabels the locations of the players by sending @xmath26 to @xmath90 , then the strategy of the player at location @xmath90 after the relabeling is the same as the strategy of the player at location @xmath26 before the relabeling . in particular , if @xmath85 is the state of the population before the relabeling , then @xmath91 is the state of the population after the relabeling , where @xmath92 . the probability that player @xmath90 is selected for death and replaced by the offspring of player @xmath93 after the relabeling is the same as the probability that player @xmath26 is selected for death and replaced by the offspring of player @xmath28 before the relabeling . thus , for this death - birth process , the probability of transitioning between states @xmath16 and @xmath17 before the relabeling is the same as the probability of transitioning between states @xmath94 and @xmath95 after the relabeling . in this sense , a relabeling of the players induces an automorphism of the markov chain defined by the process ( in the sense of definition [ def : symmetry ] ) , and the axiom states that this phenomenon should hold for _ any _ evolutionary update rule .    in order to state our main result for symmetric games , we note that an evolutionary graph , @xmath96 , in this setting consists of _ two _ graphs : @xmath39 and @xmath34 . we say that @xmath96 is regular if both @xmath39 and @xmath34 are regular . for vertex - transitivity of @xmath96 ( resp . symmetry of @xmath96 ) , we require slightly more than each @xmath39 and @xmath34 being vertex - transitive ( resp . symmetric ) ; we require them to be _ simultaneously _ vertex - transitive ( resp . symmetric ) . first of all , we need to define what an _ automorphism _ of @xmath96 is . for @xmath89 , let @xmath97 be the graph defined by @xmath98 for each @xmath26 and @xmath28 . using this action , we define an automorphism of an evolutionary graph as follows :    an _ automorphism _ of @xmath99 is an action , @xmath89 , such that @xmath100 and @xmath101 . we denote by @xmath102 the set of automorphisms of @xmath96 . we now have the definitions of vertex - transitive and symmetric evolutionary graphs :    @xmath99 is _ vertex - transitive _ if for each @xmath26 and @xmath28 , there exists @xmath103 such that @xmath104 . @xmath99 is _ symmetric _ if @xmath105 and @xmath39 is a symmetric graph .    finally , using the notion of an automorphism of @xmath96 , we have our main result :    [ thm : mainsymmetrictheorem ] consider an evolutionary matrix game on a graph , @xmath99 , with symmetric payoffs and homogeneous strategy mutations . if @xmath103 , then the states with a single mutant at vertex @xmath26 and @xmath106 , respectively , in an otherwise - monomorphic population , are evolutionarily equivalent . that is , in the notation of definition [ def : homogeneousevolutionaryprocess ] , the states @xmath24 and @xmath107 are evolutionarily equivalent for each @xmath23 . the proof of theorem [ thm : mainsymmetrictheorem ] may be found in appendix b. the proof relies on the observation that the hypotheses of the theorem imply that any two states consisting of a single @xmath4-player in a population of @xmath5-players can be obtained from one another by relabeling the players . thus , in light of the argument in remark [ axiomremark ] , relabeling the players induces an automorphism on the markov chain defined by the evolutionary game . since any relabeling of the players leaves the monomorphic states fixed , there is an evolutionary equivalence between any two such states in the sense of definition [ def : evolutionaryequivalence ] . note that this theorem makes no restrictions on the selection strength or the update rule . [ cor : mainsymmetriccorollary ] an evolutionary game on a vertex - transitive graph with symmetric payoffs and homogeneous strategy mutations is itself homogeneous .    by theorem [ thm : mainsymmetrictheorem ] , two mutants appearing on a graph might define evolutionarily equivalent states even if the graph is not vertex - transitive . for example , the tietze graph ( see * ? ? ? * ) , like the frucht graph , has @xmath37 vertices and is regular but not vertex - transitive . however , unlike the frucht graph , the tietze graph has some nontrivial automorphisms . by theorem [ thm : mainsymmetrictheorem ] , any two vertices in the tietze graph that are related by an automorphism have the property that the two corresponding single - mutant states are indistinguishable . an example of two evolutionarily equivalent states on this graph is given in fig . [ fig : tietze ] . in appendix c , for the snowdrift game with death - birth updating , we give the fixation probabilities and absorption times for all configurations of a single cooperator among defectors , which further illustrates the effects of graph symmetries on an evolutionary process . [ remark : transitivesymmetric ] for a given population size , @xmath2 , and network degree , @xmath83 , there may be many vertex - transitive graphs of size @xmath2 with degree @xmath83 . for each such graph , the fixation probability of a randomly occuring mutant is independent of where on the graph it occurs by theorem [ thm : mainsymmetrictheorem ] . however , this fixation probability depends on more than just @xmath2 and @xmath83 ; it also depends on the configuration of the network . for example , fig . [ fig : transitivesymmetric ] gives two vertex - transitive graphs of size @xmath108 and degree @xmath109 . as an illustration , consider the snowdrift game ( with payoff matrix ( [ snowdriftgame ] ) ) on these graphs with birth - death updating . if the selection intensity is @xmath110 , then the fixation probability of a single cooperator in a population of defectors is @xmath111 in ( a ) and @xmath112 in ( b ) ( both rounded to four significant figures ) . these two fixation probabilities differ for all but finitely many @xmath45 by proposition [ prop : functionofbeta ] .    until this point , our focus has been on states consisting of just a single mutant in an otherwise - monomorphic population . one could also inquire as to when any two states consisting of two ( or three , four , etc . ) mutants are evolutionarily equivalent . it turns out that that the answer to this question is simple : in general , the population must be well - mixed in order for any two @xmath113-mutant states to be evolutionarily equivalent if @xmath114 . the proof that this equivalence holds in well - mixed populations follows from the argument given to establish theorem [ thm : mainsymmetrictheorem ] ( see appendix b ) . on the other hand , if the population is not well - mixed , then one can find a pair of states with the first state consisting of two mutants on neighboring vertices and the second state consisting of two mutants on non - neighboring vertices . in general , the mutant type will have different fixation probabilities in these two states . for example , in the snowdrift game on the graph of fig . [ fig : transitivesymmetric](b ) , consider the two states , @xmath16 and @xmath17 , where @xmath16 consists of cooperators on vertices @xmath1 and @xmath115 only and @xmath17 consists of cooperators on vertices @xmath1 and @xmath38 only . if @xmath110 , then the fixation probability of cooperators under death - birth updating when starting at @xmath16 ( resp . @xmath17 ) is @xmath116 ( resp . @xmath117 ) . therefore , despite the arc - transitivity of this graph , it is not true that any two states consisting of exactly two mutants are evolutionarily equivalent . only in well - mixed populations are we guaranteed that any two such states are equivalent . one particular form of payoff asymmetry appearing in evolutionary game theory is _ ecological asymmetry _ @xcite . ecological asymmetry can arise as a result of an uneven distribution of resources . for example , in the donation game , a cooperator at location @xmath26 might provide a benefit to his or her opponent based on some resource derived from the environment . both this resource and the cost of donating it could depend on @xmath26 , which means that different players have different payoff matrices . these payoff matrices depend on both the location of the focal player and the locations of the opponents . thus , payoffs for a player at location @xmath26 against an opponent at location @xmath28 in an @xmath40-strategy  bimatrix " game @xcite are given by the asymmetric payoff matrix    @xmath118    similar to eq . ( [ eq : symmetricmatrixpayoff ] ) , the total payoff to player @xmath26 for strategy profile @xmath43 is    @xmath119    we saw in [ subsec : frequencydependentgames ] an example of a heterogeneous evolutionary game in a well - mixed population with symmetric payoffs . rather than looking at a symmetric game with heterogeneous strategy mutations , we now look at an asymmetric game with homogeneous strategy mutations . consider the ecologically asymmetric donation game on the graph of fig . [ fig : wellmixedthree ] ( both @xmath39 and @xmath34 ) with a death - birth update rule . in this asymmetric donation game , a cooperator at location @xmath26 donates @xmath120 at a cost of @xmath121 ; defectors donate nothing and incur no costs . if @xmath110 , @xmath122 , @xmath123 , and @xmath124 , then the fixation probability of a single cooperator at location @xmath1 ( see fig . [ fig : wellmixedthree](a ) ) is @xmath125 , while the fixation probability of a single cooperator at location @xmath115 ( see fig . [ fig : wellmixedthree](b ) ) is @xmath126 ( both rounded to four significant figures ) . therefore , even in a well - mixed population with homogeneous strategy mutations ( none , in this case ) , asymmetric payoffs can prevent an evolutionary game from being homogeneous . asymmetric matrix games in large populations reduce to symmetric games if selection is weak @xcite . in the limit of weak selection , @xcite establish a selection condition for asymmetric matrix games in finite graph - structured populations that extends the condition ( for symmetric games ) of @xcite :    [ asymmetricselectioncondition ] there exists a set of _ structure coefficients _ , @xmath127 , independent of payoffs , such that weak selection favors strategy @xmath128 if and only if    @xmath129    where @xmath130 , @xmath131 , @xmath132 , and @xmath133 . strictly speaking , theorem [ asymmetricselectioncondition ] is established for @xmath39 and @xmath34 undirected , unweighted , and satisfying @xmath105 . however , the proof of theorem [ asymmetricselectioncondition ] extends immediately to the case with @xmath39 and @xmath34 directed , weighted , and possibly distinct , so we make no restrictive assumptions on @xmath39 and @xmath34 in the statement of this theorem here . in the simpler case @xmath134 , condition ( [ asymmetricnetwork ] ) takes the form    @xmath135    for some collection @xmath136 . for the death - birth process with @xmath39 and @xmath34 the graph of fig . [ fig : transitivesymmetric](a ) , we calculate _ exact _ values for all of these structure coefficients ( see appendix c ) . in particular , we find that @xmath137 and @xmath138 , so vertex - transitivity does not guarantee that the structure coefficients are independent of @xmath26 and @xmath28 . for the same process on the graph in fig . [ fig : transitivesymmetric](b ) , we find that @xmath139 for each @xmath26 and @xmath28 , so these coefficients do not depend on @xmath26 and @xmath28 . ( in general , even for well - mixed populations , @xmath140 and @xmath141 need not be the same ; for the same process studied here but on the graph of fig . [ fig : wellmixedthree ] , @xmath142 and @xmath143 for each @xmath26 and @xmath28 . ) this lack of dependence on @xmath26 and @xmath28 is due to the fact that the graph of fig . [ fig : transitivesymmetric](b ) is _ symmetric _ , and it turns out to be a special case of a more general result :    [ thm : mainasymmetrictheorem ] suppose that an asymmetric matrix game with homogeneous strategy mutations is played on an evolutionary graph , @xmath99 . for each @xmath103 , @xmath144 , and @xmath145 ,    @xmath146    the proof of theorem [ thm : mainasymmetrictheorem ] may be found in appendix b. the following corollary is an immediate consequence of theorem [ thm : mainasymmetrictheorem ] :    if @xmath105 and @xmath39 is a symmetric graph ( i.e. @xmath96 is a symmetric evolutionary graph ) , then the structure coefficients are independent of @xmath26 and @xmath28 . since symmetric graphs are also regular , we have :    [ cor : spatialaverage ] if @xmath105 and @xmath39 is a symmetric graph ( i.e. @xmath96 is a symmetric evolutionary graph ) , then strategy @xmath147 is favored in the limit of weak selection if and only if @xmath148 where @xmath149 is the spatial average of the matrices @xmath150 , i.e.    @xmath151    where @xmath83 is the degree of the graph , @xmath96 . ( [ tarnitaselectioncondition ] ) is just the selection condition of @xcite for symmetric matrix games . it follows from corollary [ cor : spatialaverage ] that asymmetric matrix games on arc - transitive ( symmetric ) graphs can be reduced to symmetric games in the limit of weak selection . evolutionary games in finite populations may be split into two classes : those with absorbing states (  absorbing processes " ) and those without absorbing states . in absorbing processes , the notion of fixation probability has played a crucial role in quantifying evolutionary outcomes , but fixation probabilities are far from the only measure of evolutionary success . much of the literature on evolutionary games with absorbing states has neglected other metrics such as the time to absorption or the time to fixation conditioned on fixation occurring (  conditional fixation time " ) . this bias toward fixation probabilities has resulted in certain evolutionary processes appearing more symmetric than they actually are . we have illustrated this phenomenon using the frequency - independent moran process on graphs : the isothermal theorem guarantees that , on regular graphs , a single mutant can not distinguish between initial locations in the graph if the only metric under consideration is the probability of fixation . however , certain initial placements of the mutant may result in faster absorption times than others if the graph is regular but not vertex - transitive , and the frucht graph exemplifies this claim . the same phenomenon also holds for conditional fixation times . the frucht graph , which is a regular structure with no nontrivial symmetries ( there are no two vertices from which the graph looks " the same ) , also allowed us to show that the isothermal theorem of @xcite does not extend to frequency - dependent evolutionary games . that is , on regular graphs , the probability of fixation of a single mutant may depend on the initial location of the mutant if fitness is frequency - dependent . this claim was illustrated via a death - birth process on the frucht graph , in which the underlying evolutionary game was a snowdrift game . for @xmath53 ( strong selection ) , the fixation probability of a cooperator at vertex @xmath152 was nearly @xmath153 larger than the fixation probability of a cooperator at vertex @xmath154 . moreover , we showed that if the fixation probabilities of two initial configurations differ for a single value of @xmath54 , then they are the same for at most finitely many values of @xmath54 . in particular , these fixation probabilities differ for almost every selection strength , so our observation for @xmath53 was not an anomaly . similar phenomena are observed for frequency - dependent birth - death processes on the frucht graph , for example , and even for frequency - dependent games with the  equal gains from switching " property , such as the donation game . theorem [ thm : mainsymmetrictheorem ] is an analogue of the isothermal theorem that applies to a broader class of games and update rules . the isothermal theorem is remarkable since regularity of the population structure implies that the fixation probabilities are not only independent of the initial location of the mutant , they are the _ same _ as those of the classical moran process . our treatment of homogeneous evolutionary processes is focused on when different single - mutant states are equivalent , not when they are equivalent to the corresponding states in the classical moran process . even if the fixation probability of a single mutant does not depend on the mutant s location , other factors ( such as birth and death rates ) may affect whether or not this fixation probability is the same as the one in a well - mixed population @xcite . remark [ remark : transitivesymmetric ] , which compares the fixation probabilities for the snowdrift game on two different vertex - transitive graphs of the same size and degree , shows that the fixation probability of a single mutant  even if independent of the mutant s location  can depend on the configuration of the graph . in light of these results , the symmetry phenomena for the moran process guaranteed by the isothermal theorem do not generalize and should be thought of as properties of the frequency - independent moran process and not of evolutionary processes in general . theorem [ thm : mainsymmetrictheorem ] , and indeed most of our discussion of homogeneity , focused on symmetries of states consisting of just a _ single _ mutant . in many cases , mutation rates are sufficiently small that a mutant type , when it appears , will either fixate or go extinct before another mutation occurs @xcite . thus , with small mutation rates , one need not consider symmetries of states consisting of more than one mutant . however , if mutation rates are larger , then these multi - mutant states become relevant . our definition of evolutionary equivalence ( definition [ def : evolutionaryequivalence ] ) applies to these states , but , as expected , the symmetry conditions on the population structure guaranteeing any two multi - mutant states are equivalent are much stronger . in fact , as we argued in [ subsubsec : homogeneousevolutionarygames ] , the population must in general be well - mixed even for any pair of states consisting of _ two _ mutants to be evolutionarily equivalent . consequently , our focus on single - mutant states allowed us to simultaneously treat biologically relevant configurations ( assuming mutation rates are small ) and obtain non - trivial conditions guaranteeing homogeneity of an evolutionary process .    the counterexamples presented here could be defined on sufficiently small population structures , and thus all calculations ( fixation probabilities , structure coefficients , etc . ) are exact . however , these quantities need not always be explicitly calculated in order to prove useful : in our study of asymmetric games , we concluded that an asymmetric game on an arc - transitive ( symmetric ) graph can be reduced to a symmetric game in the limit of weak selection . ( the graph of fig . [ fig : transitivesymmetric](a ) demonstrates that vertex - transitivity alone does not guarantee that an asymmetric game can be reduced to a symmetric game in this way . ) this result was obtained by examining the qualitative nature of the structure coefficients in the selection condition ( [ asymmetricnetwork ] ) , but it did not require explicit calculations of these coefficients . therefore , despite the difficulty in actually calculating these coefficients , they can still be used to glean qualitative insight into the dynamics of evolutionary games .    on large random regular graphs , the dynamics of an asymmetric matrix game are equivalent to those of a certain symmetric game obtained as a  spatial average " of the individual asymmetric games @xcite . corollary [ cor : spatialaverage ] is highly reminiscent of this type of reduction to a symmetric game . for large populations , this result is obtained by observing that large random regular graphs approximate a bethe lattice @xcite and then using the pair approximation method @xcite to describe the dynamics . the pair approximation method is _ exact _ for a bethe lattice @xcite , so , from this perspective , corollary [ cor : spatialaverage ] is not that surprising since bethe lattices are arc - transitive . of course , a bethe lattice has infinitely many vertices , and corollary [ cor : spatialaverage ] is a finite - population analogue of this result . the term  homogeneous " is used in the literature to refer to several different kinds of population structures . this term has been used to describe well - mixed populations @xcite . for graph - structured populations ,  homogeneous graph " sometimes refers to vertex - transitive graphs @xcite . in algebraic graph theory , however , the term  homogeneous graph " implies a much higher degree of symmetry than does than vertex - transitivity ( see * ? ? ? homogeneous " has also been used to describe graphs in which each vertex has the same number of neighbors , i.e. regular graphs @xcite . in between regular and vertex - transitive graphs ,  homogeneous graph " has also referred to large , random regular graphs @xcite . as we noted , large , random regular graphs approximate bethe lattices ( which are infinite , arc - transitive graphs ) , but these approximations need not themselves be even vertex - transitive .    in many of the various uses of the term  homogeneous , " a common aim is to study the fixation probability of a randomly placed mutant . our definition of _ homogeneous evolutionary game _ formally captures what it means for two single - mutant states to be equivalent , and our explorations of the frucht graph ( in conjunction with theorem [ thm : mainsymmetrictheorem ] ) show that vertex - transitivity , and _ not _ regularity , is what the term  homogeneous " in graph - structured populations should indicate . we also demonstrated the effects of payoffs and strategy mutations on the behavior of these single - mutant states and concluded that the term homogeneous should apply to the entire process rather than to just the population structure . the homogeneity ( theorem [ thm : mainsymmetrictheorem ] ) and symmetry ( theorem [ thm : mainasymmetrictheorem ] ) results given here do not depend on the update rule , in contrast with results such as the symmetry of conditional fixation times in the moran process of @xcite or the isothermal theorem of @xcite . we now know that games on regular graphs are _ not _ homogeneous , and we know precisely under which conditions the  fixation probability of a randomly placed mutant " is well - 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011 - 0430 - 8 . j.  zukewich , v.  kurella , m.  doebeli , and c.  hauert . consolidating birth - death and death - birth processes in structured populations . _ plos one _ , 80 ( 1):0 e54639 , jan 2013 . doi : 10.1371/journal.pone.0054639 . using a method inspired by a technique of @xcite , we derive explicit expressions ( in terms of the transition matrix ) for fixation probabilities and absorption times . subsequently , we prove a simple lemma that says that markov chain symmetries preserve the set of a chain s stationary distributions . suppose that @xmath156 is a discrete - time markov chain on a finite state space , @xmath8 , that has exactly @xmath157 ( @xmath158 ) absorbing states , @xmath159 . moreover , suppose that the non - absorbing states are transient ( see * ? ? ? the transition matrix for this chain , @xmath9 , may be written as    @xmath160    where @xmath161 is the @xmath162 identity matrix and @xmath163 is the matrix of zeros ( in this case , its dimension is @xmath164 , where @xmath165 is the number of states in @xmath8 ) . this chain will eventually end up in one of the @xmath157 absorbing states , and we denote by @xmath57 the probability that state @xmath55 is reached when the chain starts off in states @xmath166 . let @xmath167 be the @xmath168 matrix of fixation probabilities , i.e. @xmath169 for each @xmath16 and @xmath26 . this matrix satisfies    @xmath170    which is just the matrix form of the recurrence relation satisfied by fixation probabilities ( obtained from a  first step " analysis of the markov chain ) . consider the matrix , @xmath171 , defined by    @xmath172    since @xmath173 for @xmath174 , we see that    @xmath175    moreover , the matrix @xmath176 must have full rank since the non - absorbing states are transient ; that is ,    @xmath177    therefore , by cramer s rule ,    @xmath178    where the notation @xmath179 means the matrix obtained by replacing the column corresponding to state @xmath16 with the @xmath26th standard basis vector , @xmath180 . thus , eq . ( [ fixationprobabilityformula ] ) gives explicit formulas for the fixation probabilities . let @xmath181 be the @xmath165-vector indexed by @xmath8 whose entry @xmath58 is the expected time until the process fixates in one of the absorbing states when started in state @xmath166 . this vector satisfies @xmath182 for @xmath183 as well as the recurrence relation @xmath184 . therefore ,    @xmath185    so , by cramer s rule ,    @xmath186    we now turn to markov chains defined by evolutionary games . before proving proposition [ prop : functionofbeta ] , we make two assumptions :    1 .   the payoff - to - fitness mapping is of the form @xmath187 , @xmath188 , @xmath189 , or @xmath190 , where @xmath54 denotes the intensity of selection and @xmath191 denotes payoff . ( of course , fitness can be defined in one of the latter three ways only if the payoffs are such that @xmath192 . ) 2 . the update probabilities are rational functions of the fitness profile of the population .    assumptions ( i ) and ( ii ) are not at all restrictive in evolutionary game theory . any process in which selection occurs with probability proportional to fitness will satisfy this rationality condition , and indeed all of the standard evolutionary processes ( birth - death , death - birth , imitation , pairwise comparison , wright - fisher , etc . ) have this property . the four payoff - to - fitness mappings are standard as well .    with assumptions ( i ) and ( ii ) in mind , we have :    each of the equalities    @xmath59    holds for either ( i ) _ every _ @xmath45 or ( ii ) at most finitely many @xmath45 . thus , if one of these equalities fails to hold for even a single value of @xmath54 , then it fails to hold for all sufficiently small @xmath60 . suppose that @xmath13 and that @xmath55 and @xmath56 are absorbing states . by eq . ( [ fixationprobabilityformula ] ) ,    @xmath193    similarly , by eq . ( [ absorptiontimeformula ] ) ,    @xmath194    assuming ( i ) and ( ii ) , eqs . ( [ equalityconditionfixation ] ) and ( [ equalityconditionabsorption ] ) are equivalent to polynomial equations in either @xmath54 or @xmath195 . either way , since nonzero polynomial equations have at most finitely many solutions , we see that the equalities @xmath196 and @xmath197 each hold for either ( i ) every @xmath54 or ( ii ) finitely many values of @xmath54 . thus , if @xmath198 ( resp . @xmath199 ) for even a single selection intensity , then these fixation probabilities ( resp . absorption times ) differ for almost every selection intensity . in particular , they differ for all sufficiently small @xmath54 . here we recall some standard notions of symmetry for graphs . although we treat directed , weighted graphs in general , throughout the main text we give several examples of _ _ un__directed and _ _ un__weighted graphs , which are defined as follows :    a graph , @xmath34 , is _ undirected _ if @xmath200 for each @xmath26 and @xmath28 . [ def : unweightedgraph ] a graph , @xmath34 , is _ unweighted _ if @xmath201 for each @xmath26 and @xmath28 .    since our goal is to discuss symmetry in the context of evolutionary processes , we first describe several notions of symmetry for graphs . in a graph , @xmath34 , the _ indegree _ and _ outdegree _ of vertex @xmath26 are @xmath202 and @xmath203 , respectively . with these definitions in mind , we recall the definition of a _ regular _ graph :    [ def : regulargraph ] @xmath34 is regular if and only if there exists @xmath204 such that    @xmath205    for each @xmath26 . if @xmath34 is regular , then @xmath83 is called the _ degree _ of @xmath34 . let @xmath88 denote the symmetric group on @xmath2 letters ; that is , @xmath88 is the set of all bijections @xmath206 . each @xmath89 extends to a relabeling action on the set of directed , weighted graphs defined by @xmath207 . in other words , any relabeling of the set of vertices results in a corresponding relabeling of the graph . the _ automorphism group of _ @xmath34 , written @xmath208 , is the set of all @xmath89 such that @xmath101 . we now recall a condition slightly stronger than regularity known as vertex - transitivity :    [ def : vertextransitivegraph ] @xmath34 is _ vertex - transitive _ if for each @xmath26 and @xmath28 , there exists @xmath209 such that @xmath104 . informally , a graph is vertex - transitive if and only if it  looks the same " from every vertex . if a graph is vertex - transitive , then it is necessarily regular . the strongest form of symmetry for graphs that we consider here is the following :    [ def : symmetricgraph ] @xmath34 is _ symmetric _ ( or _ arc - transitive _ ) if for each @xmath210 with @xmath35 and @xmath211 with @xmath212 , there exists @xmath209 such that @xmath213 and @xmath214 . a graph is symmetric if it  looks the same " from any two directed edges . arc - transitivity is typically defined for unweighted graphs , i.e. graphs satisfying @xmath215 . for the more general class of weighted graphs , we require that @xmath88 act transitively on the set of edges of @xmath34 , where  edge " means a pair @xmath216 with @xmath35 . thus , all of the edges in a symmetric , weighted graph have the same weight : otherwise , if @xmath216 and @xmath217 are edges but @xmath218 , then there would exist no permutation , @xmath191 , sending @xmath26 to @xmath219 , @xmath28 to @xmath220 , and preserving the weights of the graph . therefore , since the weights of a symmetric graph take one of two values ( @xmath221 or else the only nonzero weight ) , such a graph is essentially unweighted . in  [ subsec : generalmarkov ] , we defined two states , @xmath16 and @xmath17 , to be _ evolutionarily equivalent _ if ( i ) there exists an automorphism of the markov chain , @xmath14 , such that @xmath15 , and ( ii ) this automorphism satisfies @xmath19 for each stationary distribution , @xmath18 , of the chain . condition ( i ) , which means that @xmath16 and @xmath17 are _ symmetric _ , alone is not quite strong enough to guarantee that @xmath16 and @xmath17 have the same long - run behavior . to give an example of a symmetry of states that is not an evolutionary equivalence , we consider the neutral moran process in a well - mixed population of size @xmath61 :    [ ex : swapstrategies ] in a well - mixed population of size @xmath61 , consider the ( frequency - independent ) moran process with two types of players : a mutant type and a wild type . suppose that the mutant type is neutral with respect to the mutant ; that is , the fitness of the mutant relative to the wild type is @xmath1 . since the population is well - mixed , the state of the population is given by the number of mutants it contains , @xmath222 . consider the map @xmath11 defined by @xmath223 . states @xmath221 and @xmath38 are absorbing , and , for @xmath224 , the transition probabilities of this process are as follows :    @xmath225    it follows at once that @xmath226 preserves these transition probabilities , so @xmath226 is an automorphism of the markov chain . let @xmath227 be the probability that mutants fixate given an initial abundance of @xmath26 mutants . the states @xmath1 and @xmath115 are symmetric since @xmath228 , but it is _ not _ true that @xmath229 since @xmath230 and @xmath231 . the reason for this difference in fixation probabilities is that states @xmath1 and @xmath115 , although symmetric , are not evolutionary equivalent since @xmath226 swaps the two absorbing states of the process . in contrast to example [ ex : swapstrategies ] , processes with unique stationary distributions have the property that every symmetry of the markov chain is an evolutionary equivalence ( proposition [ prop : ergodicsymmetry ] ) . the following lemma establishes proposition [ prop : ergodicsymmetry ] :    [ lem : stationarydistribution ] if @xmath11 is a symmetry of a markov chain and @xmath18 is a stationary distribution of this chain , then @xmath232 is also a stationary distribution . in particular , if @xmath18 is unique , then @xmath19 . if @xmath9 is the transition matrix of this markov chain , then    @xmath233_{\mathfrak{s } } & = \sum_{\mathfrak{s}'\in\mathcal{s}}\phi\left(\mu\right)_{\mathfrak{s}'}\mathbf{t}_{\mathfrak{s}',\mathfrak{s } } \nonumber \\ & = \sum_{\mathfrak{s}'\in\mathcal{s}}\mu_{\phi\left(\mathfrak{s}'\right)}\mathbf{t}_{\phi\left(\mathfrak{s}'\right ) , \phi\left(\mathfrak{s}\right ) } \nonumber \\ & = \sum_{\mathfrak{s}'\in\mathcal{s}}\mu_{\mathfrak{s}'}\mathbf{t}_{\mathfrak{s}',\phi\left(\mathfrak{s}\right ) } \nonumber \\ & = \left[\mu^{t}\mathbf{t}\right]_{\phi\left(\mathfrak{s}\right ) } \nonumber \\ & = \mu_{\phi\left(\mathfrak{s}\right ) } \nonumber \\ & = \phi\left(\mu\right)_{\mathfrak{s } } , \end{aligned}\ ] ]    so @xmath234 , which completes the proof . we turn now to the proofs of our main results ( theorems [ thm : mainsymmetrictheorem ] and [ thm : mainasymmetrictheorem ] ) :    consider an evolutionary matrix game on a graph , @xmath99 , with symmetric payoffs and homogeneous strategy mutations . if @xmath103 , then the states with a single mutant at vertex @xmath26 and @xmath106 , respectively , in an otherwise - monomorphic population , are evolutionarily equivalent . that is , in the notation of definition [ def : homogeneousevolutionaryprocess ] , the states @xmath24 and @xmath107 are evolutionarily equivalent for each @xmath23 . the state space of the markov chain defined by this evolutionary game is @xmath25 , where @xmath21 is the strategy set and @xmath2 is the population size . for @xmath103 , let @xmath191 act on the state of the population by changing the strategy of player @xmath26 to that of player @xmath90 for each @xmath235 . in other words , @xmath191 sends @xmath85 to @xmath236 , which is defined by @xmath237 for each @xmath26 . therefore , for @xmath23 , we have    @xmath238    for each @xmath235 . since @xmath191 is an automorphism of the evolutionary graph , @xmath96 , we have @xmath100 and @xmath101 . moreover , @xmath191 preserves the strategy mutations since they are homogeneous . since the payoffs are symmetric , @xmath191 just rearranges the fitness profile of the population : the payoff of player @xmath26 becomes the payoff of player @xmath90 ( see eq . ( [ eq : symmetricmatrixpayoff ] ) ) , so the same is true of the fitness values . therefore , applying the map @xmath191 to @xmath25 is equivalent to applying the map on @xmath25 obtained by simply relabeling the players . since any such relabeling of the players results in an automorphism of the markov chain on @xmath25 that preserves the monomorphic absorbing states , it follows that @xmath24 and @xmath107 are evolutionarily equivalent . suppose that an asymmetric matrix game with homogeneous strategy mutations is played on an evolutionary graph , @xmath99 . for each @xmath103 , @xmath144 , and @xmath145 ,    @xmath146    let @xmath9 be the transition matrix for the markov chain defined by this process . since there are nonzero strategy mutations , this chain has a unique stationary distribution , @xmath18 . the matrix @xmath9 defines a directed , weighted graph on @xmath239 vertices that has an edge from vertex @xmath16 to vertex @xmath17 if and only if @xmath240 . if there is an edge from @xmath16 to @xmath17 , then the weight of this edge is simply @xmath241 . the ( outdegree ) laplacian matrix of this graph , @xmath242 , is defined by @xmath243 ( see * ? ? ? * ) . in terms of this laplacian matrix , @xcite show that for any vector , @xmath244 , the stationary distribution satisfies    @xmath245    for each state , @xmath16 , where @xmath246 denotes the matrix obtained from @xmath247 by replacing the column corresponding to state @xmath16 by @xmath244 . thus , if @xmath248 is the vector indexed by @xmath22 with @xmath249 being the frequency of strategy @xmath147 in state @xmath16 , then the _ average abundance _ of strategy @xmath147 is    @xmath250    since @xmath9 is a function of the payoffs , @xmath251 , we may write @xmath252 . ( @xmath253 is just an ordered tuple defined by @xmath254 for each @xmath30 , @xmath255 , @xmath26 , and @xmath28 . ) moreover , since the entries of @xmath9 are assumed to be smooth functions of @xmath253 ( see * ? ? ? * ) , @xmath256 is also a smooth function of @xmath253 by eq . ( [ eq : averageabundance ] ) and the definition of @xmath247 . we will show that for each @xmath30 and @xmath255 ,    @xmath257    for each @xmath26 and @xmath28 . the theorem will then follow from the derivations of @xmath258 , @xmath259 , and @xmath260 in @xcite since it is shown there that each @xmath261 is a function of the elements in the set    @xmath262    for @xmath30 , @xmath255 , @xmath26 , and @xmath28 fixed and @xmath263 , let @xmath264 be the function of the vector with @xmath265 at entry @xmath266 and @xmath221 in all other entries . symbolically , if @xmath267 is defined as @xmath1 if @xmath268 and @xmath221 otherwise and    @xmath269    then    @xmath270 let @xmath89 and suppose that @xmath103 ; that is , @xmath100 and @xmath101 . @xmath191 induces a map on the payoffs , @xmath253 , defined by @xmath271 . let @xmath272 denote the orbit of @xmath253 under this action , and consider the enlarged state space @xmath273 . using the markov chain on @xmath25 coming from the evolutionary process , we obtain a markov chain on @xmath274 via the transition matrix , @xmath275 , defined by    @xmath276    for @xmath277 and @xmath278 . ( we write @xmath279 to indicate the transition matrix as a function of the payoff values of the game . ) @xmath191 extends to a map on @xmath280 defined by @xmath281 . since @xmath191 preserves @xmath39 , @xmath34 , and the strategy mutations ( since they are homogeneous ) , it follows that the induced map @xmath282 is an automorphism of the markov chain on @xmath280 defined by @xmath275 . if @xmath283 is a stationary distribution for the chain @xmath275 , then , for each @xmath284 and @xmath285 ,    @xmath286    it then follows from the uniqueness of @xmath287 that there exists @xmath288 such that @xmath289 . if @xmath283 is such a stationary distribution , then , by lemma [ lem : stationarydistribution ] , @xmath290 for some other stationary distribution , @xmath291 , of the chain on @xmath280 . this equation implies that @xmath292 for each @xmath285 . therefore ,    @xmath293    for each @xmath85 and @xmath285 . consequently , since @xmath294 ,    @xmath295    as a result , we have    @xmath296 so eq . ( [ eq : partialsequation ] ) holds , which completes the proof . we now perform explicit calculations using eqs . ( [ fixationprobabilityformula ] ) and ( [ absorptiontimeformula ] ) to show that the isothermal theorem extends to neither absorption times nor frequency - dependent games . we also calculate the structure coefficients for the death - birth process on the graph of fig . [ fig : transitivesymmetric](a ) to show that corollary [ cor : spatialaverage ] does not necessarily hold for graphs that are vertex - transitive but not symmetric . consider the moran process on the frucht graph ( fig . [ fig : fruchtgraph ] ) , and suppose that the mutant type has fitness @xmath0 relative to the wild type . by the isothermal theorem of @xcite , the fixation probability of a fixed number of mutants is independent of the configuration of those mutants on the graph . for @xmath297 , the absorption times ( of configurations of a single mutant in a wild - type population ) are listed in table [ table : absorption ] . the fixation probability of a single mutant is @xmath298 for every vertex . thus , unlike fixation probabilities , absorption times depend on the initial location of the mutant . ( some of the absorption times are similar in this case , but no two are the same . ) .the absorption times of the @xmath37 initial configurations of a single mutant in a wild - type population for the moran process on the frucht graph . the fitness of the mutant relative to the wild type is @xmath297.[table : absorption ] [ cols="^,^,^",options="header " , ]
in evolutionary game theory , an important measure of a mutant trait ( strategy ) is its ability to invade and take over an otherwise - monomorphic population . 
 typically , one quantifies the success of a mutant strategy via the probability that a randomly occurring mutant will fixate in the population . however , 
 in a structured population , this fixation probability may depend on where the mutant arises . 
 moreover , the fixation probability is just one quantity by which one can measure the success of a mutant ; fixation _ time _ , for instance , is another . 
 we define a notion of homogeneity for evolutionary games that captures what it means for two single - mutant states , i.e. two configurations of a single mutant in an otherwise - monomorphic population , to be  evolutionarily equivalent " in the sense that _ all _ measures of evolutionary success are the same for both configurations . using asymmetric games , we argue that the term  homogeneous " should apply to the evolutionary process as a whole rather than to just the population structure . for evolutionary matrix games in graph - structured populations , we give precise conditions under which the resulting process is homogeneous . 
 finally , we show that asymmetric matrix games can be reduced to symmetric games if the population structure possesses a sufficient degree of symmetry .