Fixation Probabilities on Complete Star and Bipartite Digraphs

Abstract

This paper exactly formulates the 𝑘th-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.

1. Introduction

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 [2]. 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. [4]. In this framework, the structure of a population is modeled by a weighted digraph on vertices 1,2,…,𝑁, 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 [8]. 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 𝑖 is chosen for reproduction with a probability proportional to its fitness, and the resulting offspring will occupy an adjacent vertex 𝑗 with the probability 𝑤_𝑖𝑗, which represents the weight on edge ⟨𝑖,𝑗⟩, if any. The intrinsic weights of a digraph are defined this way: for a vertex 𝑖 with 𝑙 outgoing edges, let 𝑤_𝑖𝑗=1/𝑙 if edge {𝑖,𝑗} 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 𝑘 new mutants with relative fitness 𝑟 (the new mutant called advantageous if 𝑟>1, while disadvantageous if 0<𝑟<1) are introduced by placing them on 𝑘 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 𝑘th-order fixation probability, that is, the fixation probability of 𝑘 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 𝑘th-order fixation probability is 𝜌_𝑘^(𝑀)(𝑁;𝑟)=1−1/𝑟^𝑘1−1/𝑟^𝑁, (1.1) where resident individuals have fitness 1, while mutants have relative fitness 𝑟. The Moran process defines a balance between natural selection and random drift. Lieberman et al. [4] gave the approximation of the first-order fixation probability for stars with large size 𝑁 by 𝑃=1−1/𝑟²1−1/𝑟^2𝑁. (1.2) Broom and Rychtář [7] obtained the exact average fixation probability for a mutant, given by 𝜚=[𝑛²𝑟/(𝑛𝑟+1)+𝑟/(𝑛+𝑟)]/[(𝑛+1)[1+𝑛/(𝑛+𝑟)∑^𝑛−1_𝑗=1((𝑛+𝑟)/(𝑟(𝑛𝑟+1)))^𝑗]], which recovers (1.2) because for large 𝑁 we derive 𝜚≈1/(1+∑^𝑛−1_𝑗=11/𝑟^2𝑗)=(1−1/𝑟²)/(1−1/𝑟^2𝑛), where 𝑛 represents the total size of a star except the central individual. In this paper, we further extend these results by deriving the explicit 𝑘th-order fixation probability on complete star digraphs (there exists a center vertex which connects each other vertex by two directed edges) as follows: 𝑛𝜚=²𝑟/(𝑛𝑟+1)+𝑟/(𝑛+𝑟)(𝑛+1)1+𝑛/(𝑛+𝑟)∑^𝑛−1_𝑗=1((𝑛+𝑟)/𝑟(𝑛𝑟+1))^𝑗. (1.3)

The temperature of a vertex is defined as the sum of all weights that lead into that vertex, that is, the temperature of vertex 𝑗 is given by 𝑇_𝑗=∑^𝑁_𝑗=1𝑤_𝑖𝑗[8]. 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 [8]. 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 𝑘th-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. [4], by following the approach of proving analytical results for simple systems in a similar way to Broom and Rychtář [7]. 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 𝑘th-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 𝑁, denoted by 𝒮_𝑁, is a digraph with vertex set 𝒱={1,2,…,𝑁} and directed edge set ℰ={⟨1,𝑗⟩,⟨𝑗,1⟩∣𝑗=2,…,𝑁}. Figure 1 illustrates one CSD with intrinsic weights. Note that the term “CSD” means “CSD with intrinsic weights” in the sequel.

Figure 1 

One complete star digraph 𝒮6 with intrinsic weights, the central vertex is indexed 1.

The 𝑘th-order fixation probability on 𝒮_𝑁 is denoted by 𝜌_𝑘(𝑁;𝑟), which represents the probability of the event that these 𝑘 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 𝒮_𝑁, at time 𝑡, is depicted by a vector Θ(𝑡)=(𝜃₁(𝑡),𝜃₂(𝑡)), where 𝜃₁(𝑡)=1 or 0 if a mutant occupies the central vertex or not, respectively, and 𝜃₂(𝑡) denotes the number of mutants occupying vertices 2,…,𝑁. The total number of mutants at time 𝑡 is denoted by 𝜃(𝑡), that is, 𝜃(𝑡)=𝜃₁(𝑡)+𝜃₂(𝑡). The probability of the event that the mutants finally fixate by starting with Θ(0)=(𝑘₁,𝑘₂), is denoted by 𝜌_𝑘1,𝑘2(𝑁;𝑟) abbreviated to 𝜌_𝑘1,𝑘2 (without ambiguity).

2.2. Explicit Fixation Probability Formulae

Before exactly formulating the 𝑘th-order fixation probability, the following two necessary lemmas are presented.

Lemma 2.1. Consider a CSD of size 𝑁≥3 and denote by 𝑟 the relative fitness of a mutant. Then for 𝑘=1,…,𝑁−1, the difference equation system holds: 𝐩_𝑘=𝐀𝐩_𝑘−1 (2.1) where 𝐩_𝑘=(𝜌_1,𝑘,𝜌_0,𝑘)^𝑇∈ℝ², and 𝐀=𝛼1−𝛼𝛼𝛽1−𝛼𝛽, (2.2) with 𝛼∶=1+(𝑁−1)/𝑟,𝛽∶=𝑟(𝑁−1)/[𝑟(𝑁−1)+1].

Proof. This proof proceeds by calculating some conditional probabilities and by the total probability formula. Let Θ(0), at time 𝑡=0, represent the initial configuration of a population on 𝒮_𝑁, Θ(1) represent the new configuration after one step-time, and 𝑘=𝜃(0)=𝜃₁(0)+𝜃₂(0) represent the initial total number of mutants. Then two cases are discussed with respect to Θ(0).
Case 1. (The initial configuration Θ(0)=(1,𝑘−1)). In order to reduce the 𝑘 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 Θ(0)=(1,𝑘−1) to Θ(1)=(0,𝑘−1) is given by {Θ(1)=()(0)=(=()𝐏𝐫0,𝑘−1∣Θ1,𝑘−1)}𝑁−𝑘().𝑟𝑘+𝑁−𝑘 (2.3)
On the other hand, to increase 𝜔 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 Θ(0)=(1,𝑘−1) to Θ(1)=(1,𝑘) is given by {Θ(1)=()(0)=(=𝑟()𝐏𝐫1,𝑘∣Θ1,𝑘−1)}𝑁−𝑘.[(𝑟𝑘+𝑁−𝑘)(𝑁−1)] (2.4)
By employing (2.3) and (2.4), the probability of the event that the configuration stays unchanged is given by {Θ(1)=()(0)=(()𝐏𝐫1,𝑘−1∣Θ1,𝑘−1)}=1−𝑁−𝑘()−𝑟()𝑟𝑘+𝑁−𝑘𝑁−𝑘.[(𝑟𝑘+𝑁−𝑘)(𝑁−1)] (2.5)
Case 2. (The initial configuration Θ(0)=(0,𝑘)). On the one hand, in order to increase 𝑘 by one, the center individual has to be replaced by a new mutant produced by another. Therefore, the probability of going from Θ(0)=(0,𝑘) to Θ(1)=(1,𝑘) is given by 𝐏𝐫{Θ(1)=(1,𝑘)∣Θ(0)=(0,𝑘)}=𝑟𝑘().𝑟𝑘+𝑁−𝑘 (2.6)
On the other hand, to reduce 𝑘 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 Θ(0)=(0,𝑘) to Θ(1)=(0,𝑘−1) is given by {Θ(1)=()(0)=(=𝑘𝐏𝐫0,𝑘−1∣Θ0,𝑘)}.[(𝑟𝑘+𝑁−𝑘)(𝑁−1)] (2.7)
It follows from (2.6) and (2.7) that the probability of the configuration remaining unchanged is 𝐏𝐫{Θ(1)=(0,𝑘)∣Θ(0)=(0,𝑘)}=1−𝑟𝑘()−𝑘𝑟𝑘+𝑁−𝑘.[(𝑟𝑘+𝑁−𝑘)(𝑁−1)] (2.8) By the total probability formula, we derive that for 𝑘=1,…,𝑁−1, 𝜌_1,𝑘−1=𝐏𝐫{Θ(1)=(0,𝑘−1)∣Θ(0)=(1,𝑘−1)}𝜌_0,𝑘−1+𝐏𝐫{Θ(1)=(1,𝑘)∣Θ(0)=(1,𝑘−1)}𝜌_1,𝑘+𝐏𝐫{Θ(1)=(1,𝑘−1)∣Θ(0)=(1,𝑘−1)}𝜌_1,𝑘−1,𝜌_0,𝑘=𝐏𝐫{Θ(1)=(1,𝑘)∣Θ(0)=(0,𝑘)}𝜌_1,𝑘+𝐏𝐫{Θ(1)=(0,𝑘−1)∣Θ(0)=(0,𝑘)}𝜌_0,𝑘−1+𝐏𝐫{Θ(1)=(0,𝑘)∣Θ(0)=(0,𝑘)}𝜌_0,𝑘. (2.9)(2.10)
Plugging (2.3)–(2.8) into (2.9)-(2.10) and simplifying the results, we deduce 𝜌_1,𝑘=𝛼𝜌_1,𝑘−1+(1−𝛼)𝜌_0,𝑘−1,𝜌_0,𝑘=𝛽𝜌_1,𝑘+1−𝛽𝜌_0,𝑘−1, (2.11)(2.12) where 𝛼∶=1+(𝑁−1)/𝑟,𝛽∶=𝑟(𝑁−1)/[𝑟(𝑁−1)+1].
By substituting (2.11) into (2.12) and rearranging the terms, we get 𝜌_0,𝑘=𝛼𝛽𝜌_1,𝑘−1+1−𝛼𝛽𝜌_0,𝑘−1. (2.13) Equation on (2.1) follows by combining (2.11) and (2.13).

Lemma 2.2. Let 𝑁 and 𝑟 be given. Then for 1≤𝑘≤𝑁−1, the following hold 𝜌_1,𝑘−1=𝛽−(𝛼−1)𝛼^𝑘−21−𝛽^𝑘−1𝛽−(𝛼−1)𝛼^𝑁−21−𝛽^𝑁−1,𝜌_0,𝑘=𝛽1−𝛼^𝑘1−𝛽^𝑘𝛽−(𝛼−1)𝛼^𝑁−21−𝛽^𝑁−1. (2.14)

Proof. We solve the linear difference equation system (2.1) in view of the standard technique given in Elaydi [15]. The matrix 𝐀 in (2.2) has two real eigenvalues 𝜆₁=1,𝜆₂=𝛼(1−𝛽).
Furthermore, 𝜂₁=(1,1)^𝑇 and 𝜂₂=(𝛼−1,𝛼𝛽)^𝑇 are eigenvectors of 𝐀 corresponding to 𝜆₁,𝜆₂, respectively. Denote 𝐃=1𝛼−11𝛼𝛽, (2.15) and its inverse matrix 𝐃⁻¹=11−𝛼+𝛼𝛽𝛼𝛽1−𝛼−11. (2.16) Through the method of diagonalizing matrix 𝐀, we make the variable change: 𝜌_1,𝑘𝜌_0,𝑘𝑥=𝐃^𝑘𝑦_𝑘, (2.17) and thus 𝑥_𝑘𝑦_𝑘=𝐃⁻¹𝑥𝐀𝐃^𝑘−1𝑦_𝑘−1=10𝑥0𝛼1−𝛽𝑦^𝑘−1_𝑘−1=100𝛼^𝑘1−𝛽^𝑘𝑥₀𝑦₀. (2.18) Therefore, 𝜌_1,𝑘𝜌_0,𝑘=𝐃100𝛼^𝑘1−𝛽^𝑘𝐃⁻¹𝜌_1,0𝜌_0,0=𝜌_1,0()𝛼1−𝛼+𝛼𝛽𝛼𝛽−𝛼−1^𝑘1−𝛽^𝑘𝛼𝛽−𝛼^𝑘+1𝛽1−𝛽^𝑘. (2.19) Particularly, we know 1=𝜌_1,𝑁−1=𝜌_1,0×()𝛼1−𝛼+𝛼𝛽𝛼𝛽−𝛼−1^𝑁−11−𝛽^𝑁−1, (2.20) which implies 𝜌_1,0=1−𝛼+𝛼𝛽𝛼𝛽−(𝛼−1)𝛼^𝑁−11−𝛽^𝑁−1. (2.21) 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 𝑟 and 𝑁>2 be given. Then for each 𝑘=1,…,𝑁−1, the explicit 𝑘th-order fixation probability on 𝒮_𝑁 is given by 𝜌_𝑘()=𝑟𝑁;𝑟^𝑁[𝑟(𝑁−1)+1]^{𝑁−𝑘−2}𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−2−(𝑁−1+𝑟)^𝑁−2×(𝑟(𝑁−1)+1)^𝑘−𝑘𝑟^𝑘(𝑁−2)𝑟²−1(𝑁−1+𝑟)^𝑘−2−(𝑁−1+𝑟)𝑟^𝑘. (2.22)

Proof. In light of (2.21), we derive 𝜌_𝑘()=𝑘𝑁;𝑟𝑁𝜌_1,𝑘−1+1−𝑘𝑁𝜌_0,𝑘=𝑘/𝑁𝛼^𝑘−21−𝛽^𝑘−1𝛼²𝛽1−𝛽−(𝛼−1)+𝛽1−𝛼^𝑘1−𝛽^𝑘𝛽−(𝛼−1)𝛼^𝑁−21−𝛽^𝑁−1. (2.23)
By substituting 𝛼 and 𝛽 (see Lemma 2.1) into this equation, we have 𝜌_𝑘(𝑁;𝑟)=𝑘/𝑁[1+(𝑁−1)/𝑟]^𝑘−2[1/(𝑟(𝑁−1)+1)]^𝑘−1(1+(𝑁−1)/𝑟)²(𝑟(/(𝑟())𝑁−1))𝑁−1+1²−()()/(𝑟())(())𝑁−1/𝑟+𝑟𝑁−1𝑁−1+11−1+𝑁−1/𝑟^𝑘/(𝑟(𝑁−1)+1)^𝑘/𝑟(/(𝑟())−()[()](𝑁−1))𝑁−1+1𝑁−1/𝑟1+𝑁−1/𝑟^𝑁−2/(𝑟(𝑁−1)+1)^𝑁−1=𝑟^𝑁−1(𝑟(𝑁−1)+1)^𝑁−1𝑘(𝑁−1)𝑟²−1(2−𝑁)(𝑁+𝑟−1)^𝑘−2/𝑟^𝑘−1(𝑟(𝑁−1)+1)^𝑘+1+𝑟(𝑁−1)/(𝑟(𝑁−1)+1)^𝑘+1(𝑟(𝑁−1)+1)^𝑘−((𝑟+𝑁−1)/𝑟)^𝑘/(𝑁−1)𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−2−(𝑟+𝑁−1)^𝑁−2=𝑟^𝑁[𝑟(𝑁−1)+1]^{𝑁−𝑘−2}/𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−2−(𝑁−1+𝑟)^𝑁−2×(𝑟(𝑁−1)+1)^𝑘−𝑘/𝑟^𝑘(𝑁−2)𝑟²−1(𝑁−1+𝑟)^𝑘−2−((𝑁−1+𝑟)/𝑟)^𝑘. (2.24) The proof is complete.

Particularly, for the case 𝑘=1, (2.22) becomes 𝜌₁()=𝑟𝑁;𝑟^𝑁−1𝑟²−1(𝑟(𝑁−1)+1)^𝑁−3𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−2−(𝑁−1+𝑟)^𝑁−2×𝑁²+(𝑟−3)(𝑁−1)().𝑁−1+𝑟 (2.25)

Here we point out that (2.25) is just the same as the formula 𝜚=[𝑛²𝑟/(𝑛𝑟+1)+𝑟/(𝑛+𝑟)]/[(𝑛+1)[1+𝑛/(𝑛+𝑟)∑^𝑛−1_𝑗=1((𝑛+𝑟)/(𝑟(𝑛𝑟+1)))^𝑗]], which is given in Broom and Rychtář [7], because 𝑛𝜚=²𝑟/(𝑛𝑟+1)+𝑟/(𝑛+𝑟)(𝑛+1)1+𝑛/(𝑛+𝑟)∑^𝑛−1_𝑗=1(𝑛+𝑟)/𝑛𝑟²+𝑟^𝑗=𝑛²𝑟/(𝑛𝑟+1)+𝑟/(𝑛+𝑟)(𝑛+1)1+𝑟^𝑛−1(𝑛𝑟+1)^𝑛−1−(𝑛+𝑟)^𝑛−1/𝑟²−1(𝑟(𝑛𝑟+1))^𝑛−1=𝑛(𝑛+1)𝑟²+(𝑛+1)𝑛²−𝑛+1𝑟𝑟²−1[𝑟(𝑛𝑟+1)]^𝑛−1(𝑛𝑟+1)(𝑛+𝑟)(𝑛+1)𝑟^𝑛+1(𝑛𝑟+1)^𝑛−1−(𝑛+𝑟)^𝑛−1=𝑟^𝑁−1𝑟²−1[𝑟(𝑁−1)+1]^𝑁−3𝑟^𝑁[𝑟(𝑁−1)+1]^𝑁−2−[𝑁−1+𝑟]^N−2×𝑁²+(𝑟−3)(𝑁−1)(𝑁−1+𝑟)=𝜌₁(𝑁;𝑟), (2.26) where 𝑛 represents the number of all individuals except the central one, that is,  𝑛=𝑁−1.

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. 𝒮₃,𝒮₄,𝒮₅,𝒮₆,𝒮₇,𝒮₈,𝒮₉, and 𝒮₁₀ are all amplifiers of selection.

Proof. Combining (1.1) and (2.25) and simplifying, we find that it suffices to confirm that for 𝑟>1 and 𝑁=3,4,…,10, the inequality 𝜌₁(𝑁;𝑟)>𝜌₁^(𝑀)(𝑁;𝑟) holds, which is equivalent to 𝐽(𝑁,𝑟)>0, where 𝐽(𝑁,𝑟)∶=(𝑟+1)𝑟^𝑁−1𝑁²+(𝑟−3)(𝑁−1),𝑟(𝑁−1)+1^𝑁−3−(𝑁−1+𝑟)𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−2−(𝑁−1+𝑟)^𝑁−2. (2.27) By certain algebraic calculations, it is easy to derive 𝐽(3,𝑟)=𝑟³−𝑟²−𝑟+1=(𝑟−1)²()𝐽()𝑟+1>0,4,𝑟=12𝑟⁵+4𝑟⁴−8𝑟³−24𝑟²−4𝑟+20=(𝑟−1)²12𝑟³+28𝑟²𝐽()+36𝑟+20>0,5,𝑟=144𝑟⁷+72𝑟⁶+9𝑟⁵−63𝑟⁴−288𝑟³−252𝑟²=()+135𝑟+243𝑟−1²144𝑟⁵+360𝑟⁴+585𝑟³+747𝑟²+621𝑟+243>0. (2.28) By using Maple 10, we derive the polynomials 𝐽(𝑁,𝑟)/(𝑟−1)² for 𝑁=6,7,…,10, as shown in Table 1.
It follows directly from Table 1 that 𝐽(𝑁,𝑟)>0 holds for 𝑁=6,7,8,9,10, which along with (2.28) leads to this result.
By (2.28) and Table 1, we guess that for 𝑁≥3 and (2.27) have the following form: 𝐽(𝑁,𝑟)=(𝑟−1)^22𝑁−5_𝑗=0𝑎_𝑗𝑟^𝑗,𝑁≥3, (2.29) where each 𝑎_𝑗(𝑗=0,1,…,2𝑁−5) 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 𝑁 increases, as shown in Table 1.

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: 𝜌_𝑘()=1𝑁;𝑟1+(𝑟(𝑁−1))^−𝑘×11+(𝑟(𝑁−1))^𝑘−1𝑟^2𝑘𝑟1+(𝑁−1)^𝑘−𝑘(𝑁−2)𝑟²−1(𝑁−1)²𝑟^2𝑘𝑟1+𝑁−1^𝑘−2/11−𝑟^2𝑁−21+𝑟−1𝑟/1𝑁−1+𝑟^𝑁−2. (2.30) By letting 𝑟>1 and taking limits on both sides of (2.30) in 𝑁, then 𝜌_𝑘(𝑁;𝑟)→1−1/𝑟^2𝑘 as 𝑁→∞. Lieberman et al. [4] declared this result without rigorous argument. This statement tells us that for sufficiently large 𝑁, 𝒮_𝑁 is an amplifier of selection because lim_𝑁→∞𝜌₁(𝑁;𝑟)=1−1/𝑟²>1−1/𝑟=lim_𝑁→∞𝜌₁^(𝑀)(𝑁;𝑟).

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 𝑟>1. Then, 𝒮_𝑁(𝑁≥3) is an amplifier of selection.

(a) 𝑟=1.05

(b) 𝑟=1.3

(c) 𝑟=1.9

(d) 𝑟=2.7

Figure 2 

First-order fixation probabilities of CSDs and Moran process with 𝑁 by fixing relative fitness 𝑟.

In the proof of Theorem 2.3, we argue that the conjecture holds for 3≤𝑁≤10 by computing some polynomials; however, we confront complex polynomials when 𝑁 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 𝜏(𝑁)>0, depending on 𝑁, be sufficiently small. Then for any 𝑟∈(1,1+𝜏(𝑁)), 𝒮_𝑁(𝑁≥3) is an amplifier of selection.

Proof. It suffices to prove that for 𝑟∈(1,1+𝜏(𝑁)) the following inequality holds: 𝜌₁(𝑁;𝑟)>𝜌₁^(𝑀)(𝑁;𝑟), (2.31) which is equivalent to 𝜌₁(𝑁;𝑟)>𝜌₁^(𝑀)(𝑁;𝑟) by applying (1.1) and (2.25). We prove this as follows.
Let 𝑓(𝑟)=(𝑟+1)[𝑁²+(𝑁−1)(𝑟−3)][𝑟(𝑁−1)+1]^𝑁−3, then 𝑓^′(𝑟)=𝑓(𝑟)+(𝑁−1)(𝑟+1)[𝑟(𝑁−1)+1]^𝑁−3+(𝑁−3)(𝑁−1)(𝑟+1)𝑁²+(𝑁−1)(𝑟−3)[𝑟(𝑁−1)+1]^𝑁−4. (2.32) It follows from (2.27) that 𝐽(𝑁,𝑟)=𝑟^𝑁−1𝑓(𝑟)−(𝑁−1+𝑟)𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−2−(𝑁−1+𝑟)^𝑁−2 (2.33) whose first- and second-order derivatives in 𝑟 are, respectively, given by 𝜕𝐽(𝑁,𝑟)𝜕𝑟=𝑁𝑟^𝑁−1𝑓(𝑟)+𝑟^𝑁−1𝑓^′(𝑟)−𝑟^𝑁[𝑟(𝑁−1)+1]^𝑁−2−𝑁(𝑁−1+𝑟)𝑟^𝑁−1[𝑟(𝑁−1)+1]^𝑁−2+(𝑁−1)(𝑁−1+𝑟)^𝑁−2−(𝑁−2)(𝑁−1)(𝑁−1+𝑟)𝑟^𝑁[𝑟(𝑁−1)+1]^𝑁−3,𝜕²𝐽(𝑁,𝑟)𝜕𝑟²=𝑁(𝑁−1)𝑟^𝑁−2𝑓(𝑟)+2𝑁𝑟^𝑁−1𝑓^′(𝑟)+𝑟^𝑁−1𝑓^′′(𝑟)−2𝑁𝑟^𝑁−1[𝑟(𝑁−1)+1]^𝑁−2−𝑁(𝑁−1)(𝑁−1+𝑟)𝑟²(𝑁−1)+𝑟^𝑁−2−(𝑁−2)(𝑁−1)𝑟²𝑟²(𝑁−1)+𝑟^𝑁−3[()+()𝑟]+()2𝑁𝑁−12𝑁+1𝑁−2)(𝑁−1)(𝑁−1+𝑟^𝑁−3−𝑟^𝑁(𝑟(𝑁−1)+1)^𝑁−3−(𝑁−3)(𝑁−2)(𝑁−1)²(𝑁−1+𝑟)𝑟^𝑁[𝑟(𝑁−1)+1]^𝑁−4. (2.34) Through certain calculations, we derive 𝑓(1)=2𝑁²−2𝑁+2𝑁^𝑁−3,𝑓^′(1)=𝑁^𝑁−1+2(𝑁−3)(𝑁−1)𝑁²−2𝑁+2𝑁^𝑁−4, (2.35) which imply 𝜕𝐽(𝑁,𝑟)∣𝜕𝑟^𝑟=1𝜕=0,²𝐽(𝑁,𝑟)𝜕𝑟²∣_𝑟=1=2(𝑁−1)(𝑁−2)³𝑁^𝑁−3>0. (2.36) It follows from the Taylor’s theorem and the fact 𝐽(𝑁,1)=0 that 𝐽()()+()𝑁,𝑟=𝐽𝑁,1𝜕𝐽𝑁,𝑟∣𝜕𝑟^𝑟=1()+1𝑟−1𝜕2!²𝐽(𝑁,𝑟)𝜕𝑟²∣_𝑟=1(𝑟−1)²+𝑂(𝑟−1)²=(𝑁−1)(𝑁−2)³𝑁^𝑁−3(𝑟−1)²+𝑂(𝑟−1)²=(𝑟−1)²(𝑁−1)(𝑁−2)³𝑁^𝑁−3+𝑂(𝑟−1). (2.37) Therefore, there exists a small enough 𝜏(𝑁)>0 such that the inequality (𝑁−1)(𝑁−2)³𝑁^𝑁−3+𝑂(𝑟−1)>0 holds for arbitrary 𝑟∈(1,1+𝜏(𝑁)). 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 𝜌_𝑘(𝑁;𝑟) provided that 𝑘=𝑘_𝑁 depends on 𝑁. If 𝑟>1 and 𝑆_𝑁(𝑁≥3) is an amplifier of selection, then by taking limits on (2.30), we have (I)if 𝑘_𝑁→+∞, then 𝜌_𝑘𝑁(𝑁;𝑟)→1; (II) if 𝑘_𝑁→𝑘^∗<+∞, then 𝜌_𝑘𝑁(𝑁;𝑟)→1−1/𝑟^2𝑘∗. The first assertion is interesting because it demonstrates that no matter how slowly the initial number of mutants increases (e.g., proportional to [lnln𝑁]) the corresponding fixation probability will always approach one.

When 𝑟=𝑟_𝑁 depends on 𝑁, one may imagine that the asymptotic behavior of 𝜌_𝑘(𝑁;𝑟) would become much more complex. For example, given 𝑘 and suppose 𝑟=𝑟_𝑁. Then we have the assertions: (I) if 𝑟_𝑁→𝑟^∗>1, then 𝜌_𝑘(𝑁;𝑟_𝑁)→1−1/(𝑟^∗)^2𝑘; (II) if 𝑟_𝑁→1, then 𝜌_𝑘(𝑁;𝑟_𝑁)→0. Note that the first assertion follows directly from (2.30). Considering the second assertion, assume that 𝑟^𝑁_𝑁→𝛿>1, then it follows from (2.30) that lim_𝑁→∞𝜌_𝑘(𝑁;𝑟_𝑁)=lim_𝑁→∞(1−1/𝑟_𝑁^2𝑘)/(1−1/𝑟_𝑁^2𝑁−2)=0/(1−1/𝛿²)=0. Finally, in the case 𝑟^𝑁_𝑁→1, the second assertion follows naturally.

Figure 3 shows that how the 𝑘th-order fixation probability changes as the increasing number mutants 𝑘.

Figure 3 

The curves of 𝑘th-order fixation probability on 𝒮50 with respect to 𝑘 by fixing 𝑟. These curves show the values of 𝜌𝑘 dramatically converge to 1, and they increase as 𝑘 increases, which is consistent with the fact that the more initial mutants there are, the larger the fixation probability is.

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 [16]. A CBD denoted by ℬ_𝑁1,𝑁2(here 𝑁₁  and 𝑁₂ represent the sizes of two partite sets, respectively, and due to symmetry, let 𝑁₁≤𝑁₂) is a digraph with vertex set 𝒱={1,2,…,𝑁} (𝑁=𝑁₁+𝑁₂ is the total size) and edge set ℰ={⟨𝑖,𝑗⟩,⟨𝑗,𝑖⟩∣𝑖=1,2,…,𝑁₁,𝑗=𝑁₁+1,…,𝑁₁+𝑁₂}. A CBD is balanced if its two partite sets have the same size, that is, 𝑁₁=𝑁₂, otherwise it is unbalanced. The unbalance degree of ℬ_𝑁1,𝑁2 can be measured by |𝑁₂−𝑁₁|. Figure 4 depicts two CBDs (the second one’s unbalance degree is 1). The term “CBD” implies “CBD with intrinsic weights” in the sequel.

(a) ℬ𝑁1,𝑁2

(b) ℬ2,3 with intrinsic weights

Figure 4 

Two complete bipartite digraphs. (a) Shows a general CBD, while (b) shows an unbalanced one with intrinsic weights.

Without loss of generality, we suppose here that 𝑁₁≤𝑁₂. Obviously, when 𝑁₁=1, the complete bipartite digraph is just the complete star digraph 𝒮_𝑁2. 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 𝑘 new mutants with relative fitness 𝑟 are introduced by placing them on 𝑘 randomly chosen vertices. Let 𝜌_𝑘(𝑁₁,𝑁₂;𝑟) represent the 𝑘th-order fixation probability, that is, the probability of the event that these 𝑘 mutants generate a lineage that takes over the population.

At time 𝑡, the configuration of a population on ℬ_𝑁1,𝑁2 is described by a vector Ω(𝑡)=(𝑚₁(𝑡),𝑚₂(𝑡)), where 𝑚₁(𝑡) and 𝑚₂(𝑡) represent the number of mutants staying at vertices 1,2,…,𝑁₁ and vertices 𝑁₁+1,𝑁₁+2,…,𝑁₁+𝑁₂, respectively.

Let 𝑚(𝑡) represent the total number of mutants at time 𝑡, that is, 𝑚(𝑡)=𝑚₁(𝑡)+𝑚₂(𝑡), and 𝜌_𝑘1,𝑘2(𝑁₁,𝑁₂;𝑟) (without ambiguity, 𝜌_𝑘1,𝑘2) represent the probability of the event that, starting from Ω(0)=(𝑘₁,𝑘₂), 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 ℬ_𝑁1,𝑁2 with 𝑁₂≥𝑁₁≥2. Then, we have 𝜌_𝑘1,𝑘2(𝑁₁,𝑁₂;𝑟)(0≤𝑘₁≤𝑁₁,0≤𝑘₂≤𝑁₂) satisfying the recursive equation: 𝜒₁𝜌_{𝑘1−1,𝑘2}+𝜒₂𝜌_{𝑘1,𝑘2−1}+𝜒₃𝜌_{𝑘1+1,𝑘2}+𝜒₄𝜌_{𝑘1,𝑘2+1}=𝜒₁+𝜒₂+𝜒₃+𝜒₄𝜌_𝑘1,𝑘2, (3.1) where 𝜒₁∶=𝑘₁(𝑁₂−𝑘₂)/𝑁₁,𝜒₂∶=𝑘₂(𝑁₁−𝑘₁)/𝑁₂,𝜒₃∶=𝑟𝑘₂(𝑁₁−𝑘₁)/𝑁₁,𝜒₄∶=𝑟𝑘₁(𝑁₂−𝑘₂)/𝑁₂, and 𝜌_0,0=0,𝜌_𝑁1,𝑁2=1,𝜌_−1,𝑖=𝜌_𝑗,−1=𝜌_𝑁1+1,𝑖=𝜌_𝑗,𝑁2+1=0,𝑖=0,1,…,𝑁₂,𝑗=0,1,…,𝑁₁, are boundary conditions.

Proof. Let Ω(0), at time 𝑡=0, represent the initial configuration of a population on ℬ_𝑁1,𝑁2 and Ω(1) represent the new configuration after one step-time. Thus we have 𝑚₁(0)=𝑘₁,𝑚₂(0)=𝑘₂,𝑚(0)=𝑘₁+𝑘₂. It is simple to calculate the following conditional probabilities: 𝐏𝐫Ω(1)=𝑘₁+1,𝑘₂∣Ω(0)=𝑘₁,𝑘₂=𝑟𝑘₂𝑁₁−𝑘₁𝑁₁(),Ω(1)=𝑘𝑟𝑘+𝑁−𝑘𝐏𝐫₁,𝑘₂+1∣Ω(0)=𝑘₁,𝑘₂=𝑟𝑘₁𝑁₂−𝑘₂𝑁₂(,Ω(1)=𝑘𝑟𝑘+𝑁−𝑘)𝐏𝐫₁−1,𝑘₂∣Ω(0)=𝑘₁,𝑘₂=𝑘₁𝑁₂−𝑘₂𝑁₁(),Ω(1)=𝑘𝑟𝑘+𝑁−𝑘𝐏𝐫₁,𝑘₂−1∣Ω(0)=𝑘₁,𝑘₂=𝑘₂𝑁₁−𝑘₁𝑁₂(),𝑟𝑘+𝑁−𝑘 (3.2) where 𝑘=𝑘₁+𝑘₂,𝑁=𝑁₁+𝑁₂.
Therefore, the probability that the configuration does not change is given by 𝐏𝐫Ω(1)=𝑘₁,𝑘₂∣Ω(0)=𝑘₁,𝑘₂=1−𝐏𝐫Ω(1)=𝑘₁+1,𝑘₂∣Ω(0)=𝑘₁,𝑘₂−𝐏𝐫Ω(1)=𝑘₁,𝑘₂+1∣Ω(0)=𝑘₁,𝑘₂−𝐏𝐫Ω(1)=𝑘₁−1,𝑘₂∣Ω(0)=𝑘₁,𝑘₂−𝐏𝐫Ω(1)=𝑘₁,𝑘₂−1∣Ω(0)=𝑘₁,𝑘₂. (3.3) By the total probability formula, we derive 𝜌_𝑘1,𝑘2=𝐏𝐫Ω(1)=𝑘₁+1,𝑘₂∣Ω(0)=𝑘₁,𝑘₂𝜌_{𝑘1+1,𝑘2}+𝐏𝐫Ω(1)=𝑘₁,𝑘₂+1∣Ω(0)=𝑘₁,𝑘₂𝜌_{𝑘1,𝑘2+1}+𝐏𝐫Ω(1)=𝑘₁−1,𝑘₂∣Ω(0)=𝑘₁,𝑘₂𝜌_{𝑘1−1,𝑘2}+𝐏𝐫Ω(1)=𝑘₁,𝑘₂−1∣Ω(0)=𝑘₁,𝑘₂𝜌_{𝑘1,𝑘2−1}+𝐏𝐫Ω(1)=𝑘₁,𝑘₂∣Ω(0)=𝑘₁,𝑘₂𝜌_𝑘1,𝑘2. (3.4) Note that 𝜌_0,0=0,𝜌_𝑁1,𝑁2=1 and 𝜌_−1,𝑖,𝜌_𝑗,−1,𝜌_𝑁1+1,𝑖,𝜌_𝑗,𝑁2+1,𝑖=0,1,…,𝑁₂,𝑗=0,1,…,𝑁−1 are meaningless, thus we assume 𝜌_−1,𝑖=𝜌_𝑗,−1=𝜌_𝑁1+1,𝑖=𝜌_𝑗,𝑁2+1=0, 𝑖=0,1,…,𝑁₂, 𝑗=0,1,…,𝑁−1. 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 1≤𝑘≤𝑁−1, the following holds 𝜌_𝑘𝑁₁,𝑁₂=∑;𝑟^{min{𝑘,𝑁1}𝑘1=max𝑘,𝑁2−𝑁2𝑁1𝑘1𝑁2𝑘−𝑘1}_{𝑁1+𝑁2𝑘}𝜌_{𝑘1,𝑘−𝑘1}𝑁₁,𝑁₂.;𝑟 (3.5) Particularly, 𝜌₁𝑁₁,𝑁₂=𝑁;𝑟²𝑁₁+𝑁_{2𝜌120,1(𝑁,𝑁;𝑟)}+𝑁₁𝑁₁+𝑁_{2𝜌120,1(𝑁,𝑁;𝑟)}. (3.6) Sketch. The randomness of the initial configuration of mutants gives 𝐏𝐫Ω(0)=𝑘₁,𝑘−𝑘₁=^{𝑁1𝑘1𝑁2𝑘−𝑘1}_{𝑁1+𝑁2𝑘}, (3.7) where 𝑘₁ represents the number of mutants staying at vertices 1,2,…,𝑁₁.
Equation on (3.5) follows from this equation and the following total probability formula: 𝜌_𝑘𝑁₁,𝑁₂;𝑟=^{min{𝑘,𝑁1}}_{𝑘1=max𝑘,𝑁2−𝑁2}𝐏𝐫Ω(0)=𝑘₁,𝑘−𝑘₁𝜌_{𝑘1,𝑘−𝑘1}𝑁₁,𝑁₂;𝑟. (3.8)

From the above theorems, calculating 𝜌₁(𝑁₁,𝑁₂;𝑟) reduces to solving the difference equation system (3.1) with (𝑁₁+1)(𝑁₂+1)−2=𝑁₁𝑁₂+𝑁₁+𝑁₂−1 variables, because 𝜌_0,0,𝜌_𝑁1,𝑁2 are known. It is known [17] that solving a linear system in 𝑛 variables with the Gaussian elimination method would take 𝑂(𝑛²) time. So, we have the result.

Theorem 3.3. 𝜌₁(𝑁₁,𝑁₂;𝑟) is numerically obtainable in 𝑂(𝑁²₁𝑁²₂) time.
In the best case 𝜌₁(1,𝑁₂;𝑟) is computed in 𝑂(𝑁²₂) time, whereas in the worst case 𝜌₁(𝑁₁,𝑁₁;𝑟) consumes up to 𝑂(𝑁⁴₁) 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, 𝜌₁(𝑁₁,𝑁₂;𝑟) depends on three parameters. However, we will focus our attention on how 𝜌₁ is affected by 𝑁₁ and 𝑁₂ because these two parameters characterize the CBD. Therefore, we will fix the fitness 𝑟 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. [4] is referenced.

Figure 5 

Comparisons between the first-order fixation probability (FOFP) of CBDs and that of the Moran process with different fixed 𝑟. The lines and little rectangles represent the FOFPs of the Moran process and CBDs over 𝑁, respectively. This figure shows that FOFPs of CBDs are always equal to those of the corresponding Moran process with the same size 𝑁.

Experiment 2. Given 𝑟>1 and 𝒩_𝑘={2,3,…,𝑘}, we wish to know how 𝜌₁ depends on 𝑁₁ and 𝑁₂. For the purpose, we evaluate 𝜌₁ for each pair (𝑁₁,𝑁₂) with 𝑁₁,𝑁₂∈𝒩₅₀. The resulting 𝜌₁−(𝑁₁,𝑁₂) lattice graphs are plotted in Figure 6.

(a) 𝑟=1.1

(b) 𝑟=1.3

(c) 𝑟=2

(d) 𝑟=2.5

Figure 6 

The lattice graphs of 𝜌1 with respect to 𝑁1,𝑁2 by fixing 𝑟. Note that the figures are symmetric due to the symmetry of CBDs.

Experiment 3. Given 𝑁 and 𝑟>1, it is interesting to know how 𝜌₁ depends on 𝑁₁ (without loss of generality, again let 𝑁₁≤𝑁₂). Thus for each pair (𝑟,𝑁) with 𝑟∈{1.1,1.2,1.4,1.7,2} and 𝑁∈{52,102}, we evaluate 𝜌₁ for all 𝑁₁=2,…,[𝑁/2]. The resulting 𝜌₁−𝑁₁ curves are plotted in Figure 7, which suggests that 𝜌₁ declines as 𝑁₁ increases, that is, 𝜌₁(1,𝑁−1;𝑟)>𝜌₁(2,𝑁−2;𝑟)>⋯>𝜌₁𝑁2𝑁,𝑁−2;𝑟. (3.9)
In particular, 𝜌₁ attains its maximum and minimum when 𝑁₁=1 and 𝑁₁=[𝑁/2], respectively. This implies that all unbalanced CBDs under consideration are amplifiers of selection. Moreover, the more unbalanced a CBD, the more significantly it will amplify selection. On this basis, we pose the following conjecture.

(a) 𝑁=52

(b) 𝑁=102

Figure 7 

First-order fixation probability over the size of smaller partite set by fixing the total size 𝑁.

Conjecture 3.4. For any 𝑟>1 and 𝑁≥4, the inequality chain (3.9) holds.

Experiment 4. Let 𝑟>1 and 𝑁₁ be given, we expect to know how 𝜌₁ depends on 𝑁. For that purpose, for each pair (𝑟,𝑁₁) with 𝑟∈{1.2,1.6} and 𝑁₁∈{2,3,4,5} we compute 𝜌₁ for 2𝑁₁≤𝑁≤250. The resulting 𝜌₁−𝑁 curves are given in Figure 8. For comparison, then 𝜌₁−𝑁 curves for the corresponding Moran processes are also depicted in this figure. The following conjecture is posed here.

(a) 𝑟=1.2

(b) 𝑟=1.6

Figure 8 

First-order fixation probability over the total population size. This figure shows the phenomenon that for each pair (𝑟,𝑁1) under consideration, 𝜌1(𝑁1,𝑁−𝑁1;𝑟) converges. Furthermore, the limit of 𝜌1 declines as 𝑁1 increases with fixed 𝑟.

Conjecture 3.5. For any 𝑟>1 and any positive integer 𝑁₁≥2, 𝜌₁(𝑁₁,𝑁−𝑁₁;𝑟) converges. Moreover, the following inequality chain holds: lim_𝑁→∞𝜌₁(2,𝑁−2;𝑟)<lim_𝑁→∞𝜌₁(3,𝑁−3;𝑟)<lim_𝑁→∞𝜌₁(4,𝑁−4;𝑟)<⋯. (3.10)

4. Conclusions

This paper derives the explicit formulae for the 𝑘th-order fixation probabilities on complete star digraphs and applies them to study some asymptotic properties of populations on CSDs. In order to better understand how the structure of a population affects its evolution, we also chose to study how complete bipartite digraphs affect the first-order fixation probability of an advantageous mutant. For fixed population structure, a method (by solving a linear system) has been proposed to calculate this fixation probability. Numerical experiments have revealed some interesting phenomena, which need a reasonable explanation. After careful consideration, we find that, except isothermal digraphs, rooted digraphs and multiplerooted digraphs, complete star digraphs may be the only digraphs whose 𝑘th-order fixation probabilities can be expressed analytically. Therefore, a wise strategy for studying the fixation probabilities on other kinds of structures is to numerically determine them [18]. We are currently working toward this direction and also attempting to modify the model under study to adapt the evolution of the trustworthiness of large-scale distributed systems. It is also worth extending this work to, say, the cooperation on one graph, on a pair of graphs, or on hypernetworks [5, 19–23].

Acknowledgments

The authors are indebted to the anonymous referees for their much valuable advice. This paper is supported by the National Natural Science Foundation of China (no. 61170320), the Natural Science Foundation of Guangdong Province (no. S2011040002981), and the Scientific research Foundation of Guangdong Medical College (no. KY1048).