Several real-world phenomena are described in terms of an invasion process that occurs in a network and follows some propagation model. The problems considered in such settings aim to optimize the success of the invasion, quantified by some metric. For example, social media campaigns serving purposes of marketing, political advocacy, and social welfare are modeled in terms of influence spread, whereby the goal is to maximize the expected influence spread of a message as a function of a set of initial adopters Kempe, Kleinberg, and Tardos (2003); Tang, Xiao, and Shi (2014); Borgs et al. (2014); Tang, Shi, and Xiao (2015); Zhang et al. (2020); see Li et al. (2018) for a survey. Similarly, in rumor propagation Demers et al. (1987), the goal is to minimize the number of rounds until a rumor spreads to the whole population Fountoulakis, Panagiotou, and Sauerwald (2012).
A related propagation model is that of the Moran process, introduced as a model of genetic evolution Moran (1958), and the variants thereof, such as the discrete Voter model Clifford and Sudbury (1973); Liggett (1985); Antal, Redner, and Sood (2006); Talamali et al. (2021). A single mutant (e.g., a new opinion, strain, social trait, etc.) invades a population of residents scattered across the nodes of the graph. Contrariwise to influence and rumor propagation models, the Moran process accounts for active resistance to the invasion: a node carrying the mutant trait may not only forward it to its neighbors, but also lose it by receiving the resident trait from its neighbors. This model exemplifies settings in which individuals may switch opinions several times or sub-species competing in a gene pool. A key parameter in this process is the mutant fitness advantage, a real number that expresses the intensity by which mutants propagate to their neighbors compared to residents. Large favors mutants, while renders mutants and residents indistinguishable.
Success in the Moran process is quantified in terms of the fixation probability, i.e., the probability that a randomly occurring mutation takes over the whole population. As the focal model in evolutionary graph theory Lieberman, Hauert, and Nowak (2005), the process has spawned extensive work on identifying amplifiers, i.e., graph structures that enhance the fixation probability Monk, Green, and Paulin (2014); Giakkoupis (2016); Galanis et al. (2017); Pavlogiannis et al. (2018); Tkadlec et al. (2021). Still, in the real world, the probability that a mutation spreads to its neighborhood is often affected by its position: the ability to metabolise a certain sugar is advantageous to bacteria only when that sugar is present in their surroundings; likewise, people are likely to spread a viewpoint more enthusiastically if it is supported by experience from their local environment (country, city, neighborhood). By default, the Moran process does not account for such positional effects. While optimization questions have been studied in the Moran and related models Even-Dar and Shapira (2007), the natural setting of positional advantage remains unexplored. In this setting, the question arises: to which positions (i.e., nodes) should we confer an advantage so as to maximize the fixation probability?
Our contributions. We define a positional variant of the Moran process, in which the mutant fitness advantage is only realized on a subset of nodes called active nodes and an associated optimization problem, fixation maximization (FM): given a graph , a mutant fitness advantage and a budget , choose a set of nodes to activate that maximize the fixation probability of mutants. We also study the problems and that correspond to FM at the natural limits and , respectively. On the negative side, we show that FM and are (Section 3.1). One common way to circumvent -hardness is to prove that the optimization function is submodular; however, we show that is not submodular when is finite (Section 3.2). On the positive side, we first show that the fixation probability on unidrected graphs admits a FPRAS, that is, it can be approximated to arbitrary precision in polynomial time (Section 4.1). We then show that, in contrast to finite , the function is submodular on undirected graphs and thus admits a polynomial-time constant-factor approximation (Section 4.2). Lastly, regarding the limit , we show that can be solved in polynomial time on any graph (Section 4.3). Overall, while FM is hard in general, we obtain tractability in both natural limits and .
Due to space constraints, some proofs are in the Appendix A.
We extend the standard Moran process to its positional variant and define our fixation maximization problem on it.
2.1 The Positional Moran Process
Structured populations. Standard evolutionary graph theory Nowak (2006) describes a population structure by a directed weighted graph (network) of nodes, where is a weight function over the edge set
that defines a probability distribution on each node, i.e., for each , . We require that is strongly connected when is projected to the support of the probability distribution of each node. Nodes represent sites (locations) and each site is occupied by a single agent. Agents are of two types — mutants and residents — and agents of the same type are indistinguishable from each other. For ease of presentation, we may refer to a node and its occupant agent interchangeably. In special cases, we will consider simple undirected graphs, meaning that (i) is symmetric and (ii) is uniform for every node .
Active node sets and fitness. Mutants and residents are differentiated in fitness, which in turn depends on a set of active nodes. Consider that mutants occupy the nodes in . The fitness of the agent at node is:
where is the mutant fitness advantage. Intuitively, fitness expresses the capacity of a type to spread in the network. The resident fitness is normalized to , while measures the relative competitive advantage conferred by the mutation. However, the mutant fitness advantage is not realized uniformly in the graph, but only at active nodes. The total fitness of the population is .
The positional Moran process. We introduce the positional Moran process as a discrete-time stochastic process , where each
is a random variable denoting the set of nodes ofoccupied by mutants. Initially, the whole population consists of residents; at time , a single mutant appears uniformly at random. That is, is given by for each . In each subsequent step, we obtain from by executing the following two stochastic events in succession.
Birth event: A single agent is selected for reproduction with probability proportional to its fitness, i.e., an agent on node is chosen with probability .
Death event: A neighbor of is chosen with probability ; its agent is replaced by a copy of the one at .
If is a mutant and a resident, then mutants spread to , hence ; likewise, if is a resident and a mutant, then ; otherwise and are of the same type, and thus .
Related Moran processes. Our positional Moran process generalizes the standard Moran process on graphs Nowak (2006) that forms the special case of (that is, the mutant fitness advantage holds on the whole graph). Conversely, it can be viewed a special case of the Moran process with two environments Kaveh, McAvoy, and Nowak (2019). It is different from models in which agent’s fitness depends on which agents occupy the neighboring nodes (so-called frequency dependent fitness) Huang and Traulsen (2010).
Fixation probability. If such that , we say that the mutants fixate in , otherwise we say they go extinct. As is strongly connected, for any initial mutant set , the process reaches a homogeneous state almost surely, i.e., . The fixation probability (FP) of such a set is . Thus, the FP of the mutants in with active node set and fitness advantage is:
For any fixed and , the function is continuous in
and infinitely differentiable: the underlying Markov chain is finite and its transition probabilities are rational functions in, so too is a rational function in , and since the process is defined for any , has no points of discontinuity for .
Computing the fixation probability. In the neutral setting where , for any graph Broom et al. (2010). When , the complexity of computing the fixation probability is unknown even when ; it is open whether the function can be computed in . For undirected and , the problem has a fully polynomial randomized approximation scheme (FPRAS) Díaz et al. (2014); Chatterjee, Ibsen-Jensen, and Nowak (2017); Ann Goldberg, Lapinskas, and Richerby (2020), as the expected absorption time of the process is polynomial and thus admits fast Monte-Carlo simulations. We also rely on Monte-Carlo simulations for computing , and show that the problem admits a FPRAS for any when is undirected.
2.2 Fixation Maximization via Node Activation
We study the optimization problem of fixation maximization in the positional Moran process. We focus on three variants based on different regimes of the mutant fitness advantage .
Fixation maximization. Consider a graph , a mutant fitness advantage and a budget . The problem Fixation Maximization for the Positional Moran Process calls to determine an active-node set of size that maximizes the fixation probability:
As we will argue, is monotonically increasing in , hence the condition can be replaced by . The associated decision problem is: Given a budget and a threshold , determine whether there exists an active-node set of size such that . As an illustration, consider a cycle graph on nodes and two strategies: one that activates “spaced” nodes, and another that activates “contiguous” nodes. As Fig. 2 shows, depending on and , either strategy could be better.
The regime of strong selection. Next, we consider FM in the limit . The corresponding fixation probability is:
As we show later, is monotonically increasing in , and since it is also bounded by 1, the above limit exists. The corresponding optimization problem asks for the active node set that maximizes :
The regime of weak selection. Further, we consider the problem in the limit . For small positive , different sets generally give different fixation probabilities , see Fig. 3. For fixed , we can view as a function of . By the Taylor expansion of , and given that , we have
where . Since lower-order terms tend to faster than as , on a sufficiently small neighborhood of , maximizing reduces to maximizing the derivative (up to lower order terms). The corresponding optimization problem asks for the set that maximizes :
As for all , our choice of maximizes the gain of fixation probability, i.e., the difference . The set of Section 2.2 guarantees a gain with relative approximation factor tending to as , i.e., we have the following (simple) lemma.
Monotonicity. Enlarging the set or the advantage increases the fixation probability. The proof generalizes the standard Moran process monotonicity Díaz et al. (2016).
Lemma 2 (Monotonicity).
Consider a graph , two subsets and two real numbers . Then .
3 Negative Results
We start by presenting hardness results for FM and .
3.1 Hardness of Fm and
We first show -hardness for FM and . This hardness persists even given an oracle that computes the fixation probability (or ) for a graph and a set of active nodes . Given a set of mutants , let
be the fixation probability of a set of mutants under strong selection, i.e., in the limit of large fitness advantage. The following lemma states that for undirected graphs, a single active mutant with infinite fitness guarantees fixation.
Let be an undirected graph, a set of active nodes, and a set of mutants. If then .
The intuition behind Lemma 3 is as follows: consider a node . Then, as , node is chosen for reproduction at a much higher rate than any resident node. Hence, in a typical evolutionary trajectory, the neighbors of node are occupied by mutants for most of the time and thus the individual at node is effectively guarded from any threat of becoming a resident. In contrast, the mutants are reasonably likely to spread to any other part of the graph. In combination, this allows us to argue that mutants fixate with high probability as . Notably, this argument relies on being undirected. As a simple counterexample, consider being a directed triangle and . Then, for the mutants to spread to , a mutant on must reproduce. Since , a mutant on is as likely to reproduce in each round as a resident on . If the latter happens prior to the former, node becomes resident, and, with no active mutants left, we have a constant (non-zero) probability of extinction.
Due to Lemma 3, if is an undirected regular graph with a vertex cover of size , under strong selection, we achieve the maximum fixation probability by activating a set of nodes that forms a vertex cover of . The key insight is that, if the initial mutant lands on some node , then the mutant fixates if and only if it manages to reproduce once before it is replaced, as all neighbors of are in and thus Lemma 3 applies. This is captured in the following lemma.
For any undirected regular graph and , iff is a vertex cover of (and if so, the equality holds).
First, note that due to Lemma 3, the fixation probability equals the probability that eventually a mutant is placed on an active node. Due to the uniform placement of the initial mutant, the probability that it lands on an active node is . Let be the set of nodes in that have at least one neighbor not in . Thus we can write:
where is the probability that an initial mutant occupying a non-active node whose neighbors are all in spreads to any of its neighbors, i.e., reproduces before any of its neighbors replace it; reproduces with probability , while its neighbors replace it with probability , where is the degree of , hence . If is a vertex cover of , then , hence Eq. 9 yields . If is not a vertex cover of , then ; for any node , since at least one of its neighbors is not in , is strictly smaller than the probability that the mutant on reproduces before it gets replaced, which, as we argued, is , hence Eq. 9 yields . ∎
The hardness for is a direct consequence of Lemma 4 and the fact that vertex cover is even on regular graphs (see, e.g., Feige (2003)). Moreover, as is a continuous function on , we also obtain hardness for FM (i.e., under finite ). We thus have the following theorem.
and are , even on undirected regular graphs.
Finally, we note that the hardness persists even given an oracle that computes the fixation probability (or ) given a graph and a set of active nodes .
3.2 Non-Submodularity for Fm
One standard way to circumvent -hardness is to show that the optimization function is submodular, which implies that a greedy algorithm offers constant-factor approximation guarantees Nemhauser, Wolsey, and Fisher (1978); Krause and Golovin (2014). Although in the next section we prove that is indeed submodular, unfortunately the same does not hold for , even for very simple graphs.
The function is not submodular.
Our proof is by means of a counter-example. Consider , that is, is a clique on nodes. For denote by the fixation probability on when of the 4 nodes are active and mutants have fitness advantage . For fixed , the value can be computed exactly by solving a system of linear equations (one for each configuration of mutants and residents). Solving the three systems using a computer algebra system, we obtain , and . Hence , thus submodularity is violated. ∎
For , the three systems can even be solved symbolically, in terms of . It turns out that the submodularity property is violated for , where . In contrast, for the submodularity property holds for all (the verification is straightforward, though tedious).
4 Positive Results
We now turn our attention to positive results for fixation maximization in the limits of strong and weak selection.
4.1 Approximating the Fixation Probability
We first focus on computing the fixation probability. Although the complexity of the problem is open even in the standard Moran process, when the underlying graph is undirected, the expected number of steps until the standard (non-positional) process terminates is polynomial in Díaz et al. (2014), which yields a FPRAS. A straightforward extension of that approach applies to the expected number of steps in the positional Moran process.
Given an undirected graph on nodes, a set and a real number , we have .
As a consequence, by simulating the process multiple times and reporting the proportion of runs that terminated with mutant fixation, we obtain a fully polynomial randomized approximation scheme (FPRAS) for the fixation probability.
Given a connected undirected graph , a set and a real number , the function admits a FPRAS.
4.2 Fixation Maximization under Strong Selection
Here we show that the function is submodular. As a consequence, we obtain a constant-factor approximation for in polynomial time.
Lemma 6 (Submodularity).
For any undirected graph , the function is submodular.
Consider any set of active nodes. Then the positional Moran process with mutant advantage is equivalent to the following process:
While , perform an update step as if .
If , terminate and report fixation.
Indeed, as long as no active node hosts a mutant, all individuals have the same (unit) fitness and the process is indistinguishable from the Moran process with . On the other hand, once any one active node receives a mutant, fixation happens with high probability by Lemma 3. All in all, the fixation probability in the limit can be computed by simulating the neutral process () until either (extinction) or (fixation).
To prove the submodularity of , it suffices to show that for any two sets we have:
Consider any fixed trajectory . We say that a subset is good with respect to , or simply good, if there exists a time point such that . It suffices to show that, for any , there are at least as many good sets among , as they are among and . To that end, we distinguish three cases based on how many of the sets , are good.
Both of them: Since is good, both and are good and we get .
One of them: If is good we conclude as before (we get ). Otherwise is good, hence at least one of , is good and we get .
None of them: We have . ∎
Given an undirected graph and integer , let be the solution to , and the set returned by a greedy maximization algorithm. Then .
Consequently, a greedy algorithm approximates the optimal fixation probability within a factor of
, provided it is equipped with an oracle that estimates the fixation probabilitygiven any set . For the class of undirected graphs, Lemma 5 and Theorem 3 yield a fully polynomial randomized greedy algorithm for with approximation guarantee .
4.3 Fixation Maximization under Weak Selection
Here we show that can be solved in polynomial time. Recall that given an active set , we can write , while calls to compute a set which maximizes the coefficient across all sets that satisfy . We build on the machinery developed in Allen et al. (2017).
In Lemma 7 below, we show that for a certain function . The bottleneck in computing is solving a system of linear equations, one for each pair of nodes. This can be done in matrix-multiplication time for Alman and Williams (2021). By virtue of the linearity of , the optimal set is then given by choosing the top- nodes in terms of . Thus we establish the following theorem.
Given a graph and some integer , the solution to can be computed in time, where is the matrix multiplication exponent.
For brevity, we denote the nodes of by and we denote by the probability that, if is selected for reproduction, then the offspring migrates to .
Let be a graph with nodes . Consider a function defined by , where is the solution to the linear system given by together with
and is the solution to the linear system
The intuition behind the quantities , , and from Lemma 7 is as follows: Consider the Moran process on with . Then for , the value is in fact the mutant fixation probability starting from the initial configuration Allen et al. (2021). Indeed, at any time step, the agent on will eventually fixate if (i) is not replaced by its neighbors and eventually fixates from the next step onward, or (ii) spreads to a neighbor and fixates from there. The first event happens with rate , while the second event happens with rate . (We note that for undirected graphs, the system has an explicit solution Broom et al. (2010).)
The values for also have intuitive meaning: They are equal to the expected total number of steps during the Moran process (with ) in which is mutant and is not, when starting from a random initial configuration . To sketch the idea behind this claim, suppose that in a single step, a node spreads to node (this happens with rate ). Then the event “ is mutant while is not” holds if and only if was mutant but was not, hence the product . Similar reasoning applies to the product . The term in the numerator comes out of the probability that the initial mutant lands on (in which case indeed is mutant and is not).
Finally, the function also has a natural interpretation: The contribution of an active node to the fixation probability grows vs. at to the extent that a mutant is likely to be chosen for reproduction, spread to a resident neighbor , and fixate from there; that is, by the total time that is mutant but a neighbor is not (), calibrated by the probability that, when chosen for reproduction, propagates to that neighbor (), who thereafter achieves mutant fixation (); this growth rate is summed over all neighbors and weighted by the rate at which reproduces when .
Here we report on an experimental evaluation of the proposed algorithms and some other heuristics. Our data set consists of 110 connected subgraphs of community graphs and social networks of the Stanford Network Analysis Project Leskovec and Krevl (2014). These subgraphs where chosen randomly, and varied in size between 20-170 nodes. Our evaluation is not aimed to be exhaustive, but rather to outline the practical performance of various heuristics, sometimes in relation to their theoretical guarantees.
In particular, we use 7 heuristics, each taking as input a graph, a budget , and (optionally) the mutant fitness advantage . The heuristics are motivated by our theoretical results and by the related fields of influence maximization and evolutionary graph theory.
Random: Simply choose nodes uniformly at random. This heuristic serves as a baseline.
High Degree: Choose the nodes with the largest degree.
Centrality: Choose the nodes with the largest betweenness centrality.
Temperature: The temperature of a node is defined as ; this heuristic chooses the nodes with the largest temperature.
Vertex Cover: Motivated by Lemma 4, this heuristic attempts to maximize the number of edges with at least one endpoint in . In particular, given a set , let
The heuristic greedily maximizes , i.e., we start with and perform update steps
Weak Selector: Choose the nodes that maximize using the (optimal) weak-selection method.
Lazy Greedy: A simple greedy algorithm starts with , and in each step chooses the node to add to that maximizes the objective function. For and this process requires to repeatedly evaluate (or ) for every node . This is done by simulating the process a large number of times (recall Lemma 5), and becomes a computational bottleneck when we require high precision. As a workaround we suggest a lazy variant of the greedy algorithm Minoux (1978), which is faster but requires submodularity of the objective function. In effect, this algorithm is a correct implementation of the greedy heuristic in the limit of strong selection (recall Lemma 6), while it may still perform well (but without guarantees) for finite .
In all cases ties are broken arbitrarily. Note that the heuristics vary in the amount of information they have about the graph and the invasion process. In particular, Random has no information whatsoever, High Degree only considers the direct neighbors of a node, Temperature considers direct and distance-2 neighbors of a node, Centrality and Vertex Cover consider the whole graph, while Weak Selector and Lazy Greedy are the only ones informed about the Moran process.
For each graph , we have chosen values of corresponding to , and of its nodes, and have evaluated the above heuristics in their ability to solve (strong selection) and (weak selection). We have not considered other values of as evaluating precisely via simulations requires many repetitions and becomes slow.
Strong selection. We start with the case of . Since different graphs and budgets generally result in highly variant fixation probabilities, in order to get an informative aggregate measure of each heuristic, we divide the fixation probability it obtains by the maximum fixation probability obtained across all heuristics for the same graph and budget. This normalization yields values in the interval , and makes comparison straightforward. Fig. 4 shows our results. We see that the Lazy Greedy algorithm performs best for small budgets (, ), while its performance is matched by Vertex Cover for budget . The high performance of Lazy Greedy is expected given its theoretical guarantee (Theorem 3
). We also observe that, apart from Weak Selector, the other heuristics perform quite well on many graphs, though there are several cases on which they fail, appearing as outlier points in the box plots. As expected, our baseline Random heuristic performs quite poorly; this result indicates that the node activation sethas a significant impact on the fixation probability. Finally, recall that Weak Selector is optimal for the regime of weak selection (). The fact that Weak Selector underperforms for strong selection () indicates an intricate relationship between fitness advantage and fixation probability.
Weak selection. We collect the results on weak selection in Fig. 5, using the normalization process above. Since the derivative satisfies (Section 4.3), Lazy Greedy is optimal and coincides with Weak Selector, and is thus omitted from the figure. Naturally, the Weak Selector always outperforms others, while the Random heuristic is weak. The other heuristics have mediocre performance, with a clear advantage of Centrality over the rest, which becomes clearer for larger budget values.
We introduced the positional Moran process and studied the associated fixation maximization problem. We have shown that the problem is in general, but becomes tractable in the limits of strong and weak selection. Our results only scratch the surface of this new process, as several interesting questions are open, such as: Does the strong-selection setting admit a better approximation than the one based on submodularity? Can the problem for finite be approximated within some constant-factor? Are there classes of graphs for which it becomes tractable?
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Appendix A Appendix
Fix pairs and . The key property is that, for any two configurations and any node , we have : Indeed, the inequality is strict if and only if . As a consequence, Lemma 5 from Díaz et al. (2016) applies and yields a desired coupling between the processes with parameters and . ∎
Due to Lemma 2, it suffices to prove the statement for when is a singleton set. Our proof is by a stochastic domination argument. In particular, we construct a simple Markov chain and a coupling between and the Moran process on from such that the probability of a random walk starting from a particular start state of has at least probability of getting absorbed in a final state .
Let , i.e., equals 2 plus the number of neighbors of . consists of the following set of states
The transition probability function is as follows, for constants that depend on but not on .
while for every state , the remaining probability mass is added as a self-loop probability .
We now sketch the coupling between and the positional Moran process on . The coupling guarantees the following correspondence between the states of and the configuration of the Moran process.
For the state , we have .
For the state , we have (i) , and (ii) for every , we have .
For each state , with , we have (i) , (ii) for every , we have , and (iii) .
We argue that the transition probabilities preserve the above correspondence.
While , the probability for a resident neighbor of to reproduce and place an offspring on is bounded by . On the other hand, the probability that reproduces in successive rounds until it places a mutant offspring to each of its neighbors can be exponentially small in but independent of , as, since , the fitness of is at least a fraction of the total population fitness regardless of the remaining nodes. It follows that, from a configuration with , the probability that turns resident before every neighbor of turns mutant is upper-bounded by , for some constant that depends on but not on .
Given any configuration with , the probability that a resident reproduces is at most . Similarly, the probability that a mutant reproduces and replaces a resident is at least . Thus, the probability that the latter event happens before the former event is lower-bounded by some constant that depends on but not on .
Thus, we have a coupling in which the probability that a random walk in starting at gets absorbed in is at least as large as . It is straightforward to verify that the probability of the former event tends to as . Thus , as desired. ∎
Following Lemma 4, the hardness for follows directly by reducing the question of whether a graph has a vertex cover of of size to the problem of whether can achieve fixation probability of at least . For the hardness of FM, recall that for any , the function is continuous on . A crude but slightly more detailed analysis in the proof of Lemma 4 shows that if we have an edge with , then the fixation probability from is
In turn, if is not a vertex cover of then
Due to the continuity of , there exists a large enough such that if has a vertex cover of size then . On the other hand, due to the monotonicity (Lemma 2), if has no vertex cover of size , then we have
Thus for such a fitness advantage , we can achieve fixation probability iff has a vertex cover of size . ∎
We follow the proof strategy of Theorem 11 from Díaz et al. (2014). Given a set of nodes occupied by mutants, consider a potential function defined by , where is the number of edges incident to . Note that , since and for each we have . Moreover, given a configuration at time-point , let be a random variable that measures the increase of the potential function in a single step. We make two claims:
: Let be a set of edges such that is occupied by a mutant and by a resident, and let be the total fitness of the population. In a single step, the set of nodes occupied by mutants changes either by a mutant replacing a resident neighbor, or by a resident replacing a mutant neighbor. Thus
where the last inequality holds since the fitness of any mutant is at least as large as that of any neighboring resident.
If then : If the population is not homogeneous then there exists at least one edge with and . With probability the agent at is selected for reproduction and with probability its offspring migrates to , which changes the potential function by .
By Item 1, the potential function gives rise to a sub-martingale. Moreover, the function is bounded by and by Item 2, we have . Thus, the standard martingale machinery of drift analysis applies Kötzing and Krejca (2019). Namely, the rescaled function satisfies the conditions of the upper additive drift theorem (with initial value at most and step-wise drift at least ). The expected time until termination is thus at most
Given a set , we write to indicate that (and otherwise). Given any configuration , we write to indicate that (and otherwise). We also define a function and we let
be its expected change in a single step of the Moran process with fitness advantage at nodes . Let . Then we can write
Finally, the derivative of at is
where in the last equality we used that and that the second sum vanishes, since all the terms that involve any fixed sum up to (due to Eq. 11).
Now consider the neutral positional Moran process () starting with for . For any and , consider an event defined as “at time we have and ” and let be the total expected time spent with being a mutant and not. Then by (McAvoy and Allen, 2021, Theorem 1) we have