Research Article | Open Access
Fixation Probabilities on Complete Star and Bipartite Digraphs
This paper exactly formulates the th-order fixation probabilities on complete star digraphs (CSDs), which extend the results from Broom and Rychtář (2008). By applying these probability formulae, some asymptotic properties on CBDs are analyzed, and certain CSDs are determined to be amplifiers of selection for arbitrary relative fitness larger than 1, while all the CSDs are proved to be amplifiers of selection for fixed relative fitness slightly larger than 1. A numerical method for fixed population structure (by solving a linear system) is developed to calculate the fixation probabilities on complete bipartite digraphs (CBDs), and some conjectures are finally given through simulations.
As a newly emerging branch of population dynamics, evolutionary graph theory studies the evolution of structured populations and explores the effect of population structure on evolutionary dynamics [1–6]. Primarily, evolutionary dynamics were investigated on homogeneous infinite populations however, populations in the real world are neither infinite nor homogeneously mixed . In recent years, there is a growing interest in the investigation of evolutionary dynamics on spatial structures or populations with certain nonhomogeneous structure [5, 7–10]. The study of evolutionary dynamics on graphs was popularized by Lieberman et al. . In this framework, the structure of a population is modeled by a weighted digraph on vertices , which are occupied by individuals (residents and mutants). The graph can describe the architecture of cells in a multicellular organism and represent spatial structure among animals or plants in an ecosystem . Moreover, graphs can also represent relationships in a social network of humans, which means that the dynamics describes cultural selection and the spread of new inventions and ideas. It is supposed that every individual of the population occupies a unique vertex of the graph. In each iteration, a random individual is chosen for reproduction with a probability proportional to its fitness, and the resulting offspring will occupy an adjacent vertex with the probability , which represents the weight on edge , if any. The intrinsic weights of a digraph are defined this way: for a vertex with outgoing edges, let if edge exists. In this paper, we focus our attention on the structures of this kind of digraphs with intrinsic weights.
Consider a homogeneous population on a weighted graph whose individuals all have baseline fitness 1. Each individual is chosen as the reproducing one with probability proportional to its fitness. Suppose new mutants with relative fitness (the new mutant called advantageous if , while disadvantageous if ) are introduced by placing them on randomly chosen vertices of the graph. These mutants have a certain chance of fixation, that is, to generate a lineage that takes over the population. It is an issue in population dynamics to find the th-order fixation probability, that is, the fixation probability of mutants, in a population [7, 11, 12].
An unstructured population can be modeled by a complete digraph where all edges have the same weight. The evolution of an unstructured population is often modeled by the Moran process, whose th-order fixation probability is where resident individuals have fitness 1, while mutants have relative fitness . The Moran process defines a balance between natural selection and random drift. Lieberman et al.  gave the approximation of the first-order fixation probability for stars with large size by Broom and Rychtář  obtained the exact average fixation probability for a mutant, given by , which recovers (1.2) because for large we derive , where represents the total size of a star except the central individual. In this paper, we further extend these results by deriving the explicit th-order fixation probability on complete star digraphs (there exists a center vertex which connects each other vertex by two directed edges) as follows:
The temperature of a vertex is defined as the sum of all weights that lead into that vertex, that is, the temperature of vertex is given by . If all the vertices have the same temperature, then a graph is isothermal. It is known that a structured population has the same first-order fixation probability as the corresponding Moran process if and only if the structure is an isothermal digraph . A structure is referred to as an amplifier of selection (resp., suppressor of selection) if the first-order fixation probability of one advantageous mutant on this structure is greater than (resp., less than) that for the corresponding Moran process. An important issue in evolutionary graph theory is to answer whether a given structure is an amplifier of selection or a suppressor of selection [4, 13, 14].
This paper proceeds as follows. In Section 2, we formulate the exact th-order fixation probability on complete star graphs, then apply it to answer whether a given complete star graph is an amplifier of selection or not. In Section 3, we obtain the first-order fixation probability on complete bipartite digraphs (all vertices are divided into two sets, and each vertex is connected to every vertex in the other set by a directed edge) by solving a linear system through numerical methods, and pose several conjectures based on the simulations results.
In this paper, we further develop the theory of evolutionary processes on graphs first developed by Lieberman et al. , by following the approach of proving analytical results for simple systems in a similar way to Broom and Rychtář . Some useful results advance the formal underpinnings of the modeling of evolution on graphs, since nearly no attempt is made to prove general theoretical results in studies of evolution on graphs.
2. On Complete Star Digraphs
In this section, we obtain explicit formulae for the th-order fixation probabilities on complete star digraphs and explore their properties and applications in nature and society.
2.1. Basic Concepts and Notations
A complete star digraph (CSD) is a digraph with a single central vertex such that (I) there exists an edge from the center vertex to each peripheral vertex; (II) there exists an edge from each peripheral vertex to the center vertex; (III) there exist no other edges. Therefore, a CSD of size , denoted by , is a digraph with vertex set and directed edge set . Figure 1 illustrates one CSD with intrinsic weights. Note that the term “CSD” means “CSD with intrinsic weights” in the sequel.
The th-order fixation probability on is denoted by , which represents the probability of the event that these mutants generate a lineage that takes over the whole population.
For technical reasons, we need the following notations in the sequel.
The configuration of a population on , at time , is depicted by a vector , where or 0 if a mutant occupies the central vertex or not, respectively, and denotes the number of mutants occupying vertices . The total number of mutants at time is denoted by , that is, . The probability of the event that the mutants finally fixate by starting with , is denoted by abbreviated to (without ambiguity).
2.2. Explicit Fixation Probability Formulae
Before exactly formulating the th-order fixation probability, the following two necessary lemmas are presented.
Lemma 2.1. Consider a CSD of size and denote by the relative fitness of a mutant. Then for , the difference equation system holds: where , and with .
Proof. This proof proceeds by calculating some conditional probabilities and by the total probability formula. Let , at time , represent the initial configuration of a population on , represent the new configuration after one step-time, and represent the initial total number of mutants. Then two cases are discussed with respect to .
Case 1. (The initial configuration ). In order to reduce the mutants by one, a resident individual must be selected to reproduce and, meanwhile the center mutant (staying at vertex 1) has to be chosen for death, whose conditional on the resident’s selection, happens with probability 1. Therefore, the conditional probability of going from to is given by
On the other hand, to increase by one, the center mutant has to be chosen for reproduction, and one resident individual connected to the center vertex must be selected to die. Thus the conditional probability of going from to is given by
By employing (2.3) and (2.4), the probability of the event that the configuration stays unchanged is given by
Case 2. (The initial configuration ). On the one hand, in order to increase by one, the center individual has to be replaced by a new mutant produced by another. Therefore, the probability of going from to is given by
On the other hand, to reduce by one, the center resident individual has to be chosen for reproduction and its offspring must replace one mutant. Therefore, the probability of going from to is given by
It follows from (2.6) and (2.7) that the probability of the configuration remaining unchanged is By the total probability formula, we derive that for ,
Plugging (2.3)–(2.8) into (2.9)-(2.10) and simplifying the results, we deduce where .
By substituting (2.11) into (2.12) and rearranging the terms, we get Equation on (2.1) follows by combining (2.11) and (2.13).
Lemma 2.2. Let and be given. Then for , the following hold
Proof. We solve the linear difference equation system (2.1) in view of the standard technique given in Elaydi . The matrix in (2.2) has two real eigenvalues .
Furthermore, and are eigenvectors of corresponding to , respectively. Denote and its inverse matrix Through the method of diagonalizing matrix , we make the variable change: and thus Therefore, Particularly, we know which implies Equation on (2.14) follows from substituting (2.21) into (2.19). The proof is complete.
Next, we present the main theorem of this section.
Theorem 2.3. Let and be given. Then for each , the explicit th-order fixation probability on is given by
Particularly, for the case , (2.22) becomes
2.3. Applications and Properties of the Fixation Probability
Theorem 2.3 allows us to calculate the fixation probability for a given CSD, so here we use it to answer whether a given CSD is an amplifier of selection or not. We have the following result.
Theorem 2.4. , and are all amplifiers of selection.
Proof. Combining (1.1) and (2.25) and simplifying, we find that it suffices to confirm that for and , the inequality holds, which is equivalent to , where
By certain algebraic calculations, it is easy to derive
By using Maple 10, we derive the polynomials for , as shown in Table 1.
It follows directly from Table 1 that holds for , which along with (2.28) leads to this result.
By (2.28) and Table 1, we guess that for and (2.27) have the following form: where each is a nonnegative integer. However, it seems very difficult to prove this inequality in its general form even by the mathematical softwares, since we confront complicated polynomials as increases, as shown in Table 1.
Another utility of Theorem 2.3 is to study the asymptotic properties of the fixation probabilities on complete star digraphs. For that purpose, we rewrite (2.22) in the following form: By letting and taking limits on both sides of (2.30) in , then as . Lieberman et al.  declared this result without rigorous argument. This statement tells us that for sufficiently large , is an amplifier of selection because .
Figure 2 shows how the first-order fixation probability goes to the limit. For comparative purposes, the first-order fixation probability for the Moran process is also given in this figure. By Theorem 2.4 and a close look at Figure 2, we pose the conjecture: let . Then, is an amplifier of selection.
In the proof of Theorem 2.3, we argue that the conjecture holds for by computing some polynomials; however, we confront complex polynomials when becomes large. Thus, it seems very difficult to generally prove the conjecture. Fortunately, here we derive the following weak but useful result through rigorous reasoning.
Theorem 2.5. Let , depending on , be sufficiently small. Then for any , is an amplifier of selection.
Proof. It suffices to prove that for the following inequality holds:
which is equivalent to by applying (1.1) and (2.25). We prove this as follows.
Let , then It follows from (2.27) that whose first- and second-order derivatives in are, respectively, given by Through certain calculations, we derive which imply It follows from the Taylor’s theorem and the fact that Therefore, there exists a small enough such that the inequality holds for arbitrary . The proof is complete.
Theorem 2.5 shows that no matter the size of a complete star digraph, it is always an amplifier of selection if the mutant’s relative fitness is slightly larger than 1.
In the following, let us examine the asymptotic behavior of provided that depends on . If and is an amplifier of selection, then by taking limits on (2.30), we have (I)if , then ; (II) if , then . The first assertion is interesting because it demonstrates that no matter how slowly the initial number of mutants increases (e.g., proportional to ) the corresponding fixation probability will always approach one.
When depends on , one may imagine that the asymptotic behavior of would become much more complex. For example, given and suppose . Then we have the assertions: (I) if , then ; (II) if , then . Note that the first assertion follows directly from (2.30). Considering the second assertion, assume that , then it follows from (2.30) that . Finally, in the case , the second assertion follows naturally.
Figure 3 shows that how the th-order fixation probability changes as the increasing number mutants .
3. Fixation Probabilities on Complete Bipartite Digraphs
This section gives a recursive equation regarding the fixation probabilities on complete bipartite digraphs and obtains some results through a numerical method by simulation.
3.1. Basic Concepts and Notations
A complete bipartite digraph (CBD) is a digraph whose vertices are partitioned into two partite sets so that there is no edge connecting any two vertices in the same partite set, and there is an directed edge connecting each vertex in one partite set to each vertex in the other partite set . A CBD denoted by (here and represent the sizes of two partite sets, respectively, and due to symmetry, let ) is a digraph with vertex set ( is the total size) and edge set . A CBD is balanced if its two partite sets have the same size, that is, , otherwise it is unbalanced. The unbalance degree of can be measured by . Figure 4 depicts two CBDs (the second one’s unbalance degree is 1). The term “CBD” implies “CBD with intrinsic weights” in the sequel.
(b) with intrinsic weights
Without loss of generality, we suppose here that . Obviously, when , the complete bipartite digraph is just the complete star digraph . A CBD is an isothermal digraph if and only if it is balanced, thus the first-order fixation probability on a balanced CBD is already known by the isothermal theorem.
For technical reasons, the following notations are needed in the sequel.
Consider a homogeneous population on a CBD where all individuals have fitness 1. Suppose that new mutants with relative fitness are introduced by placing them on randomly chosen vertices. Let represent the th-order fixation probability, that is, the probability of the event that these mutants generate a lineage that takes over the population.
At time , the configuration of a population on is described by a vector , where and represent the number of mutants staying at vertices and vertices , respectively.
Let represent the total number of mutants at time , that is, , and (without ambiguity, ) represent the probability of the event that, starting from , the mutants finally fixate.
3.2. A Numerical Method
Here we will give an approach for calculating the fixation probability on a CBD and thus three theorems are established as follows.
Theorem 3.1. Consider a CBD with . Then, we have satisfying the recursive equation: where , and ,, are boundary conditions.
Proof. Let , at time , represent the initial configuration of a population on and represent the new configuration after one step-time. Thus we have . It is simple to calculate the following conditional probabilities:
Therefore, the probability that the configuration does not change is given by By the total probability formula, we derive Note that and , are meaningless, thus we assume , , . Then (3.1) follows immediately by plugging (3.2) and (3.3) into (3.4) and simplifying. The proof is complete.
Theorem 3.2. For , the following holds
Sketch. The randomness of the initial configuration of mutants gives
where represents the number of mutants staying at vertices .
Equation on (3.5) follows from this equation and the following total probability formula:
From the above theorems, calculating reduces to solving the difference equation system (3.1) with variables, because are known. It is known  that solving a linear system in variables with the Gaussian elimination method would take time. So, we have the result.
Theorem 3.3. is numerically obtainable in time.
In the best case is computed in time, whereas in the worst case consumes up to calculation time.
3.3. Numerical Experiments
Based on the theory established above, this section aims to experimentally reveal how a CBD, affects the first-order fixation probability (3.6) of an advantageous mutant. The experiments are done using Matlab.
At first sight, depends on three parameters. However, we will focus our attention on how is affected by and because these two parameters characterize the CBD. Therefore, we will fix the fitness in the subsequent experiments.
Experiment 1. To verify the theory’s correctness and guarantee the effectiveness of the following experiments, this experiment illustrates (see Figure 5) that the fixation probability of balanced CBDs is equal to that of the corresponding Moran process. Here, the isothermal theorem of Lieberman et al.  is referenced.
Experiment 2. Given and , we wish to know how depends on and . For the purpose, we evaluate for each pair