Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating

Populations evolve by accumulating advantageous mutations. Every population has some spatial structure that can be modeled by an underlying network. The network then influences the probability that new advantageous mutations fixate. Amplifiers of selection are networks that increase the fixation probability of advantageous mutants, as compared to the unstructured fully-connected network. Whether or not a network is an amplifier depends on the choice of the random process that governs the evolutionary dynamics. Two popular choices are Moran process with Birth-death updating and Moran process with death-Birth updating. Interestingly, while some networks are amplifiers under Birth-death updating and other networks are amplifiers under death-Birth updating, so far no spatial structures have been found that function as an amplifier under both types of updating simultaneously. In this work, we identify networks that act as amplifiers of selection under both versions of the Moran process. The amplifiers are robust, modular, and increase fixation probability for any mutant fitness advantage in a range r ∈ (1, 1.2). To complement this positive result, we also prove that for certain quantities closely related to fixation probability, it is impossible to improve them simultaneously for both versions of the Moran process. Together, our results highlight how the two versions of the Moran process differ and what they have in common.


Introduction
Moran process is a classic stochastic process that models natural selection in populations of asexually reproducing individuals, especially when new mutations are rare [1,2].It is January 29, 2024 1/23 arXiv:2401.14914v1[q-bio.PE] 26 Jan 2024 commonly used to understand the fate of a single new mutant, as it attempts to invade a population of indistinguishable residents.Eventually, the new mutation will either fixate on the whole population, or it will go extinct.It is known that when the invading mutant has relative fitness advantage r > 1 as compared to the residents, this fixation probability tends to a positive constant 1 − 1/r as the population size N grows large.On spatially structured populations, fixation probability of an invading mutant can both increase or decrease.In the framework of evolutionary graph theory [3,4], the spatial structure is represented by a graph (network) in which nodes (vertices) correspond to individual sites, and edges (connections) correspond to possible migration patterns.Each edge is assigned a weight that represents the strength of the connection.Such network-based spatial structures can represent island models, metapopulations, lattices, as well as other arbitrarily complex structures [5][6][7][8][9].Spatial structures that increase the fixation probability of a randomly occurring advantageous mutant beyond the constant 1 − 1/r are called amplifiers of selection [10].The logic behind the name is that living on such a structure effectively amplifies the fitness advantage that the mutants has, as compared to living on the unstructured (well-mixed) population.Identifying amplifiers is desirable, since they could potentially serve as tools in accelerating the evolutionary search, especially when new mutations are rare [11,12].
When run on a spatial structure, Moran process can be implemented in two distinct versions.They are called Moran Birth-death process and Moran death-Birth process.In the Moran Birth-death process, first an individual is selected for reproduction with probability proportional to its fitness, and the offspring then replaces a random neighbor.In contrast, in the Moran death-Birth process, first a random individual dies and then its neighbors compete to fill up the vacant site (see Fig. 1).Both the Moran Bd-updating [1,3,13] and the Moran dB-updating [14][15][16][17] have been studied extensively.While essentially identical on the unstructured population, the two versions of the process yield different results when run on most spatial structures [18][19][20].Moran Birth-death and death-Birth processes on a population structure.a, Each node is occupied by a resident with fitness 1 (blue), or a mutant with fitness r ≥ 1 (red).Thicker edges denote higher edge weights (stronger interactions).b, In Moran Birth-death process, a random individual reproduces, and the produced offspring migrates along a random edge.c, In Moran death-Birth process, a random individual dies, and the vacancy is filled by a random neighbor.In both cases, edges with higher weight are selected more often, and fitness plays a role in the Birth step but not in the death step.
residents co-exist in the population, we perform discrete time steps that change the state of (at most) one node at a time.
There are two versions of the Moran process (see Fig. 1).In the Moran Birth-death process, we first select an individual to reproduce (randomly, proportionally to the fitness of the individual), and then the offspring migrates along one adjacent edge (randomly, proportionally to the weight of that edge) to replace the neighbor.Formally, denoting by F = u f (u) the total fitness of the population, node u gets selected for reproduction with probability f (u)/F , and then it replaces a neighbor v with probability In contrast, in the Moran death-Birth process, we first select an individual to die (uniformly at random), and then the neighbors compete to fill in the vacancy (randomly, proportionally to the edge weight and the fitness of the neighbor).Formally, node v dies with probability 1/N and it gets replaced by a node u with probability We note that in both versions we capitalize the word "Birth" to signify that fitness plays a role in the birth step (and not in the death step).

Fixation probability and Amplifiers
If the graph G N that represents the population structure is connected then the Moran process eventually reaches a "homogeneous state", where either all nodes are occupied by mutants (we say that mutants fixated ), or all nodes are occupied by residents (we say that mutants went extinct).Given a graph G N , a mutant fitness advantage r ≥ 1, and a set S ⊆ V of nodes initially occupied by mutants, we denote by ρ Bd r (G N , S) the fixation probability, that is, the probability that mutants eventually reach fixation, under Moran Birth-death process.We are particularly interested in the fixation probability of a single mutant who appears at a node selected uniformly at random.We denote this fixation probability under uniform initialization by We define ρ dB r (G N , S) and ρ dB r (G N ) analogously.In this work we focus on population structures that increase the fixation probability of invading mutants.The base case is given by an unweighted complete graph K N that includes all edges and represents an unstructured, well-mixed population.It is known [4,20,21] that Given a graph G N and a mutant fitness advantage r ≥ 1, we say that G N is a Bd r -amplifier if ρ Bd r (G N ) > ρ Bd r (K N ).We define dB r amplifiers analogously, that is, as those graphs G N that satisfy ρ dB r (G N ) > ρ dB r (K N ).Similarly, suppressors are graphs that decrease the fixation probability as compared to the complete graph.

Results
First, we present three negative results that illustrate that the two worlds of Birth-death and death-Birth updating often present contradictory requirements when it comes to enhancing the fixation probability of a single newly occurring mutant.Nevertheless, as our main contribution in the positive direction, we then present population structures that are both Bd r -amplifiers and dB r -amplifiers for a range of mutant fitness advantages r ∈ (1, 1.2).January 29, 2024 4/23

Negative results
In this section, we present results that suggest that finding simultaneous Bd r -and dB ramplifiers is not easy.First, we show empirically that known amplifiers for one process are suppressors for the other process.Second, we show that in the neutral regime (r = 1), any fixed vertex is a "good" starting vertex for the mutant in at most one of the two processes.Finally, we show that for any starting vertex, the chance of not dying immediately can be enhanced in at most one of the two processes (see below for details).

Known amplifiers for one process
In this section we examine spatial structures that are known to amplify under one of the two versions of the Moran process, in order to see whether they amplify under the other version of the Moran process (spoiler alert: they don't).First, we consider the smallest known unweighted dB-amplifier [17], which is a certain graph on N = 11 nodes (see Fig. 2).We call the graph D 11 .The graph D 11 is an extremely weak dB r -amplifier in a range of approximately r ∈ (1, 1.00075), where it increases the fixation probability by a factor less than 1.0000001× (see [17,  .= 0.997.Next, we examine the star graph S 11 on 11 vertices which, to our knowledge, is the strongest unweighted amplifier for Bd-updating at this population size.The Star graph is a clear Bd r -amplifier for r ∈ (1.01, 1.1), but an equally clear dB r -suppressor in that range.
The situation is reversed for the Fan graph F 11 [32].While the Fan graph clearly functions as an amplifier under the dB-updating when r ∈ (1.01, 1.1), it lags behind the baseline given by the complete graph under the Bd-updating.Neutral regime (r = 1) The second negative result pertains to the case of neutral mutations (r = 1).Recall that ρ Bd r (G N , v) and ρ dB r (G N , v) denote the fixation probabilities when the initial mutant appears at node v.The following theorem states that for neutral mutations (r = 1), no initial mutant node increases the fixation probability both for Birth-death and death-Birth updating.January 29, 2024 5/23 Theorem 1.Let G N be a graph and v an initial mutant node.Then either The idea behind the proof is that for neutral evolution there are explicit formulas for fixation probabilities ρ Bd r (G N , v) and ρ dB r (G N , v) on any undirected graph G N [34,35].The result then follows by applying Cauchy-Schwarz inequality.See Supplementary Information for details.In Supplementary Information, we also note that Theorem 4 does not generalize to the case when instead of having one initial mutant node we start with an initial subset S of k ≥ 2 nodes occupied by mutants.

Immediate extinction and forward bias
In order to present our third and final negative result, we need to introduce additional notions and notation.When tracking the evolutionary dynamics on a given graph G N with a given mutant fitness advantage r ≥ 1, it is often useful to disregard the exact configuration of which nodes are currently occupied by mutants, and only look at how many nodes are occupied by mutants.
One example of this is the celebrated Isothermal theorem [3] which states that once N and r are fixed, the fixation probability under the Moran Birth-death process on any regular graph is the same.Here, a graph is regular if each node has the same total weight of adjacent edges.Examples of regular graphs include the complete graph, the cycle graph, or any grid graph with periodic boundary condition.
The intuition behind the proof of the Isothermal theorem is that for any regular graph R N , the Moran Birth-death process can be mapped to a random walk that tracks just the number of mutants, instead of their exact positions on the graph.It can be shown that this random walk has a constant forward bias, that is, the probabilities p + (resp.p − ) that the size of the mutant subpopulation increases (resp.decreases) satisfy p + /p − = r, for any number of mutants in any particular mutant-resident configuration.A natural approach to construct amplifiers is thus to construct graphs for which this forward bias satisfies an inequality p + /p − ≥ r for the Moran Birth-death process and an analogous inequality for the Moran death-Birth process.Our final negative result shows that this goal can not be achieved already in the first step.
Formally, consider the Moran Birth-death process on a graph G N with a single initial mutant placed at node u.Let γ Bd r (G N , u) be the probability that the first reproduction event that changes the size of the mutant subpopulation is the initial mutant reproducing (as opposed to the initial mutant being replaced by one of its neighbors).In other words, γ Bd r (G N , u) is the probability that the first step that changes the configuration of the mutants does not eliminate the initial mutant, leaving the options of later mutant extinction or mutant fixation.
For the complete graph K N (and any single mutant node) it is not hard to show that γ Bd r (K N ) = γ Bd r (K N , u) = r/(r + 1) for any node u.Moreover, by a slight extension of the Isothermal theorem, we have γ Bd r (R N , u) = r/(r + 1) for any regular graph R N and any node u.For Moran death-Birth process, we define γ dB r (G N , u) and γ dB r (K N ) analogously.To construct a graph that is both a Bd-and a dB-amplifier, a natural approach is to look for a graph and an initial mutant node u such that γ Bd r (G N , u) > γ Bd r (K N ) and γ dB r (G N , u) > γ dB r (K N ).However, the following theorem states that no such graphs exist.
Theorem 2. Let G N be a graph, u an initial mutant node, and r ≥ 1.Then either January 29, 2024 6/23 The proof relies on the notion of the temperature of a node.Formally, given a graph G N = (V, E) and a node u ∈ V , its temperature T (u) is defined as where the sum goes over all the neighboring nodes v of u in G N .The temperature of a node represents the rate at which the node is being replaced by its neighbors in the Moran Birth-death process when r = 1.Nodes with high temperature are replaced often, whereas nodes with low temperature are replaced less frequently.Building on this, it is straightforward to show that if a node u has above-average temperature, then γ Bd r (G N , u) < γ Bd r (K N ), that is, in Moran Birth-death process with a single mutant at u the forward bias is lower than the forward bias on a complete graph.To complete the proof, we then show that for any node u with below-average temperature, we have γ dB r (G N , u) < γ dB r (K N ).Our proof of the latter claim uses Jensen's inequality for a certain concave function.See Supplementary Information for details.

Positive result
Despite the above negative results, in this section we identify population structures A N that substantially amplify the fixation probability under both Birth-death updating and death-Birth updating when the number N of nodes is sufficiently large.
The structures A N are composed of two large chunks A Bd and A dB that are connected by a single edge, see Fig. 3a for an illustration.The chunk A dB is a Fan graph [32], which is to our knowledge the strongest currently known dB-amplifier.The chunk A Bd could be any of the many strong Bd-amplifiers.For definiteness, in Fig. 3a we use a Fan-like structure with a nodes in a central hub and b blades of two nodes each surrounding it.The single connecting edge has a very low edge weight so that the two chunks interact only rarely.For population size N = 1001, the resulting weighted graph is both a Bd r -amplifier and a dB r -amplifier for any r ∈ (1, 1.09), see Fig. 3b.
Similarly, we identify large population structures that serve as both Bd r -amplifiers and dB r -amplifiers for any r ∈ (1, 1.2).
Theorem 3 (Simultaneous Bd-and dB-amplifier).For every large enough population size N there exists a graph A N such that for all r ∈ (1, 1.2) we have In what follows we provide intuition about the proof of Theorem 6.The fully rigorous proof is relegated to Supplementary Information.Let e be the edge connecting the two chunks, u its endpoint in A Bd , and v its endpoint in A dB .
First, observe that since e has a low weight, the two chunks evolve mostly independently.This means that, with high probability, each chunk resolves to a homogeneous state in between any two interactions across the chunks.In particular, if the initial mutant appears in the chunk where it is favored (e.g. if it appears in the chunk A Bd when Bd-updating is run), the mutants fixate on that chunk with reasonable probability.If that occurs, we say that mutants are "half done".
Once the mutants are half done, the next relevant step occurs when the two chunks interact.There are two cases.Either a mutant at u reproduces and the offspring migrates along e to v, or a resident at v reproduces and the offspring migrates along e to replace the mutant at u.In both cases, the individual (mutant or resident) who "invades" the other half eventually either succeeds in spreading through that half, or they fail at doing that.If the latter occurs, we are back at the situation in which mutants are half January 29, 2024 7/23 done and the situation repeats.By bounding all the relevant probabilities, we show that once half done, mutants are overwhelmingly likely to fixate, as opposed to going extinct.
We highlight an interesting phenomenon that occurs in our proof.As we run the evolutionary dynamics, we can look at the flow along the connecting edge e. Thanks to the edge weights, it turns out that the direction of the flow along e flips depending on whether we run the Moran Birth-death process or the Moran death-Birth process.In particular, under the Bd-updating the edge e is used mostly in the direction from u to v.That is, many individuals migrate from u to v, whereas few individuals migrate from v to u.Under dB-updating the situation reverses.That is, many individuals migrate from v to u, whereas few of them migrate from u to v. Thus, under the Bd-updating the A Bd chunk is effectively upstream of the chunk A dB , whereas under the dB-updating the A dB chunk is effectively upstream of the chunk A Bd .This asymmetry is a key factor that contributes to the fact that once the mutants are half done, they are likely to fixate on the whole graph (see Fig. 4).
What remains in the proof is to balance out the sizes of the two chunks.For small r > 1, the strongest known dB-amplifiers are roughly 3  2 × stronger than the Complete graph (in terms of the fixation probability).Thus, in order to achieve amplification under dB-updating, we need the chunk A dB to take up at least 2/3 of the total population size.The chunk A Bd then takes up at most 1/3 of the total population size.In order to achieve Bd-amplification, fixation probability on A Bd under Bd-updating must therefore be at least 3× larger than that on the Complete graph.Interestingly, a Star graph is not strong enough to do that (for r ≈ 1 and large population size N it is only roughly 2× stronger than the Complete graph), but sufficiently strong Bd-amplifiers do exist (e.g.any superamplifier).

Discussion
Population structure has a profound impact on the outcomes of evolutionary processes and, in particular, on the probability that a novel mutation achieves fixation [3,36].January 29, 2024 8/23 Interactions between A Bd and A dB .a, The edge weights in the chunks A Bd (red) and A dB (blue) are shown as a function of t (here t ≫ 1 is large).For each of two versions of the Moran process, the rates at which the offspring migrate from u to v and from v to u can be calculated and are listed in the table.b, Under Birth-death updating, the migration rate p u→v from u to v is roughly t× larger than the migration rate p v→u from v to u, so the chunk A Bd is upstream of the chunk A dB , and a mutant who has fixated over A dB is likely to fixate over A dB too.c, In contrast, under death-Birth updating we have p v→u ≈ t • p u→v , hence the chunk A dB is upstream of A Bd .
Population structures that increase the fixation probability of beneficial mutants, when compared to the case of a well-mixed population, are known as amplifiers of selection.
Somewhat surprisingly, to tell whether a specific spatial structure is an amplifier or not, one needs to specify seemingly minor details of the evolutionary dynamics.The well-studied Moran process comes in two versions, namely Moran process with Birth-death updating and Moran process with death-Birth updating.While many spatial structures are amplifiers under the Bd-updating [21], only a handful of amplifiers under the dB-updating are known [31].Moreover, none of the dB-amplifiers that we checked amplify under the Bd-updating.
In this work we help explain this phenomenon by proving mathematical results which illustrate that the two objectives of amplifying under the Bd-updating and amplifying under the dB-updating are often contradictory.Thus, one might be tempted to conclude that perhaps there are no population structures that amplify in both worlds, that is, regardless of the choice of the underlying dynamics (Bd or dB).Nevertheless, we proceed to identify population structures that serve as amplifiers of selection under both Bd-updating and dB-updating.
The amplifiers we identify in this work have several interesting features.First, they are robust in the sense that they amplify selection under both the Bd-updating and the dB-updating.Second, they provide amplification for any mutant fitness advantage r in a range r ∈ (1, 1.2), which covers many realistic values of the mutant fitness advantage, and the amplification is non-negligible (for instance, for r = 1.05 the fixation probability increases by 14% and 44%, respectively.see Fig. 3).Third, the amplifiers are modular.That is, they consist of two large chunks that serve as building blocks and that interact rarely.For definiteness, in this work we specified the two chunks and their relative sizes, but each chunk can be replaced by an alternative building block and the relative sizes can be altered.For example, the best currently known dB-amplifiers amplify by a factor of 1.5× for r ≈ 1 and continue to amplify for r in a range r ∈ (1, φ), where φ = 1 2 ( √ 5 + 1) ≈ 1.618 is the golden ratio [32].If better dB-amplifiers are found, they can be used as a building block in place of one of the chunks to improve the range r ∈ (1, 1.2) for which the resulting structure amplifies in both worlds.January 29, 2024 9/23 In this work, our objective was to increase the fixation probability of an invading mutant in both worlds (Bd-updating and dB-updating).An interesting direction for future work is to optimize other quantities in both worlds.
One such quantity is the duration of the process until fixation occurs [37][38][39].For example, achieving short fixation times in combination with increasing the fixation probability does not appear to be easy.Our proofs rely on the existence of small edge weights to separate the time scales at which different stages of the process happen.While using more uniform edge weights might still lead to the same outcome, the proofs would need to become more delicate.A possible approach to identify structures that serve as fast amplifiers in both worlds would be to find unweighted amplifiers, because then the time would be guaranteed to be at most polynomial [40,41].The first step in this direction would be to identify large and substantially strong unweighted dB-amplifiers.There are promising recent results in this direction [31].
Looking beyond fixation time, there are other relevant quantities such as the recently introduced rate at which beneficial mutations accumulate [42].Existing research suggests that the two versions of the Moran process behave quite differently in terms of the fixation probability [21], but quite similarly in terms of the fixation time [40,41].Which of those two cases occurs for other relevant quantities remains to be seen.This is a supplementary information to the manuscript Amplifiers of selection for the Moran process with both Birth-death and death-Birth updating.It contains formal proofs of the theorems listed in the main text.

Preliminaries
Given an undirected graph G N = (V, E) on N nodes, the degree of a node u, denoted deg(u), is the number of neighbors of u in G N .When the edges are weighted, we define the degree deg(u) = (u,v)∈E w(u, v) as the sum of the weights of all the adjacent edges.As a direct extension of [34,35] we obtain the following formula for fixation probability under neutral drift (r = 1).For completeness, we include a proof.
Lemma 1 (Fixation probability on edge-weighted undirected graphs when r = 1).Let G N = (V, E) be an edge-weighted undirected graph on N nodes and S ⊂ V any set of vertices occupied by mutants.Then .
Proof.Let p u→v be the probability that, in a single step, an individual at node u produces an offspring that replaces an individual at node v.For Birth-death updating, it suffices to check that for any subset S ⊂ V of mutant nodes and any edge (u, v) connecting a mutant node u ∈ S and a non-mutant node v ̸ ∈ S we have .
Since for Birth-death updating and r = 1 we have p u→v = 1 N • w(u,v) deg(u) , both sides rewrite as , and so the claim is proved.Likewise, for death-Birth updating it suffices to check that .
Since for death-Birth updating and r = 1 we have , this time both sides rewrite as .
The proof of our positive result relies on three existing results.For convenience, we list them here.First, there exist unweighted graphs called Incubators that are strong amplifiers under Birth-death updating [29, Theorem 2].Lemma 2. There exists a family of graphs A Bd N such that for all r > 1, we have January 29, 2024 14/23 Second, there exist edge-weighted graphs called Separated Hubs that are substantial amplifiers under death-Birth updating [32,Theorem 3].Lemma 3.There exists a family of graphs A dB N such that for all r > 1, we have .
Third, the evolutionary dynamics terminates polynomially quickly in terms of the population size N , under both the Birth-death updating [40,Theorem 9] and the death-Birth updating [41, Theorem 1].Lemma 4. Fix r > 1.For Bd and dB process on an undirected graph with N vertices with the highest ratio between edge weights 1 ε , the expected fixation time is in O( N 4 ε ).
2 Negative result 2 In this section, we show that one fixed neutral mutant cannot have a better fixation probability in both processes than on a complete graph.This means that even if we can choose the starting position, we are not guaranteed to increase the fixation probability for both processes.
Theorem 4. Let G N be a graph and v an initial mutant node.Then one of the following three cases occurs: Proof.First, note that ρ Bd r=1 (K N ) = ρ dB r=1 (K N ) = 1/N .Next, recall the known formulas for the fixation probability on undirected graphs under neutral drift (see Lemma 1 and [34,35]], namely: u∈V deg(u) .
As the final ingredient, note that for any N non-negative numbers x 1 , . . ., x N we have a bound 1 This follows e.g. from the inequality between the arithmetic and harmonic mean of numbers x 1 , . . ., x N (called AM-HM), or from Cauchy-Schwarz inequality.Moreover, the equality occurs if and only if If equalities occur everywhere then deg Under Birth-death updating, the effect persists for r ≥ 1. (Data obtained by numerically solving the underlying Markov chains.) The following example illustrates that there exists a graph and a subset S = {u, v} of k = 2 nodes, such that the fixation probability starting from mutants at both u and v is strictly greater than fixation probability starting from k = 2 mutant nodes on a well-mixed population, both for the Birth-death and for the death-Birth updating.
The intuition behind the result is that node u is a really good initial mutant node for Birth-death updating, and node v is a really good initial mutant node for death-Birth updating.Together, they form an above-average set of two mutant nodes, even when compared to a complete graph with two initial mutants.

Negative result 3
In this section, we prove that for any fixed vertex, in the first step, the ratio between increasing and decreasing the number of mutants cannot be better than in the complete graph in both processes.This means we cannot find a vertex from which both processes spread better than in the complete graph.To achieve amplification for both processes, we know that some vertices will be better for Bd and some for dB amplification.
Theorem 5. Let G N be a graph, u an initial mutant node, and r ≥ 1.Then one of the following three cases occurs: deg v the so-called temperature of node u, that is, the rate at which node u is replaced by its neighbors in the neutral case.
Denote by p + Bd,r = p + Bd,r (G N , u) the probability that in a single step of the Moran Birth-death process the mutant reproduces, and by p − Bd,r = p − Bd,r (G N , u) the probability that it gets replaced by a resident.Denoting the total fitness by F = N + (r − 1) we have In particular, in the complete graph K N each node has temperature 1, and thus If T (u) ≥ 1 then r/(r + T (u)) ≤ r/(r + 1) and hence γ Bd r (G N , u) ≤ γ Bd r (K N ) with equality if and only if T (u) = 1.From now on, suppose T (u) ≤ 1.
Consider Moran death-Birth process and define the quantities p + dB,r = p + dB,r (G N , u) and p − dB,r = p − dB,r (G N , u) as above.Then .
In particular, for the complete graph K N and any its node u we have .
Hence in order to prove γ dB r (G N , u) ≤ γ dB r (K N ), it suffices to prove .
We rearrange When r = 1, the desired claim reduces precisely to T (u) ≤ 1. Suppose r > 1, that is r − 1 > 0, and consider a function f : (0, ∞) → (0, ∞) defined by f (x) = x (r−1)x+1 .Then f is concave and increasing, therefore by Jensen's inequality we have Finally, since the function f is increasing, using bounds T (u) ≤ 1 and |N (u)| ≤ N − 1, the right-hand side is at most as desired.For the equality to occur everywhere, we must in particular have T (u) = 1, in which case the other equality γ Bd r (G N , u) = γ Bd r (K N ) holds too.
January 29, 2024 17/23 In this section, we prove the main positive result which states that there exists an undirected, edge-weighted graph that is simultaneously an amplifier of selection for Birth-death Moran process and for death-Birth Moran process (under uniform mutant initialization).We first bound the number of steps until fixation or extinction for both processes (Bd and dB) and any graph.Second, we show that for any graph, there is a good starting vertex where a mutant has fixation probability at least 1 N .Then we construct the graph and we prove that it is indeed an amplifier for both processes.

Auxiliary statements
Lemma 5.For Bd and dB process for any r on an undirected graph with N vertices with the ratio between edge weights at most 1 ε , the probability that the process is not completed after O(N 5 /ε) steps is in O( 12 N ).Proof.From Lemma 4, we can take constant c such that for both processes and all graphs with N vertices, the expected time is at most cN 4 /ε.From Markov's inequality, the probability that the process takes more than 2cN 4 /ε steps is at most 1 2 .If the process does not finish, the expected time is again cN 4 /ε.That means we can take N epochs of size 2cN 4 /ε each, and the probability that the process does not finish in any epoch is at most 1 2 N .Lemma 6.For any graph G N with N vertices any r ≥ 1, and a process p ∈ {Bd, dB} there exists a vertex v such that ρ p r (G N , v) ≥ 1 N .Proof.It suffices to prove the statement for r = 1, since increasing the mutant fitness advantage r increases its fixation probability [37,Theorem 6].
In the neutral case (r = 1), we have v∈V ρ r=1 (G N , v) = 1, thus there exists at least one vertex with fixation probability at least 1  N .Note that in some cases, no starting vertex v satisfies both ρ Bd r (G N , v) ≥ 1/N and ρ dB r (G N , v) ≥ 1/N simultaneously.An example is a Star graph S 3 on N = 3 vertices with center c and leaves l 1 , l 2 when r = 1.Then ρ Bd r (S 3 , c) = 1/5 < 1/3 and ρ dB r (S 3 , l 1 ) = ρ dB r (S 3 , l 2 ) = 1/4 < 1/3.In fact, later we prove that in the neutral case no starting vertex is a strict improvement under both Bd and dB, see Theorem 4.

Construction
For given N and γ ∈ (0, 1), we describe how to construct graph A N,γ .We show that for some γ, this graph is an amplifier for both processes for r ∈ (1, 1.2).The graph A N,γ has two parts.The first part is a graph A Bd (1−γ)N (from Lemma 2) on (1 − γ)N vertices, the second part is a graph A dB γN (from Lemma 3) on γN vertices.Let ε be the smallest weight among edges when both graphs are independently scaled such that the largest edge weight is 1.
We will connect the two parts by a single edge.To that end, we select a vertex v from A Bd (1−γ)N such that the fixation probability starting from v in A Bd (1−γ)N in dB-process is at least 1 N , (such vertex exists from Lemma 6).Similarly, we select a vertex v ′ from A dB γN such that the fixation probability starting from v ′ in the graph under Bd-process is at least 1 N , (existence follows from Lemma 6).Then, we connect v and v ′ by an edge of weight w = ε 3 N 9 .Finally, we scale all edges in the first part A Bd (1−γ)N by a factor of ε N 3 .That is, the heaviest edge in A dB γN has weight 1, and the heaviest edge in A Bd Before we turn to the main proof, we show several properties of the graph we A N,γ we have just constructed.The first property is that the two parts A Bd (1−γ)N and A dB γN interact so rarely that most of the time they interact, the population on either part is already homogeneous (all mutants or all residents).Then we show Lemma 8 and Lemma 9.The lemmas show that in both processes, the probability of an individual reproducing over the edge between v and v ′ is unbalanced and in both processes, the individual in the respective amplifier is more likely to spread the the other graph.
Lemma 7.For any N , γ, both processes, and randomly placed mutant in A N,γ with the probability of at least mutants become extinct or fixate on their part of A N,γ before any reproduction over edge v, v ′ .
Proof.First, we bound the probability that edge v, v ′ is selected in both processes and then we use the union bound.
For Bd, the edge v, v ′ is used either by (i) selecting the individual at v and spreading over v, v ′ , or (ii) selecting the individual at v ′ and spreading over v ′ , v. Event (i) happens with probability at most For dB, the edge v, v ′ is used either if (i) individual at v dies and is replaced individual at v ′ , or (ii) individual at v ′ dies and is replaced by individual at v. Event (i) happens with probability at most 1 N • rε 3 /N 9 ε 3 /N 9 +ε 2 /N 3 < rε N 7 .Event (ii) happens with probability at most 1 N • rε 3 /N 9 ε 3 /N 9 +ε < rε 2 N 10 .The sum of these probabilities is at most 2ε N 7 .From Lemma 5, we know that with high probability the process ends in O(N 5 /ε) steps.In every step the probability of using edge v, v ′ is at most 2ε N 7 , that gives probability of using v, v ′ at most O( 1N 2 ) at first N 5 /ε steps from union bound.Since the probability that the process does not end during these steps is also in O( 1N 2 ), we have that the randomly placed mutant resolves on one part of the graph before using edge v, v ′ with a probability at least 1 − O( 1 N 2 ) Lemma 8.In the graph A N,γ under the Bd process, if edge v, v ′ is used, then with probability at least 1 − r 2 N 2 occupant of v spreads to v ′ .Proof.At one step, individual at v spreads to v ′ with probability at least The opposite event, v spreading to v ′ happens with probability at least 1 − r 2 N 2 .Lemma 9.In the graph A N,γ under the dB process, if edge v, v ′ is used, then with probability at least 1 − r 2 N 2 occupant of v ′ spreads to v. Proof.At one step, individual at v ′ spreads to v with probability at least

b
Fig 1.Moran Birth-death and death-Birth processes on a population structure.a, Each node is occupied by a resident with fitness 1 (blue), or a mutant with fitness r ≥ 1 (red).Thicker edges denote higher edge weights (stronger interactions).b, In Moran Birth-death process, a random individual reproduces, and the produced offspring migrates along a random edge.c, In Moran death-Birth process, a random individual dies, and the vacancy is filled by a random neighbor.In both cases, edges with higher weight are selected more often, and fitness plays a role in the Birth step but not in the death step.

Fig 2 .
Fig 2. Known amplifiers are suppressors for the other process.a, We consider four graphs on N = 11 nodes, namely the Complete graph K 11 , the star graph S 11 , the Fan graph F 11 , and the smallest known undirected suppressor D 11 (see [17]).b, Under Bd-updating, the only amplifier for r ∈ {1.01, . . ., 1.1} is the Star graph S 11 .c, Under dB-updating, the only amplifier for r ∈ {1.01, . . ., 1.1} is the Fan graph F 11 .Values computed by numerically solving the underlying Markov chains.

Fig 3 .
Fig 3. Simultaneous Bd-and dB-amplifier A N .a, The graph A N is composed of two large chunks A Bd and A dB that are connected by a single edge.The chunk A dB is a Fan graph on f nodes.The chunk A Bd is a fan-like graph with a vertices in a central hub and b blades of two nodes each.The total population size is N = a + 2b + f (here a = b = 5, f = 11, and N = 26).The edge weights are defined such that different circled units within the chunks interact only rarely, and the chunks themselves interact even more rarely.b, Here we consider graph A N with population size N = 1001 and (a, b, f ) = (30, 85, 801).The fixation probabilities under Bd-and dB-updating are computed by numerically solving the underlying Markov chain.We find that the inequality ρ Bd r (A N ) > ρ Bd r (K N ) is satisfied for r ∈ (1, 1.09) and the inequality ρ dB r (A N ) > ρ dB r (K N ) is satisfied for r ∈ (1, 1.2).In particular, at r = 1.05 the ratios satisfy ρ Bd r (A N )/ρ Bd r (K N ) > 1.44 and ρ dB r (A N )/ρ dB r (K N ) > 1.14.

Fig 5 .
Fig 5. Mutant subset that amplifies for both Bd and dB.a, With two neutral mutants (r = 1) on a complete graph K N , the fixation probability is equal to 2/N under both Birth-death and death-Birth updating.b, When two neutral mutants initially occupy vertices u and v of the so-called dart graph D 5 , the fixation probability under both Birth-death and death-Birth updating is increased.c, As r increases above roughly r ≈ 1.24, the fixation probability on the Dart graph under death-Birth updating drops below the reference value of two mutants on a complete graph K 5 .Under Birth-death updating, the effect persists for r ≥ 1. (Data obtained by numerically solving the underlying Markov chains.) (1−γ)N has weight ε N 3 .Observe that the scaling of edges in A Bd (1−γ)N does not influence the fixation time.