Self-loops in evolutionary graph theory: Friends or foes?

Evolutionary dynamics in spatially structured populations has been studied for a long time. More recently, the focus has been to construct structures that amplify selection by fixing beneficial mutations with higher probability than the well-mixed population and lower probability of fixation for deleterious mutations. It has been shown that for a structure to substantially amplify selection, self-loops are necessary when mutants appear predominately in nodes that change often. As a result, for low mutation rates, self-looped amplifiers attain higher steady-state average fitness in the mutation-selection balance than well-mixed populations. But what happens when the mutation rate increases such that fixation probabilities alone no longer describe the dynamics? We show that self-loops effects are detrimental outside the low mutation rate regime. In the intermediate and high mutation rate regime, amplifiers of selection attain lower steady-state average fitness than the complete graph and suppressors of selection. We also provide an estimate of the mutation rate beyond which the mutation-selection dynamics on a graph deviates from the weak mutation rate approximation. It involves computing average fixation time scaling with respect to the population sizes for several graphs.

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Competing Interests
On behalf of all authors, disclose any competing interests that could be perceived to bias this work.This statement will be typeset if the manuscript is accepted for publication.Selection operates during the birth event, and it is represented by the capital letter, B. Death occurs randomly with uniform probability, and it is represented by the small letter, d.The first letter of a shorthand represents a global event where every individual of the population participates.The second letter of the shorthand represents a local event where it is only the individuals neighboring to the individual selected in the first event participate.More details on the types of evolutionary update rules in spatially structured populations can be found in [21].
To study mutation-selection dynamics on graphs, we use a modified version of Moran Bd updating where mutations appear with probability µ, see Fig. 1.One Moran Bd with mutation update step can be described as follows: 1. Birth: First, an individual at node i is selected with probability proportional to its fitness, fi j fj , to reproduce.

Mutation:
The offspring is either identical to the parent individual with probability 1 − µ or it is a mutant with probability µ.If the offspring turn out to be a mutant, its fitness f is sampled from the mutant fitness distribution, ρ(f , f ) with f being the parent's fitness.Birth-death (Bd) updating with mutation.Here, an example of the single time step of the Moran Bd with mutation is shown.First an individual is selected with probability proportional to its fitness to give birth to an offspring.The offspring either resembles the parent with probability 1 − µ, or mutates with probability µ.In case the mutation takes place, the offspring fitness f is then sampled from the distribution ρ(f , f ) with f being the parent's fitness.In the figure, we have shown the case when mutation takes place.The mutant offspring will then replaces one of the individuals neighboring the parent individual, or the parent individual itself via the self-loop.The choice is made at random with probability proportional to the outgoing weight from the parent node.Here, we have shown the case when the parent individual is replaced by the offspring via the self-loop.The stronger the self-loop, more likely it is for the parent to be replaced by its offspring.
At the level of graphs where each node is occupied by an individual, self-loops were introduced as mathematical objects [13].But they make clear sense at the level where each node of a graph is occupied by a population, i.e., here a graph is a population of populations [22], the weak migration rate regime dynamics can be understood as the 3 Amplification in the low mutation rate regime In Ref. [16], the Moran Bd mutation-selection dynamics was studied in the low mutation rate regime.In this regime, a newly appeared mutant either reaches fixation or goes extinct before the next mutant appears in the population.For low mutation rates, the population is effectively monomorphic throughout the mutation-selection dynamics and thus, the dynamics can be modelled as a random walk problem through fitness space where a steady-state is attained at long times.The steady-state for a graph G subjected to a low mutation rate mutation-selection dynamics with mutant's fitness f drawn from a continuous distribution ρ(f , f ) can be computed as Here, φ T G (f , f ) is the fixation probability of a mutant with fitness f , in a background of wild-type population with fitness f under temperature mutant initialisation scheme T .The fixation probabilities entering the steady-state expression 1 are temperature initialised, because when a new mutant appears in a homogenous population, according to the Moran Bd updating stated in Sec. 2, it is more likely to appear on the high temperature nodes.
The fixation probability of a mutant with fitness f on the complete graph with background fitness f is given by [10] Using the above expression for the fixation probability and the Eq. 1, we obtain the average steady-state fitness for the complete graph with uniform mutant fitness distribution, As expected, amplifiers of selection attain a higher steady-state average fitness than the well-mixed population.On the other hand, suppressors of selection attain lower steady-state fitness than the well-mixed population, see Ref. [16] for a formal proof.
However, a suppressor of fixation, a structure that has lower fixation probabilities than the complete graph regardless of the mutant fitness values, attains higher average fitness in the mutation-selection balance than the complete graph.This happens because of its ability to reject mutants more efficiently than the complete graph, compensating for its poor ability to fix beneficial mutants.These structures can also attain higher fitness than amplifiers of selection in the steady-state.Therefore, amplifiers of selection are not the only structures that adapt better than the well-mixed populations in the long-term evolutionary dynamics.criterion where a mutant appearing in the population should either reach fixation or go extinct before the next mutation appears.But with large fixation times, a new mutant can appear while the previous mutation is still under way towards fixation or extinction, and thus leads to effects like clonal interference [29,30].
Inside the low mutation rate regime, the steady-state average fitness of the population remains unchanged on decreasing the mutation rate further as the steady-state is independent of the mutation rate in this regime.However, the average steady-state fitnesses of structures decrease as the mutation rate is increased beyond the low mutation rate regime.Therefore outside the weak mutation rate regime, it not clear how amplifiers of selection, suppressors of selection, suppressors of fixation, and the well-mixed population are ordered in terms of their average steady-state fitness.To analyze this, we simulate the Moran Birth-death with mutation process for the self-looped star graph (weighted), the complete graph, the cycle graph, the star graph, and the directed line with self-loops.These graphs are shown in Fig. 2 B. Notice that without self-loops, nodes of the directed graphs that have no incoming links are frozen during the mutation-selection dynamics, their states remain the same throughout the dynamics.Therefore, we focus instead on a structure where self-loops are added to all the nodes of the directed line to facilitate their participation in the evolutionary dynamics.
The self-looped star graph is a piecewise amplifier of selection [13] for finite population size.Only in the limit N → ∞, it is a true amplifier of selection.The complete graph, and the cycle graph are isothermal graphs [1].Under temperature initialisation, for finite N , the star graph is suppressor of fixation [13,16].The directed line with self-loops is a suppressor of selection [10].From Fig. 2 A, we find that in the low mutation rate regime, the steady-state average fitness is highest for the self-looped star graph and the star graph, slightly lower for the complete and the cycle graph, and much lower for the self-looped directed line.
In Ref. [26], it has been shown that the temperature initialised star graph has a lower effective rate of evolution compared to the complete graph.However from Fig. 2 A, we see that the star graph attains higher steady-state average fitness than the complete graph.Therefore, a structure that speeds up evolution does not necessarily lead to higher fitness in the long-term evolutionary dynamics.Similarly, a structure that slows down evolution does not necessarily lead to lower fitness in the long-term evolutionary dynamics.Although at low mutation rates, the self-looped star graph outperforms all other graphs by attaining the highest steady state fitness, outside the low mutation rate regime, it performs poorly.On increasing the mutation rates, the star with self-loops not only attains lower steady-state fitness than the complete graph, but also the directed line with self-loops, a suppressor of selection.The main reason for this poor adaptation of the self-looped star graph outside the low mutation rate regime are self-loops.We explore this in detail in the following section.
We conclude this section by providing an estimate for the threshold mutation rate µ th beyond which the dynamics is considered to be outside the low mutation rate regime.It is given by where τ T 1 (r) and τ T 1 (r) are the temperature initialised average fixation and extinction time, respectively, of a mutant with fitness r relative to the wild-type.Eq. 4 follows from the criterion for the dynamics to be in the low mutation rate regime.Recall that the criterion for an evolutionary dynamics to be in the low mutation rate regime is that the time between any two successive mutations should be larger than the time to fixation or extinction (whatever is higher for a given pair of mutant and wild-type March 22, 2023 5/31 Mutation rate threshold, µ th .(A) The steady-state average fitnesses obtained using the Moran Birth-death mutation-selection dynamics simulations for the self-looped (weighted) star graph, an amplifier of selection, the star graph, a suppressor of fixation, the self-looped directed line, a suppressor of selection, the cycle graph, an isothermal graph, and the complete graph are shown via circles as a function of mutation rates.(B) We mostly work with these five graphs through out the manuscript.Solid horizontal lines in panel A represent steady-state average fitnesses for different graphs obtained under the low mutation rate approximation, Eq. 1.The arrows mark the mutation rates beyond which the low mutation rate approximation is violated for respective graphs.The graphs with higher average fixation time is expected to deviate earlier, see Eq. 4. (C) The average fixation time scaling with N is shown for different graphs.Solid lines are the analytical results whereas circles represent Moran Bd simulations.For larger N , it gets computationally expensive to work with microscopic Moran Bd simulations, in such cases we use a Gillespie algorithm, shown via triangles.For details on the Gillespie algorithm, refer to App. 7.2.3.(Parameters: (A) population size, N = 10, uniform mutant fitness distribution, i.e., ρ(f , f ) = neutrality was discussed in Refs.[35].For the case of self-looped directed line, we found that the average fixation time decreases as the mutant relative fitness is increased, whereas, the average extinction time increases with increasing mutant fitness, see Fig. 8.However, for a given fitness domain, it is the average fixation time corresponding to the lowest possible mutant's relative fitness that decides the µ th for the self-looped directed line graph.The star with self-loops has the lowest mutation threshold, as it has the highest fixation time.At neutrality, for large N the average fixation time for the self-looped star graph scales as N 5 , whereas for the star graph it scales as N 3 .The average fixation time scaling for the complete graph is N 2 , and the average fixation time scaling for the cycle graph is N 3 .For the self-looped directed line graph, the average fixation time scaling deciding µ th is, N 2 (1 + 1/r).The scalings for µ th for the above-mentioned structures are simply the inverse of the average fixation time scalings mentioned in Tab.D of the Fig. 2. The scaling relations are derived in App.7.2.

Self-loops and high mutation rate regime
Under the Moran Bd update scheme, an offspring always replaces one of the parent's neighbours -unless the parent node is self-looped.For an individual occupying a self-looped node, the offspring can replace the individual with a finite probability.Thus, self-loops effectively decrease the fitness of the parent individual, as the parent cannot spread its offspring freely into the population.The extent of this effect on the parent's fitness depends on the weight of the self-loops.This suggests that the fitness of a highly advantageous strain can be decreased by placing it on a self-looped node with negligible outward flowing weight to the neighbouring nodes [20].Under Bd updating, the fixation probability of a mutant on a structured population with the weight matrix, w, decreases as the diagonal weights of the matrix are increased [36].
For update schemes like bD and dB, and a given structure with the weight matrix w, it is necessary to have self-loops (w ii > 0) for all i, in order to have a fixation probability of mutants on that structure that is equivalent to a birth death process (of any type) on the self-looped complete graph with every link having equal weight [36].
Self-loops also fix some issues for the bD and dB dynamics that seem to make them unattractive from a modelling perspective [37]: One problem with the bD updating is that a mutant with fitness tending to zero can have a finite fixation probability.On the other hand, for the dB updating, an infinitely fit mutant can have a fixation probability smaller than one.Self-loops fix these issues.
In order for a structure to be a strong amplifier (a spatial structure where the fixation of a beneficial mutant is guaranteed), self-loops have been proven to be necessary [20], both under the uniform and the temperature initialisation.Though the concept of strong amplifiers is defined for infinite N , the self-loops also play a quintessential role in generating amplifiers of finite N [20].Intuitively, for a structure to be an amplifier of selection, it should have a sufficient number of cold temperature nodes so that the mutants are less likely to get replaced by wild-type individuals [28] and thus, a mutant type can persist in the population for longer time and spread its offspring into the population.This is where self-loops come into play, they help in creating more of these cold nodes and hence, amplifying selection.Consequently, self-loops contribute substantially in attaining higher fitness in the mutation-selection balance [16].
However as seen in Fig. 2 A, the steady-state average fitness of the self-looped star, an amplifier of selection, decreases fitness as the mutation rate is increased beyond the mutation threshold.Outside the low mutation rate regime, clonal interference starts to play an important role in the evolutionary dynamics.Therefore, to systematically investigate the effects of self-loops on evolutionary dynamics, we need to analyze the dynamics on structured populations for higher mutation rates.While this can be March 22, 2023 7/31 studied by simulations, it is challenging to obtain analytical insights for arbitrary mutation rates µ.Thus, in addition to simulations we study another -biologically not relevant -extreme of the high mutation rate limit, i.e., µ → 1.While this seems to be an irrelevant limit, its analysis reveals some crucial properties of evolutionary dynamics that are already relevant for much lower mutation rates.

Sampling fitness from the uniform distribution
In the limit µ → 1, every time a parent reproduces, the offspring is a mutant.We start with a uniform mutant fitness distribution, ρ(f , f ) =

Reference graph-complete graph with self-loops
When studying evolutionary dynamics on structured populations, the results are always compared with the dynamics on a reference graph.The standard choice in Evolutionary graph theory for the reference graph is the complete graph (without self-loops).For example, for the case of fixation probabilities and for the mutation-selection dynamics under mutation rates, the complete serves as the reference graph.However, for high mutation rates, instead of the complete graph, we choose the self-looped complete graph as a reference.This is because every node of the self-looped complete graph has an equal chance of being replaced by a mutant offspring during every birth event.This also implies that after a sufficiently long time, the states of the nodes would be completely uncorrelated in space and time.The coarse-grained evolutionary dynamics satisfies a master equation where each offspring's fitness f is chosen randomly from the with f being the parent's fitness.
The probability density function corresponding to population's state, P SC (f , t) changes as where the subscript SC stands for the self-looped complete graph, and, is the fitness state of the population of size N .By assuming detailed balance [38], i.e.
and the normalisation condition df P * (f ) = 1, we find the steady-state for the high mutation rate dynamics on the self-looped complete graph Here, average fitness of a node.This follows from the symmetry of the graph.Using the explicit form of the uniform mutational jump density function in Eq. 7, we obtain , which is independent of f .At very high mutation rates, the self-looped complete graph is totally blind to the fitness advantage/disadvantage of a mutant.Therefore, for the self-looped complete graph the average steady-state fitness with µ = 1, and the uniform mutational distribution is independent of the population size.
Similarly, the expression for the standard deviation of the steady-state fitness of the population is given by Var(f ) * where Because the fitness values are independently and identically distributed the population level variance equals where var(f ) * is the variance of a node.Simplifying the above equations, we obtain the standard deviation of the steady-state fitness as fmax−fmin √

12N
. With this, we are now ready to discuss the evolutionary dynamics on various self-looped graphs.We find a very good agreement for the steady-state statistics between the analytics and the simulations.The thick line represents the analytical average fitness, while the shaded grey area represents the standard deviation around the average.Symbols and error bars show simulations.In the steady-state, on average the self-looped complete graph attains the midpoint of the fitness domain, as the fitness dynamics for each individual node of the population becomes uncorrelated in the fitness space and time.The steady-state average fitness is also independent of the population size.The fluctuations in the steady-state however depends on the population size and decreases with the increase in population size as 1/ √ N (Parameters: f min = 0.1, f max = 10, number of independent realisations is equal to 2000, mutant fitness distribution, ρ(f , f ) = 1 fmax−fmin ).

Self-looped directed line beats the looping star
In this section, we study the high mutation rate dynamics, µ → 1, on the self-looped directed line and the weighted self-looped star graph.To recall, the self-looped directed March 22, 2023 9/31 line is a suppressor of selection [10,16], whereas, the (weighted) self-looped star graph is an amplifier of selection.In the low mutation rate dynamics, the self-looped weighted star attains higher steady-state fitness than the self-looped directed line.However, it is unclear what happens in the high mutation rate regime, which is far from a fixation-like dynamics.Simulating the Moran Bd dynamics with µ = 1 for these two graphs, we find that the weighted self-looped star attains lower steady-state fitness not only than the self-looped complete graph, but also the self-looped directed line, see Fig. 4. Nodewise analysis of the star graph with self-loops and the directed line with self-loops.Here, the average fitness trajectories for each node of the self-looped star graph (shown in panel A) and the self-looped directed line (shown in panel B are shown.Thick lines represent average fitness trajectories at the population level, whereas, thin lines represent average fitness trajectories for the nodes.The effect of self-loops on a node's fitness depends on the incoming and outgoing weight flowing out of that node.In panel A, self-loops have the least effect on the central node because of relatively higher incoming and outgoing weight.As a result, the central node attains higher average steady-state fitness than the leaf nodes.In panel B, the root node of the directed line has the lowest steady-state average fitness because of the absence of an incoming link to the root node.(Parameters: N = 10, µ = 1, f min = 0.1, f max = 10, number of independent realisations is equal to 2000, mutant fitness distribution, ρ(f , f ) = 1 fmax−fmin .For the directed line with self-loops, every outgoing link from a node (including the self-loop) has the same weight.For the self-looped star graphs, the weights of the links follows Eq. ( 22), such that λ = 1/(N − 1) and δ = 1/(N − 1) 2 .)For the case of (weighted) self-looped star graph, from the Fig. 4 A, all the leaf nodes attain the same steady-state fitness.This is expected due to symmetry reasons.The central node, node 0, stands out, and has the highest fitness.This is because the fitness decreases effects of the self-loop is minimised by the vast number of outgoing (incoming) links from (to) the central node.
A self-loop affects the node's steady-state fitness depending on the node's connections to other nodes.As an example, the root node 0 of the directed line attains the lowest steady-state fitness among all other nodes, Fig. 4 B. This is because the only incoming link to node 0 is the self-loop.In a mutation-selection dynamics, a self-loop leads to the decrease in the long-term fitness of a node.This can be understood by the following argument: If a given node is currently occupied by a highly fit individual, it is more likely that during the next Moran Bd update this particular node is selected to reproduce.If this node is self-looped, assuming small outgoing weight to other nodes for now, with high probability the mutated offspring replaces its parent via the self-loop.If the mutated offspring is again very fit, this offspring will again be more likely to be selected to reproduce, and thus, repeating the cycle.This process will repeat until the March 22, 2023 10/31 node's fitness decreases.Therefore, self-loops make it harder for highly fit individuals to persist in the population.
On the other hand, incoming and outgoing links decrease the stated negative effect of self-loops.When a highly fit individual occupying a self-looped node is selected to give birth, its mutated offspring can be placed on a neighbouring node if the parent node has a substantial outgoing weight to other nodes.This decreases the participation of the self-loops in the update process, and leading to diminished effects of the self-loops.The role of incoming links is more subtle.Incoming links make node's fitness state more randomised in accordance with the mutational jump distribution.In the long run, for the case of uniform distribution, the mean of the fitness states attained by an individual node solely via the incoming links is the mid point of fitness domain.
Thus, depending on the mutational fitness jump distribution, incoming links can have beneficial or detrimental effects on a node's fitness.For the case of uniform distribution, compared to self-loops, incoming links have beneficial effect on the population's fitness as adding self-loops decreases the population's fitness below the mid-point of the fitness domain.These arguments explain, why the end node of the directed line has higher steady-state fitness than the root node, but lower fitness than the bulk nodes (node 9 in Fig. 4 B).The incoming link to the end node decreases the self-loops effect by making the node's fitness more randomised.However, the absence of an outgoing link from the end node makes the negative impact of self-loop still substantial.The steady-state fitness for the node 1 is an interesting case because being a bulk node, its steady-state fitness is lower than other bulk nodes fitnesses.It is because the incoming link to node 1 does not reach its full potential in randomising the fitness, because the incoming link gets activated only when the root node 0 is selected to reproduce.But since the root node has the lowest fitness, it is less likely to be selected during the update steps.

Sampling fitness from the Gaussian distribution
Until now, in the Moran Bd with mutation update scheme, the mutant's fitness has been sampled from a uniform distribution.However, an offspring's fitness being completely uncorrelated with the parent's fitness is an extreme assumption.Therefore, to find out how robust are the negative effects of the self-loops seen previously, we study the evolutionary dynamics with the fitness of a mutant offspring sampled from the truncated Gaussian distribution on the fitness domain [f min , f max ].At a given point of the dynamics, the Gaussian distribution is centered around the parent's fitness with a standard deviation of σ around the mean.From Fig. 5 A, we see that adding self-loops decreases the steady-state fitnesses for all the graphs.The effect of adding self-loops is the smallest for the complete graph.
This is what we have also observed for the case of uniform mutation fitness distribution.On increasing the σ from 0.1 to 1, compared to other graphs, the self-looped star experiences a considerable decrease in the steady-state average fitness, Fig. 5 B. For σ = 1, the self-looped star graph, an amplifier of selection, attains lower steady-state fitness than the self-looped directed line, a suppressor of selection.For very large σ, we recover the uniform distribution limit, as expected, see Fig. 5 C, where all the non-self looped graphs attain the same mutation-selection balance.The average fitness in this case is higher than that of the self-looped complete graph.All self-looped graphs have lower average fitness than the self-looped complete graph.We refer to Appendix 7.3 for more details on the high mutation rate dynamics for the non self-looped graphs.
Overall, the average steady-state fitness for different graphs increases as σ is decreased.This trend agrees with the intuition that low σ distributions provides directionality to the evolutionary dynamics towards higher fitness values.However, not all the self-looped graphs are affected by this directionality equally.The steady-state average fitness for the heterogenous self-looped graph like, the self-looped star, March 22, 2023 11/31 < l a t e x i t s h a 1 _ b a s e 6 4 = " e 9 j / S 9 x E X l J F p b 1 3 m D x X W i s 6 G t A = " > A A A B 7 3 i c b V B N S w M x E J 3 U r 1 q / q h 6 9 B I v g q e y K q B e h 6 M V j B f s < l a t e x i t s h a 1 _ b a s e 6 4 = " I z e U E g S j z Q 5 e Z U D n + u T 4 6 f C + y j A = " > A A A B 8 H i c b V B N S w M x E J 3 U r 1 q / q h 6 9 B I v g q e y K q B e h 6 M V j B f s  (A) When mutant fitness is sampled from the Gaussian (truncated) distribution with σ = 0.1, we find that adding self-loops decreases the population fitness in all the graphs.(B) Increasing the σ from 0.1 to 1, the average fitness in the steady-state goes down for many graphs.The effect of increasing the σ is largest in the heterogenous star graphs and smallest in the more homogenous structure like the complete graph.(C) We recover the uniform mutant fitness distribution case for very large σ, here σ = 10.In this case, all the non-self looped graphs attain the same steady-state.All self-looped graphs have lower average steady-state fitness than a non-self looped graph and the self-looped complete graph (Parameters: N = 10, µ = 1, f min = 0.1, f max = 10, number of independent realisations is equal to 2000).decreases substantially on increasing σ, compare Fig. 5 A, B. In contrast, regular self-looped structures like the self-looped complete graph and the self-looped cycle graph, do not experience such a sharp fitness decrease, see again Fig. 5 A, B.
In nutshell, from Fig. 5, we can say that the fitness decreasing effects of the self-loops are not arising for a uniform mutant fitness distribution only.

Discussion
Amplifiers of selection [1,10] are fascinating spatial structures.These structures can speed up evolution [8] by enhancing the fixation of beneficial mutants.A randomly generated connected spatial structure, under Moran Bd updating with uniform initialisation, is very likely to be an amplifier of selection [9].Due to their ability to amplify selection and ubiquity, amplifiers of selection have been in the focus of research recently [13,20,28,[39][40][41][42][43].Not only at the fixation time scales, but also for the long-term weak mutation rate mutation-selection dynamics, amplifiers of selection perform better than the well-mixed population by attaining higher average fitness [16].Since mutations are occurring during reproduction, the fixation probabilities entering the steady-state distribution (mutation-selection balance) for spatial structures are temperature initialised.It has been shown that for structures to amplify selection under temperature initialisation, self-loops are important [20].
While self-loops help structures perform better in the low mutation rate regime, outside the low mutation rate regime, self-looped graphs do not attain higher average fitness than the well-mixed population.In fact, outside the weak mutation rate regime, amplifiers of selection can even perform worse than suppressors of selection in terms of maximizing fitness.An example is shown in the Fig. 2, where the self-looped star graph, an amplifier of selection, attains lower fitness in the mutation-selection balance than the complete and self-looped directed line, a suppressor of selection.To further investigate the effect of self-loops, we have worked in the extreme mutation probability regime, µ → 1.The idea was to remove other effects from the evolutionary dynamics, and focus solely on the effect of self-loops on the adaptation of a spatially structured population.March 22, 2023 12/31 The insights we obtain working in the high mutation regime can be useful for the intermediate mutation rate regime as well.While we worked with extremely high mutation rates, high mutation rate studies are not uncommon.One of the celebrated theories dealing with high mutation rate mutation-selection dynamics is the quasispecies theory [44,45].It is a deterministic theory used to study mutation-selection dynamics in infinite well-mixed population.Its variants have also been used to study finite well-mixed populations [46,47].However, the effect of spatial structure remains to be analyzed in the quasispecies theory.Our work is takes step in that direction.
While studying high mutation rate mutation-selection dynamics, the self-looped complete graph naturally serves as the reference graph instead of the complete graph.
The fitness dynamics for the self-looped complete graph is random in the fitness space and time, i.e., at a given time, the fitness state of a node is independent of its fitness states in the past and instantaneous the fitness state of its neighbours.We found that self-loops have a strong fitness-decreasing effect on a node having lower outgoing and incoming weight.In the limit µ → 1, we found that the non-self-looped graphs attain higher steady-state fitness than their self-looped counterparts.Maybe more surprisingly, all the non-self-looped graphs attain the same average fitness in the mutation-selection balance.All self-looped graphs attain lower steady-state fitness than the complete graph.
We also observed the fitness-decreasing effects of the self-loops for the case where the mutant's fitness is sampled from a Gaussian distribution.Thus, the fitness-decreasing effects of the self-loops are not an artefact of a uniform mutant fitness distribution.
We also provide a heuristic measure of the low mutation rate thresholds, µ th , the mutation rate beyond which the evolutionary dynamics is outside the low mutation rate regime.The mutation rate threshold µ th for a graph, depends on the average fixation times and the extinction times of mutants on that graph.As expected, structures with higher fixation times have lower mutation rate thresholds.Therefore, compared to the complete graph and suppressors of selection, amplifiers of selection show deviation for the low mutation rate approximation at lower mutation rates.For a majority of the spatial structures, these thresholds are estimated using the structures' near-neutrality average fixation time scaling with population size.For the directed line with self-loops, the average fixation time grows monotonically with the decrease in mutant's fitness and therefore, µ th is computed from the average fixation time of a mutant with least possible relative fitness for a fitness domain.In this work, we have derived the large N average fixation time scalings for several graphs which in return give the µ th scalings.
The knowledge of these thresholds prevents one from running heavy simulations deep in the low mutation regime.As in the low mutation rate regime, the steady-state statistics is independent of the mutation rate, it is sufficient to access the steady-state via simulations by going slightly below the computed mutation rate thresholds but not deep into the low mutation rate regime.Due to higher sojourn times, it is expected for a self-looped graph to have higher average fixation time for a mutant than its non self-looped counterpart.This however needs a further detailed investigation.
Amplifiers of selection have been in the focus of EGT.However, their promising aspects to optimise fixation of fit mutants are somewhat limited to short-term time scales, where they come with the caveat that they tend to have long fixation times [8,39].In the long-term mutation-selection dynamics, it has been shown in Ref. [16] that suppressors of fixation have the potential to perform better than the amplifiers of selection.This is because of the ability of the suppressor of fixation to reject deleterious mutations more efficiently compensating for its poor probability of fixation for beneficial mutations.Moreover, outside the low mutation rate regime, we see that the temperature initialised star graph, a suppressor of fixation, takes over the self-looped star graph, an amplifier of selection, and maintain higher average fitness in the steady-state throughout the mutation rate regime.However, the reason for the star March 22, 2023 13/31 graph to take over the self-looped star outside the weak mutation rate regime is not clear and requires further investigation.In conclusion, we suggest to broaden the scope of evolutionary graph theory to other structures and to move its focus away from amplifiers of selection.

Appendix
7.1 Kolmogorov's Criterium In the section 5.1.1,we have used the detailed balance condition.Here, we justify the use of detailed balance by proving that the stochastic process at hand is indeed reversible.To do so we make use of Kolmogorov's criterium [48].According to this criterium, a Markov chain on a fitness space spanned by f is reversible if and only if: for any finite set of ordered fitness states The basic idea behind the Kolmogorov's criterium relies on the fact that a reversible Markov chain has zero probability current in the steady-state.In our case, Since, ρ(f i , f i ) = 1 fmax−fmin , the transition probabilities are independent of fitness.
Thus, Kolmogorov's criterium in Eq. 10 is satisfied and the Markov chain for the self-looped complete graph presented in the Sec.5.1.1 is reversible.

Mutation rates threshold and Fixation times
Here we derive the expressions for the average fixation times of a mutant, τ 1 on various network topologies like the self-looped star, star, complete, cycle and the self-looped directed line.

Star graphs
To compute the fixation time for the star graph and self-looped weighted star graph, we use the method of solving recursions inspired from Ref. [49].To start, we write down the recursion satisfied by τ • i , the average fixation time with i mutants in the leaves and a mutant in the center node.We denote this state by (•, i).Similarly, τ • i , is the average fixation time starting with the state (•, i), i.e., i mutants in the leaves and a wild-type individual in the central node.
where, (i) φ • i is the fixation probability with the initial state being (•, i), (ii) φ • i is the fixation probability with the initial state (•, i).
(iii) T (vi) T •• i,i is the transition probability from the state (•, i) to the state (•, i).
The recursions in Eq. 12 satisfy the boundary conditions: φ • 0 = 0 and τ • n = 0.These recursions can be simplified further by dividing the recursion one by T

and recursion two by
Introducing and we finally have, Here, is the conditional transition probability from the state (•, i) to the state (•, i + 1), with the condition that the number of mutants changes.
(ii) π •• i,i is the conditional transition probability from the state (•, i), to the state (•, i), given that the number of mutants changes.
(iii) π •• i,i is the conditional transition probability from the state (•, i) to the state (•, i), with the condition that the number of mutants changes.
(iv) π •• i,i−1 is the conditional transition probability from the state (•, i), to the state (•, i − 1), given that the number of mutants changes.
Solving the recursions 16 using boundary conditions φ • 0 = 0 and τ • n = 0 we get where, March 22, 2023 15/31 and The expressions for φ • i and φ • i are derived in Ref. [49], Now, The self-looped (weighted) star graph is defined by the weighted adjacency matrix with 0 < λ ≤ 1 and 0 < δ ≤ 1.Here, w ij is the weight of the link directed from node i to node j with the center being node number 0. With this, the transition probabilities for a weighted self-looped star graph for the transitions from state (•, i) are The related conditional transition probabilities are Similarly, the transition probabilities for the transitions from state (•, i) are The corresponding conditional transition probabilities are We can use these probabilities along with Eq.21 to obtain the temperature initialised fixation probability and the average fixation time for the self-looped star graph, τ T .In the following, we define the temperature for the center and leaf nodes.The central node temperature is March 22, 2023 16/31 and the leaf node temperature is The temperature initialised fixation probability for the self-looped star graph is The temperature initialised average fixation time for the self-looped star graph is Substituting λ = 1 n and δ = 1 n 2 in the above equation, we get the temperature initialised average fixation time for the self-looped weighted star graph.Setting λ = δ = 1 yields the temperature initialised average fixation time for the standard star graph.
To compute the average extinction time, we use symmetry arguments in Eqs.17, 18, 19, 20, and, 21.With this, we replace Doing so, we obtain φ where φ• i is the extinction probability of mutants starting with the state (•, i) node, and is equal to 1 − φ • i .Similarly, the average extinction time starting with the state ( where The average extinction time starting in state ( Finally, using Eqs.37, 34, the temperature initialised average extinction time of a mutant on the looping star graph is The circles represent Moran Bd simulations.Firstly, we observe that for both the graphs, the average fixation time of a mutant is higher than its extinction time, regardless of the mutant's relative fitness.Secondly, the average fixation time peaks near neutrality for both of the graphs.Therefore, according to Eq. 4, µ th for the star graphs scales as the inverse average fixation time at neutrality.Because the fixation of a mutant takes longer on the self-looped star graph, the weak mutation rate approximation is more restrictive for the self-looped star graph than the star graph.(Parameters: N = 10, wild-type fitness, f = 1, and the number of independent realisations conditioned on mutant's fixation or extinction are 2000.) From Fig. 6, we see that the average fixation time of a mutant is higher than the average extinction time of the mutant regardless of its relative fitness.Moreover, the fixation time peaks near neutrality.Therefore, according to the Eq. 4, the µ th for the stars graphs is the inverse of the average fixation times at neutrality.In the next section, we derive the scaling of τ T 1 at neutrality with respect to the population size N for the star graphs.

Scaling of the average fixation time with population size for the star graphs at neutrality
While the approach used above to compute the fixation and the extinction time on star graph has many merits like extension to the frequency dependent selection case, it is not March 22, 2023 18/31 straightforward to use this approach to derive the exact formula even at neutrality.Therefore, to computethe scaling relation for the fixation time on star graphs, we use a method inspired from Ref. [50].To start, we recast the recursion Eqs. 13 into the form Here, we have replaced, because the conditional transition probabilities are independent of the number of mutants, see Eqs. 24.The horizontal arrows in the subscript of π represent change in the number of mutants in the leaf nodes, right arrow for the increase, and left arrow for the decrease in the number of mutants.The vertical arrows in the subscript of π represent change in individual type at the central node, upward arrow for the change from the wild-type to mutant type, and downward arrow for the change from the mutant type to the wild-type.We also use the shorthand notations Here, t • i is the average time spent in the state (•, i) (the sojourn time of state (•, i)) and t • i is the sojourn time of state (•, i).Shifting the index i to i − 1 in recursion Eq. 39, and solving for φ • i τ • i gives, Now we substitute this relation for φ • i τ • i in the recursion Eq. 40, and obtain, Recursion Eqs.43, and 44, can be written in a matrix representation as, The matrix equation can be further simplified, Remember that we want to compute the scaling for τ T , and for that we need to solve the above matrix equation for τ • 0 , and τ • 1 .The first thing that we need to calculate for For purpose of illustrations, we have chosen N = 4. Broadly speaking, there are three categories of extinction trajectories.(i) The case where the initial mutant goes extinct without spreading in the population.This would be a one time step extinction process, shown by arrow leading from the boxed initial state to the wild-type state, highlighted in grey.(ii) The second category corresponds to the case, where the initial mutant spreads, but the mutant goes extinct before the terminal node is ever occupied by a mutant.This would contain all the paths that go from the boxed state via two mutants to the grey highlighted state without going through the states highlighted in red.(iii) The third category refers to the case, where the initial mutant spreads and reach the state highlighted in red.After the terminal node is occupied by the mutant type, the number of mutants then starts to decrease from the left (shown via the trajectory marked with blue arrows).This third category is especially relevant when the mutant's relative fitness is very high.We make use of this argument to approximate the extinction time for the self-looped directed line by computing the time covered by the blue arrowed trajectory.

Non self-looped graphs and universality at equilibrium
Here, we study the evolutionary dynamics, considering both directed as well as undirected graphs, but without self-loops.
In the absence of self-loops, the average steady-state fitness for all the graphs, whether directed or not, is the same, see Fig. 10 A. We hypothesise that all the graphs where every node has a finite temperature attain the same steady-steady fitness in the mutation-selection balance.All the nodes for these graphs become indistinguishable in their fitness in the steady-state.The steady-state fitness attained by these graphs in the mutation-selection balance is higher than the self-looped complete graph.For these graphs, the fitnesses of nodes in the steady-state are not completely uncorrelated in time as opposed to nodes of the self-looped complete graph where the fitness states of nodes are entirely uncorrelated in time.To test our hypothesis, we analysed a few variants of the directed line and burst graph in Fig. 10 B and C, where each node has a finite temperature.To achieve this, we add a link from node 1 to the root node of the directed line yielding a molded directed line.Similarly, the modified burst is constructed by adding a link from the leaf node to the center in a burst graph.In both the cases, we find that the steady-state fitness attained by these two variant graphs in the mutation-selection balance is the same as that of the complete graph and hence, the other non-self looped graphs considered in Fig. 10 A. All nodes of these graphs become indistinguishable in their fitness in the steady-state.We further check the validity of our hypothesis by varying the population size.In Fig. 10 Review the instructions link below and PLOS Computational Biology's competing interests policy to determine what information must be disclosed at submission.We have no competing interests Data and Code Availability From the time of publication, Authors are required to make fully available and without restriction all data and computational code underlying their findings.Please see our PLOS Data Policy page for detailed policy information, and our Code Sharing page for specific information on code sharing.A Data Availability Statement, detailing where the data (and code, if applicable) can be accessed, is required at first submission.Insert your Data Availability Statement in the box below.The statement you provide will be published in the article, if accepted.All relevant data are within the manuscript and its Supporting Information files.Powered by Editorial Manager® and ProduXion Manager® from Aries Systems CorporationPlease see the Data Reporting section of our submission guidelines for instructions on what you need to include in your Data Availability Statement.If the data and code are all contained in your submission files, please state: All relevant data are within the manuscript and its Supporting Information files.PLOS allows rare exemptions to address legal and ethical concerns.If you have legal or ethical restrictions, please detail these in your Data Availability Statement below for the Journal team to consider.

3 . 1 µ
Fig 1.Birth-death (Bd) updating with mutation.Here, an example of the single time step of the Moran Bd with mutation is shown.First an individual is selected with probability proportional to its fitness to give birth to an offspring.The offspring either resembles the parent with probability 1 − µ, or mutates with probability µ.In case the mutation takes place, the offspring fitness f is then sampled from the distribution ρ(f , f ) with f being the parent's fitness.In the figure, we have shown the case when mutation takes place.The mutant offspring will then replaces one of the individuals neighboring the parent individual, or the parent individual itself via the self-loop.The choice is made at random with probability proportional to the outgoing weight from the parent node.Here, we have shown the case when the parent individual is replaced by the offspring via the self-loop.The stronger the self-loop, more likely it is for the parent to be replaced by its offspring.
the marginal probability density function for the node i to have fitness f i .The marginal probability density function also satisfies the normalisation condition df p * (f ) = 1.The average steady-state fitness of the self-looped complete graph in terms of the individual node's average steady-state fitness satisfies f * = f i * , i.e. the average fitness of the population is the same as the March 22, 2023 8/31

Fig 3 .
Fig 3.  Reference graph: complete graph with self-loops.Here, the mutation-selection dynamics is studied for the self-looped complete graph with µ → 1.We find a very good agreement for the steady-state statistics between the analytics and the simulations.The thick line represents the analytical average fitness, while the shaded grey area represents the standard deviation around the average.Symbols and error bars show simulations.In the steady-state, on average the self-looped complete graph attains the midpoint of the fitness domain, as the fitness dynamics for each individual node of the population becomes uncorrelated in the fitness space and time.The steady-state average fitness is also independent of the population size.The fluctuations in the steady-state however depends on the population size and decreases with the increase in population size as 1/ √ N (Parameters: f min = 0.1, f max = 10, number of independent realisations is equal to 2000, mutant fitness distribution, ρ(f , f ) = Fig 4.  Nodewise analysis of the star graph with self-loops and the directed line with self-loops.Here, the average fitness trajectories for each node of the self-looped star graph (shown in panel A) and the self-looped directed line (shown in panel B are shown.Thick lines represent average fitness trajectories at the population level, whereas, thin lines represent average fitness trajectories for the nodes.The effect of self-loops on a node's fitness depends on the incoming and outgoing weight flowing out of that node.In panel A, self-loops have the least effect on the central node because of relatively higher incoming and outgoing weight.As a result, the central node attains higher average steady-state fitness than the leaf nodes.In panel B, the root node of the directed line has the lowest steady-state average fitness because of the absence of an incoming link to the root node.(Parameters: N = 10, µ = 1, f min = 0.1, f max = 10, number of independent realisations is equal to 2000, mutant fitness distribution, ρ(f , f ) = O s 6 l 9 U z + / P K 7 W b P I 4 i H M E x n I I P l 1 C D O 6 h D A y h I e I Z X e E M a v a B 3 9 D F v L a B 8 5 h D + A H 3 + A A W l j + g = < / l a t e x i t > = 10 Average fitness of the population Time < l a t e x i t s h a 1 _ b a s e 6 4 = " t 4 W Y b g

Fig 5 .
Fig 5. Sampling mutant's fitness from the Gaussian.(A) When mutant fitness is sampled from the Gaussian (truncated) distribution with σ = 0.1, we find that adding self-loops decreases the population fitness in all the graphs.(B) Increasing the σ from 0.1 to 1, the average fitness in the steady-state goes down for many graphs.The effect of increasing the σ is largest in the heterogenous star graphs and smallest in the more homogenous structure like the complete graph.(C) We recover the uniform mutant fitness distribution case for very large σ, here σ = 10.In this case, all the non-self looped graphs attain the same steady-state.All self-looped graphs have lower average steady-state fitness than a non-self looped graph and the self-looped complete graph (Parameters: N = 10, µ = 1, f min = 0.1, f max = 10, number of independent realisations is equal to 2000).

Fig 6 .
Fig 6.Average extinction and fixation time for the self-looped star graph and the standard star graph.Here, we plot the average extinction and fixation time of a mutant for the self-looped (weighted) star graph (panel A) and the star graph (panel B) as a function of mutant's relative fitness.Solid lines corresponds to the analytic results, Eqs. 30, 38.The circles represent Moran Bd simulations.Firstly, we observe that for both the graphs, the average fixation time of a mutant is higher than its extinction time, regardless of the mutant's relative fitness.Secondly, the average fixation time peaks near neutrality for both of the graphs.Therefore, according to Eq. 4, µ th for the star graphs scales as the inverse average fixation time at neutrality.Because the fixation of a mutant takes longer on the self-looped star graph, the weak mutation rate approximation is more restrictive for the self-looped star graph than the star graph.(Parameters: N = 10, wild-type fitness, f = 1, and the number of independent realisations conditioned on mutant's fixation or extinction are 2000.)

Fig 9 .
Fig 9. Paths to extinction.Here, the possible mutant extinction routes are shown for the self-looped directed line when the initial mutant appears on a non-root node.For purpose of illustrations, we have chosen N = 4. Broadly speaking, there are three categories of extinction trajectories.(i) The case where the initial mutant goes extinct without spreading in the population.This would be a one time step extinction process, shown by arrow leading from the boxed initial state to the wild-type state, highlighted in grey.(ii) The second category corresponds to the case, where the initial mutant spreads, but the mutant goes extinct before the terminal node is ever occupied by a mutant.This would contain all the paths that go from the boxed state via two mutants to the grey highlighted state without going through the states highlighted in red.(iii) The third category refers to the case, where the initial mutant spreads and reach the state highlighted in red.After the terminal node is occupied by the mutant type, the number of mutants then starts to decrease from the left (shown via the trajectory marked with blue arrows).This third category is especially relevant when the mutant's relative fitness is very high.We make use of this argument to approximate the extinction time for the self-looped directed line by computing the time covered by the blue arrowed trajectory.
••  i,i±1is the transition probability from the state (•, i) to the state (•, i ± 1), (iv) T •• i,i±1 is the transition probability from the state (•, i) to the state (•, i ± 1), is the transition probability from the state (•, i) to the state (•, i).
D, we see that all the non self-looped graph attain the same steady-state fitness for all population sizes.
Another interesting observation is that the steady-state fitness balance of the non self-looped graphs decreases with increasing N .This is interesting, because with March 22, 2023 26/31