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The analysis of bi-level evolutionary graphs
Available online at www.sciencedirect.com
BioSystems 90 (2007) 897–902
The analysis of bi-level evolutionary graphs Pei-ai Zhang a , Pu-yan Nie a,b,∗ , Dai-qiang Hu a , Fei-yan Zou c a Department of Mathematics, Jinan University, Guangzhou 510632, PR China Institute of Industrial Economics, Jinan University, Guangzhou 510632, PR China Institute of Reproductive Immunology, Jinan University, Guangzhou 510632, PR China b
c
Received 13 December 2006; received in revised form 31 May 2007; accepted 31 May 2007
Abstract Evolutionary graphs (EGs), in which evolutionary dynamic is arranged on a graph, were initially proposed by Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312–316] in the biological field and many biological phenomena are successfully explained. EGs on two levels (or bi-level EGs), based on some biological phenomena, are considered in this paper. The bi-level EGs are compared with the one-rooted EGs in two cases. One has the identical numbers of the followers, the other with the same numbers of total individuals. Then, some properties of the bi-level EGs are obtained. It is showed that bi-level EGs are more stable, and the bi-level EGs with just two leaders are the most stable, if they have identical followers respectively. The bi-level EGs theory can successfully explain the phenomena of symbiosis in biology. © 2007 Elsevier Ireland Ltd. All rights reserved. Keywords: Evolutionary graphs; Fixation probability; Fitness; Stability; Game theory
1. Introduction Evolutionary graphs (EGs) were initially introduced by Lieberman et al. (2005). Evolutionary graph theory is a nice measure to implement evolutionary dynamic, where individuals in a population are posed on a graph, the weighted edges denoting reproductive rates which govern how often individuals place offspring into adjacent vertices. When a mutant appears in this population, its fixation probability is the probability that this mutant takes over or enters the whole population. It can be affected by the number of the individuals, the fitness of the mutant, and the structure of the population, where the fitness reflects the fit degree of the mutant. After a mutant enters/appears in a population, this mutant may choose to
∗ Corresponding author at: Institute of Industrial Economics, Jinan University, Guangzhou 510632, PR China. E-mail address: [email protected] (P.-y. Nie).
leave the group. A population with lower fixation probability is consequently more stable than a population with higher fixation probability. The stability of a graph refers to the degree of stabilization. If a population or species has higher degree of stability, it has more probability to survive when there is a mutant. Lieberman et al. (2005) introduced and analyzed respectively the evolution on the isothermal structure, Kstar structure and directed cycle graphs. They determined the fixation probabilities of the mutants and gave the outcome of how evolutionary games can depend entirely on the structure of the graph. An EG is isothermal if each vertex has the same sum of weights incoming the vertex. Nie (2007) investigated EGs on two levels. The upper level is a star EG. The lower level is an isothermal EG. Nie showed that the fixation probabilities of the mutants of the EGs on two levels are lower than those of the star structure, such that the EGs on two-level are more stable. The Moran process describes stochastic evolution of a finite population of constant size. In each time step, an
0303-2647/$ – see front matter © 2007 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2007.05.008
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P.-a. Zhang et al. / BioSystems 90 (2007) 897–902
individual is chosen with a probability proportional to its fitness; the weights of the outgoing edges determine the probability that the corresponding neighbor will be replaced by the offspring. In the paper of Traulsen et al. (2005), a stochastic bi-level selection based on a hierarchy of Moran process was discussed. In the paper of Ohtsuki et al. (2006), the EGs of the structured cycles, spatial lattices, random regular graphs, random graphs and scale-free networks are reconsidered and a condition that the cooperation will overwhelm over defector was obtained. This approach has become extremely popular for understanding the structure of populations in the social world. Santos et al. (Santos et al., 2006; Santos and Pacheco, 2006) studied the scale-free networks of the prisoner’s dilemma problem. They showed that for all dilemmas, increasing heterogeneity favors the emergence of cooperation, such that long-term cooperative behavior easily resists short-term noncooperative behavior. Moreover, they showed cooperation depends on the intricate ties between individuals in scale-free populations. Li (2006) investigated the scale-free evolution of the traffic flow. He showed the distribution of the connections is largely related to the traffic flow states. Taylor and Nowak (2006) researched the evolutionary game theory with non-uniform interaction rates. Zhou et al. (2005) studied the discrete hierarchical organization of social group sizes. We will study a family of bi-level EGs (Fig. 1). The individuals on the upper level are conveniently called the leaders, which is modeled as an isothermal subgroup, while individuals on the lower level are called the followers. If there is an individual in the upper and multiple followers, one-rooted EG is formed (Fig. 2). The twolevel graph is different from those of Nie (2007) and Traulsen et al. (2005). The one-rooted graphs and the bi-level graphs are exceedingly common in the realm of biology (Traulsen et al., 2005; Santos et al., 2006; Santos
Fig. 1. A bi-level EG ρ2 (3,3,r).
Fig. 2. A One-rooted EG ρ2 (3,3,r).
and Pacheco, 2006; Michael, 2003) and social groups (Taylor and Nowak, 2006; Zhou et al., 2005; Bala and Goyal, 2000). Since the fixation probability of a dynamics graph is opposite to the stability of the graph. The EG theory can be used to study the stability of a constructed population. Establishing the bi-level EGs is not only the theoretical extension, but also the application impetus. The following biological phenomena motivate us to establish bi-level EGs. There are many examples of symbiosis in nature, which are of the bi-level structure. Mutualism is a positive reciprocal relationship between two individuals of different species. Lichens are symbiotic associations between mycobiont and photobiont. The mycobiont protects the photobiont from exposure to intense sunlight and desiccation and absorbs mineral nutrients from the underlying surface or from minute traces of atmospheric contaminants. The photobiont synthesises organic nutrients from carbon dioxide and produces ammonium from N2 gas, by nitrogen fixation in the case of cyanobacteria. In the lichens, the mycobiont depends on and controls the photobiont. Therefore the mycobiont forms the upper level and the photobiont is in the lower level. Commensalism is also of bi-level structure. Clownfishes live within the waving mass of tentacles of sea anemones. The clownfishes are protected from predators. Perhaps this relationship borders on mutualism because the clownfishes actually may attract other fishes on which the anemone can feed. The sea anemone’s tentacles quickly paralyze and seize other fishes as prey. The bi-level EGs given in the above appear in bones (Giotopoulos et al., 2006). There are many hemopoiesis stem cells in a bone as an organ. The hemopoiesis stem cell is a sort of somatic stem cells with infinity ability of renewal in medullas. Each of them may create red cells, leukocytes, hematoblasts, lymphocytes etc. They can be regarded as identical, which forms the upper level subEG. Meanwhile the cells that are produced by the stem cells can be looked as the followers, which come into
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being the lower level sub-EG. The upper level and the lower level are of the one-rooted structure. In this paper, we regard the EG with multiple leaders as a bi-level EG. The rest of the paper is organized as follows: The main results are given in Section 2. Some biological remarks are presented in Section 3.
of the lower level, the lower level has n + 1 individuals actually. Suppose that the fixation probability of A(m, r1 , s1 ) is ρ(A), and the fixation probability of B(n, r2 , s2 ) is ρ(B). Thus, according to the conditional probability, the fixation probability of the whole bi-level EG is:
2. Main results
For the bi-level EG that we discussed above, since the upper level and the lower level are independent, (3) can be rewritten as:
Throughout this paper, we always assume that the fitness of a resident is 1 and the fitness of the mutant is r. We denote ρ2 (n, m, r) as the fixation probability of a mutant in an EG, m leaders and n followers, where n is much large than m, namely, n m > 0. If m = 1, it is one-rooted with n + 1 individuals. From the result (Lieberman et al., 2005), we know the corresponding fixation probability is: 1 ρ1 (n, r) = ρ2 (n, 1, r) = . (1) n+1 2.1. EGs with the identical followers Firstly, we give the fixation probability of the bi-level EG which is given in the above section. Lemma 2.1. For the bi-level EGs with m leaders and n followers, if a mutant enters the graph with the fitness r, the fixation probability of the mutant is: 1 − r−1 1 ρ2 (n, m, r) = . 1 − r −m n + 1
(2)
Proof. Let A(m, r1 , s1 ) be the upper level with m leaders, fitness r1 and structure s1 , and B(n, r2 , s2 ) be the lower level with n followers, fitness r2 and structure s2 . Since the upper level as a whole is also an individual
ρ(AB) = ρ(A)ρ(B|A).
ρ(AB) = ρ(A)ρ(B).
(3)
(4)
From the discussion above we know that A(m, r1 , s1 ) is isothermal so that: ρ(A) =
1 − r1−1 , 1 − r1−m
while; 1 . n+1 (4) induces; ρ(B) =
1 − r −1 1 , 1 − r −m n + 1 which is just the fixation probability of the bi-level EG or ρ2 (n, m, r). The result is therefore obtained and the proof is complete. ρ(AB) = ρ(A)ρ(B) =
When m = 1, ρ2 (n, m, r) = 1/n + 1 = ρ1 (n + 1, r). It seems that a one-rooted EG is a special bi-level EG, with just one leader. In the later, we will discuss that the one-rooted EG ρ1 (n, r) is not a real bi-level EG. In the following, we will give an example of the bi-level EG. Example 2.1. The following Fig. 3a is a bi-level EG. A1 , B1 , C1 are the leaders, which forms the upper level,
Fig. 3. (a) A bi-level EG with 3 leaders and 7 followers; (b) the corresponding one-rooted EG.
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P.-a. Zhang et al. / BioSystems 90 (2007) 897–902
while A2 , B2 , C2 , D2 , E2 , F2 , G2 are the followers, which forms the lower level. The bi-level EG can be denoted as ρ2 (7, 3, r), where r is the corresponding fitness. Fig. 3b is the corresponding one-rooted EG, denoted as ρ1 (8, r). We have explained the bi-level EGs. Then we will compare the bi-level EGs with one-rooted EGs, when they have identical followers. Theorem 2.2. There are two EGs ρ1 (n1 , r1 ), ρ2 (n2 , m2 , r2 ) as discussed in the above. If n1 = n2 = n, m2 ≥ 2, then ρ1 (n1 , r1 ), ρ1 (n, r1 ) > ρ2 (n, m2 , r2 ). Proof. For r2 ∈R, we have: 0<
1 − r2−1 < 1, 1 − r2−m
The above inequality is zero if and only if r = 1 and m = 2. The conclusion of the lemma is therefore obtained and the proof is complete. In the above, we compared the bi-level EGs with the one-rooted EGs of the identical numbers of the followers. In the following, we will discuss the case that they are of the same numbers of individuals. Theorem 2.4. Suppose that there are two EGs ρ1 (n1 , r1 ), ρ2 (n2 , m2 , r2 ) as above. If the total numbers of the two groups are identical (i.e. n1 + 1 = n2 + m2 ), n1 m2 ≥ 2 and r1 = r1 = r∈(0,1), we have ρ1 (n1 , r) > ρ2 (n2 , m2 , r). Proof. Since ρ2 (n2 , m2 , r) = ρ2 (n1 − m2 , m2 , r) = ρ(m2 , r)ρ1 (n1 − m2 , r) = (1 − r −1 /1 − r −m2 )(1/n1 − m2 + 1), from Lemma 2.3, we know that ρ(m2 , r) is a strictly monotone increase function in r. Because;
which induces: 1 − r2−1 1 1 < . −m 2 1 − r2 n + 1 n+1
lim ρ(m2 , r) =
This is just ρ2 (n, m2 , r2 ) < ρ1 (n, r1 ) and the result is obtained. The above result tells us that if the numbers of the followers are identical, the fixation probabilities of the bi-level EGs are lower such that they are more stable than an unstructured EG. 2.2. EGs with the same individuals Here we will discuss the case that the total individuals on EGs are the same, i.e. n1 + 1 = m2 + n2 . We will give a necessary lemma before we give the conclusion. Lemma 2.3. For the isothermal EG, ρ(m, r) is a strictly monotone increase function in r.
r→1
1 , m2
(3)
and n1 m2 ≥ 2, it is easy to obtain the following inequalities: ρ2 (n2 , m2 , r) < lim ρ2 (n2 , m2 , r) r→1−
=
1 1 1 < = ρ1 (n1 , r). m 2 n1 − m 2 + 1 n1 + 1
The conclusion of the theorem is obtained and the proof is therefore complete. Theorems 2.2 and 2.4 show the fixation probabilities of the individuals in bi-level EGs are lower than those of the one-rooted EGs in two cases. Thus, the bi-level EGs are more stable.
Proof. It can be calculated directly that:
2.3. Some properties of the bi-level EGs
dρ(m, r) dr
In the above, we have compared the bi-level EGs with the one-rooted EGs. We will further consider some properties of the bi-level EGs.
= = =
r−2 (1 − r −m ) − mr −m−1 (1 − r −1 ) (1 − r −m )2 rm−2 (r
+ r2
− 1)(1 + r (r m − 1)2
+ · · · + r m−1
− m)
rm−2 (r−1)((r − 1)+(r 2 − 1)· · ·+(r m−1 −1)) ≥ 0, (r m −1)2
where: dρ(m, r) dρ(m, r) m−2 | = lim = . r→1 dr dr 2(m − 1)
Theorem 2.5. Suppose that there is a bi-level EG ρ2 (n, m, r) as above. If m1 < m2 n, and r∈(0,1), then ρ2 (n, m1 , r) < ρ2 (n, m2 , r). Proof. Since for r∈(0,1) and m ≥ 2, ∂ρ2 (n, m, r) −mr−m−1 (1 − r −1 ) = ∂m (1 − r −m )2 (n + 1) =
−mr m−2 (r − 1) > 0. (r m − 1)2 (n + 1)
P.-a. Zhang et al. / BioSystems 90 (2007) 897–902
Thus, for m ≥ 2, ρ2 (n, m, r) is a monotone increase function in m. Therefore, for m1 < m2 and r∈(0,1), we have ρ2 (n, m1 , r) < ρ2 (n, m2 , r). Theorem 2.5 discusses the fixation probabilities of the bi-level EGs increase when numbers of the leaders increase. In the following, we will discuss the change of the bi-level EGs when the numbers of the followers increase. Theorem 2.6. Suppose that there is a bi-level EG ρ2 (n, m, r) as above. If 1 ≤ m n1 < n2 and r∈(0,1), we then have ρ2 (n1 , m, r) > ρ2 (n2 , m, r). Proof. Since for r∈(0,1), ∂ρ2 (n, m, r) (1 − r −1 ) < 0. =− ∂n (1 − r −m )(n + 1)2 Thus, ρ2 (n, m, r) is a monotone decrease function in n. Therefore, the conclusion is immediately obtained. Theorem 2.6 illustrates that the fixation probabilities of the bi-level EGs decrease when the number of the followers increase. The fixation probability of a bi-level EG is closely related to the fitness of the mutant. The following theorem will study the change of the fixation probabilities of the bi-level EGs with respect to the fitness of the mutants. Theorem 2.7. Suppose that there is a bi-level EG ρ2 (n, m, r) as above. If 0 < r1 < r2 <1, 2 ≤ m n, then ρ2 (n, m, r1 ) < ρ2 (n, m, r2 ). Proof. This conclusion can be obtained directly from (2) and Lemma 2.3. Theorems 2.2 and 2.4 inform us that the fixation probabilities of the bi-level EGs are lower than those of the one-rooted, either of the identical numbers of followers or of the same size. Since the fixation probability of a dynamics graph is opposite to the stability of the graph, the bi-level EGs are more stable. Theorems 2.5, 2.6 and 2.7 show that the fixation probabilities of the bi-level EGs increase with the increase of the numbers of the leaders, the decrease of the numbers of the followers and the increase of the fitness of the mutants. Thus, under the same other conditions, the fixation probability of a bi-level EG with just two leader is the least such that it is the most stable. Combining the structured populations, the above results illustrate that the fixation probability of a structured population is lower than that of an undivided population, such that they are more stable. There are different species in nature that often form mutualistic associations. For example, aphids are associated with
901
ants. The association is based on what is generally assumed to be mutual beneficial services. Aphids provide the ants with honeydew and the ants defend the aphids from predators. These associations will increase the fitness of both ants and aphids. In other words, the fitness of the invaders will be relatively smaller. Thus, the fixation probabilities of the invaders will be smaller or the structured population is more stable. 3. Concluding remarks We have compared the bi-level EGs with the onerooted EGs, and some formal properties of the bi-level EGs have been obtained in the previous section. In this final section, we present some biologically oriented remarks. It seemed the one-rooted EG is a bi-level EG with just one leader, the fixation probability should be lower. In fact, this is not true. The reason is that a one-rooted EG has not the bi-level structure. The bi-level EGs come from structured population. According to the results of bi-level EGs, we conclude that the fixation probability of a structured population is lower than that of an undivided population. It is well known that the mutant in the medulla causes some diseases, such as myeloproliferative disorders, leukemia or multiple myeloma. But the bi-level structure of the cells in the bone depresses the occurrence of the diseases. Symbiosis plays an important role in the biological fields. Mutualism is a positive reciprocal relationship between two individuals of different species. Mycobiont and photobiont cannot growth well in some hard environment solely, but Lichens can grow well. One important reason is their symbiotic relations, which is of bi-level structure. Commensalism is also of bi-level structure. The clownfishes are protected from predators. Meanwhile, the clownfishes may attract other fishes for the anemone. It is known that the fixation probability of the bi-level EG is lower. Therefore the clownfishes and the sea anemones can fit the environment better. Acknowledgements Sincere thanks to the anonymous reviewer for his helpful suggestions. This work is partially supported by National Natural Science Foundation of China (No. 10372036, No. 10501019). References Bala, V., Goyal, S., 2000. A noncooperative model of network formation. Econometrica 68, 1181–1230.
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Giotopoulos, G., McCormick, C., Cole, C., 2006. DNA methylation during mouse hemopoietic differentiation and radiation-induced leukemia. Exp. Hematol. 34, 1462–1470. Li, K.P., 2006. Scale-free properties in traffic system. Commun. Theor. Phys. 46, 374–380. Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312–316. Michael, M.A., 2003. Fixation probability and time in subdivided populations. Genetics 164, 767–779. Nie, P.Y., 2007. Evolutionary graphs on two levels, Ars Combinatoria, in press. Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M.A., 2006. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505.
Santos, F.C., Pacheco, J.M., 2006. A new route to the evolution of cooperation. J. Evol. Biol. 19, 726–733. Santos, F.C., Pacheco, J.M., Lenaerts, T., 2006. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl. Acad. Sci. U.S.A. 103, 3490–3494. Taylor, C., Nowak, M.A., 2006. Evolutionary game dynamics with non-uniform interaction rates. Theor. Popul. Biol. 69, 243– 252. Traulsen, A., Sengupta, A.M., Nowak, M.A., 2005. Evolutionary game on two levels. J. Theor. Biol. 235, 393–401. Zhou, W.X., Sornette, D., Hill, R.A., et al., 2005. Discrete hierarchical organization of social group sizes. Proc. R. Soc. B-Biol. Sci. 272, 439–444.
BioSystems 90 (2007) 897–902
The analysis of bi-level evolutionary graphs Pei-ai Zhang a , Pu-yan Nie a,b,∗ , Dai-qiang Hu a , Fei-yan Zou c a Department of Mathematics, Jinan University, Guangzhou 510632, PR China Institute of Industrial Economics, Jinan University, Guangzhou 510632, PR China Institute of Reproductive Immunology, Jinan University, Guangzhou 510632, PR China b
c
Received 13 December 2006; received in revised form 31 May 2007; accepted 31 May 2007
Abstract Evolutionary graphs (EGs), in which evolutionary dynamic is arranged on a graph, were initially proposed by Lieberman et al. [Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312–316] in the biological field and many biological phenomena are successfully explained. EGs on two levels (or bi-level EGs), based on some biological phenomena, are considered in this paper. The bi-level EGs are compared with the one-rooted EGs in two cases. One has the identical numbers of the followers, the other with the same numbers of total individuals. Then, some properties of the bi-level EGs are obtained. It is showed that bi-level EGs are more stable, and the bi-level EGs with just two leaders are the most stable, if they have identical followers respectively. The bi-level EGs theory can successfully explain the phenomena of symbiosis in biology. © 2007 Elsevier Ireland Ltd. All rights reserved. Keywords: Evolutionary graphs; Fixation probability; Fitness; Stability; Game theory
1. Introduction Evolutionary graphs (EGs) were initially introduced by Lieberman et al. (2005). Evolutionary graph theory is a nice measure to implement evolutionary dynamic, where individuals in a population are posed on a graph, the weighted edges denoting reproductive rates which govern how often individuals place offspring into adjacent vertices. When a mutant appears in this population, its fixation probability is the probability that this mutant takes over or enters the whole population. It can be affected by the number of the individuals, the fitness of the mutant, and the structure of the population, where the fitness reflects the fit degree of the mutant. After a mutant enters/appears in a population, this mutant may choose to
∗ Corresponding author at: Institute of Industrial Economics, Jinan University, Guangzhou 510632, PR China. E-mail address: [email protected] (P.-y. Nie).
leave the group. A population with lower fixation probability is consequently more stable than a population with higher fixation probability. The stability of a graph refers to the degree of stabilization. If a population or species has higher degree of stability, it has more probability to survive when there is a mutant. Lieberman et al. (2005) introduced and analyzed respectively the evolution on the isothermal structure, Kstar structure and directed cycle graphs. They determined the fixation probabilities of the mutants and gave the outcome of how evolutionary games can depend entirely on the structure of the graph. An EG is isothermal if each vertex has the same sum of weights incoming the vertex. Nie (2007) investigated EGs on two levels. The upper level is a star EG. The lower level is an isothermal EG. Nie showed that the fixation probabilities of the mutants of the EGs on two levels are lower than those of the star structure, such that the EGs on two-level are more stable. The Moran process describes stochastic evolution of a finite population of constant size. In each time step, an
0303-2647/$ – see front matter © 2007 Elsevier Ireland Ltd. All rights reserved. doi:10.1016/j.biosystems.2007.05.008
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P.-a. Zhang et al. / BioSystems 90 (2007) 897–902
individual is chosen with a probability proportional to its fitness; the weights of the outgoing edges determine the probability that the corresponding neighbor will be replaced by the offspring. In the paper of Traulsen et al. (2005), a stochastic bi-level selection based on a hierarchy of Moran process was discussed. In the paper of Ohtsuki et al. (2006), the EGs of the structured cycles, spatial lattices, random regular graphs, random graphs and scale-free networks are reconsidered and a condition that the cooperation will overwhelm over defector was obtained. This approach has become extremely popular for understanding the structure of populations in the social world. Santos et al. (Santos et al., 2006; Santos and Pacheco, 2006) studied the scale-free networks of the prisoner’s dilemma problem. They showed that for all dilemmas, increasing heterogeneity favors the emergence of cooperation, such that long-term cooperative behavior easily resists short-term noncooperative behavior. Moreover, they showed cooperation depends on the intricate ties between individuals in scale-free populations. Li (2006) investigated the scale-free evolution of the traffic flow. He showed the distribution of the connections is largely related to the traffic flow states. Taylor and Nowak (2006) researched the evolutionary game theory with non-uniform interaction rates. Zhou et al. (2005) studied the discrete hierarchical organization of social group sizes. We will study a family of bi-level EGs (Fig. 1). The individuals on the upper level are conveniently called the leaders, which is modeled as an isothermal subgroup, while individuals on the lower level are called the followers. If there is an individual in the upper and multiple followers, one-rooted EG is formed (Fig. 2). The twolevel graph is different from those of Nie (2007) and Traulsen et al. (2005). The one-rooted graphs and the bi-level graphs are exceedingly common in the realm of biology (Traulsen et al., 2005; Santos et al., 2006; Santos
Fig. 1. A bi-level EG ρ2 (3,3,r).
Fig. 2. A One-rooted EG ρ2 (3,3,r).
and Pacheco, 2006; Michael, 2003) and social groups (Taylor and Nowak, 2006; Zhou et al., 2005; Bala and Goyal, 2000). Since the fixation probability of a dynamics graph is opposite to the stability of the graph. The EG theory can be used to study the stability of a constructed population. Establishing the bi-level EGs is not only the theoretical extension, but also the application impetus. The following biological phenomena motivate us to establish bi-level EGs. There are many examples of symbiosis in nature, which are of the bi-level structure. Mutualism is a positive reciprocal relationship between two individuals of different species. Lichens are symbiotic associations between mycobiont and photobiont. The mycobiont protects the photobiont from exposure to intense sunlight and desiccation and absorbs mineral nutrients from the underlying surface or from minute traces of atmospheric contaminants. The photobiont synthesises organic nutrients from carbon dioxide and produces ammonium from N2 gas, by nitrogen fixation in the case of cyanobacteria. In the lichens, the mycobiont depends on and controls the photobiont. Therefore the mycobiont forms the upper level and the photobiont is in the lower level. Commensalism is also of bi-level structure. Clownfishes live within the waving mass of tentacles of sea anemones. The clownfishes are protected from predators. Perhaps this relationship borders on mutualism because the clownfishes actually may attract other fishes on which the anemone can feed. The sea anemone’s tentacles quickly paralyze and seize other fishes as prey. The bi-level EGs given in the above appear in bones (Giotopoulos et al., 2006). There are many hemopoiesis stem cells in a bone as an organ. The hemopoiesis stem cell is a sort of somatic stem cells with infinity ability of renewal in medullas. Each of them may create red cells, leukocytes, hematoblasts, lymphocytes etc. They can be regarded as identical, which forms the upper level subEG. Meanwhile the cells that are produced by the stem cells can be looked as the followers, which come into
P.-a. Zhang et al. / BioSystems 90 (2007) 897–902
899
being the lower level sub-EG. The upper level and the lower level are of the one-rooted structure. In this paper, we regard the EG with multiple leaders as a bi-level EG. The rest of the paper is organized as follows: The main results are given in Section 2. Some biological remarks are presented in Section 3.
of the lower level, the lower level has n + 1 individuals actually. Suppose that the fixation probability of A(m, r1 , s1 ) is ρ(A), and the fixation probability of B(n, r2 , s2 ) is ρ(B). Thus, according to the conditional probability, the fixation probability of the whole bi-level EG is:
2. Main results
For the bi-level EG that we discussed above, since the upper level and the lower level are independent, (3) can be rewritten as:
Throughout this paper, we always assume that the fitness of a resident is 1 and the fitness of the mutant is r. We denote ρ2 (n, m, r) as the fixation probability of a mutant in an EG, m leaders and n followers, where n is much large than m, namely, n m > 0. If m = 1, it is one-rooted with n + 1 individuals. From the result (Lieberman et al., 2005), we know the corresponding fixation probability is: 1 ρ1 (n, r) = ρ2 (n, 1, r) = . (1) n+1 2.1. EGs with the identical followers Firstly, we give the fixation probability of the bi-level EG which is given in the above section. Lemma 2.1. For the bi-level EGs with m leaders and n followers, if a mutant enters the graph with the fitness r, the fixation probability of the mutant is: 1 − r−1 1 ρ2 (n, m, r) = . 1 − r −m n + 1
(2)
Proof. Let A(m, r1 , s1 ) be the upper level with m leaders, fitness r1 and structure s1 , and B(n, r2 , s2 ) be the lower level with n followers, fitness r2 and structure s2 . Since the upper level as a whole is also an individual
ρ(AB) = ρ(A)ρ(B|A).
ρ(AB) = ρ(A)ρ(B).
(3)
(4)
From the discussion above we know that A(m, r1 , s1 ) is isothermal so that: ρ(A) =
1 − r1−1 , 1 − r1−m
while; 1 . n+1 (4) induces; ρ(B) =
1 − r −1 1 , 1 − r −m n + 1 which is just the fixation probability of the bi-level EG or ρ2 (n, m, r). The result is therefore obtained and the proof is complete. ρ(AB) = ρ(A)ρ(B) =
When m = 1, ρ2 (n, m, r) = 1/n + 1 = ρ1 (n + 1, r). It seems that a one-rooted EG is a special bi-level EG, with just one leader. In the later, we will discuss that the one-rooted EG ρ1 (n, r) is not a real bi-level EG. In the following, we will give an example of the bi-level EG. Example 2.1. The following Fig. 3a is a bi-level EG. A1 , B1 , C1 are the leaders, which forms the upper level,
Fig. 3. (a) A bi-level EG with 3 leaders and 7 followers; (b) the corresponding one-rooted EG.
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while A2 , B2 , C2 , D2 , E2 , F2 , G2 are the followers, which forms the lower level. The bi-level EG can be denoted as ρ2 (7, 3, r), where r is the corresponding fitness. Fig. 3b is the corresponding one-rooted EG, denoted as ρ1 (8, r). We have explained the bi-level EGs. Then we will compare the bi-level EGs with one-rooted EGs, when they have identical followers. Theorem 2.2. There are two EGs ρ1 (n1 , r1 ), ρ2 (n2 , m2 , r2 ) as discussed in the above. If n1 = n2 = n, m2 ≥ 2, then ρ1 (n1 , r1 ), ρ1 (n, r1 ) > ρ2 (n, m2 , r2 ). Proof. For r2 ∈R, we have: 0<
1 − r2−1 < 1, 1 − r2−m
The above inequality is zero if and only if r = 1 and m = 2. The conclusion of the lemma is therefore obtained and the proof is complete. In the above, we compared the bi-level EGs with the one-rooted EGs of the identical numbers of the followers. In the following, we will discuss the case that they are of the same numbers of individuals. Theorem 2.4. Suppose that there are two EGs ρ1 (n1 , r1 ), ρ2 (n2 , m2 , r2 ) as above. If the total numbers of the two groups are identical (i.e. n1 + 1 = n2 + m2 ), n1 m2 ≥ 2 and r1 = r1 = r∈(0,1), we have ρ1 (n1 , r) > ρ2 (n2 , m2 , r). Proof. Since ρ2 (n2 , m2 , r) = ρ2 (n1 − m2 , m2 , r) = ρ(m2 , r)ρ1 (n1 − m2 , r) = (1 − r −1 /1 − r −m2 )(1/n1 − m2 + 1), from Lemma 2.3, we know that ρ(m2 , r) is a strictly monotone increase function in r. Because;
which induces: 1 − r2−1 1 1 < . −m 2 1 − r2 n + 1 n+1
lim ρ(m2 , r) =
This is just ρ2 (n, m2 , r2 ) < ρ1 (n, r1 ) and the result is obtained. The above result tells us that if the numbers of the followers are identical, the fixation probabilities of the bi-level EGs are lower such that they are more stable than an unstructured EG. 2.2. EGs with the same individuals Here we will discuss the case that the total individuals on EGs are the same, i.e. n1 + 1 = m2 + n2 . We will give a necessary lemma before we give the conclusion. Lemma 2.3. For the isothermal EG, ρ(m, r) is a strictly monotone increase function in r.
r→1
1 , m2
(3)
and n1 m2 ≥ 2, it is easy to obtain the following inequalities: ρ2 (n2 , m2 , r) < lim ρ2 (n2 , m2 , r) r→1−
=
1 1 1 < = ρ1 (n1 , r). m 2 n1 − m 2 + 1 n1 + 1
The conclusion of the theorem is obtained and the proof is therefore complete. Theorems 2.2 and 2.4 show the fixation probabilities of the individuals in bi-level EGs are lower than those of the one-rooted EGs in two cases. Thus, the bi-level EGs are more stable.
Proof. It can be calculated directly that:
2.3. Some properties of the bi-level EGs
dρ(m, r) dr
In the above, we have compared the bi-level EGs with the one-rooted EGs. We will further consider some properties of the bi-level EGs.
= = =
r−2 (1 − r −m ) − mr −m−1 (1 − r −1 ) (1 − r −m )2 rm−2 (r
+ r2
− 1)(1 + r (r m − 1)2
+ · · · + r m−1
− m)
rm−2 (r−1)((r − 1)+(r 2 − 1)· · ·+(r m−1 −1)) ≥ 0, (r m −1)2
where: dρ(m, r) dρ(m, r) m−2 | = lim = . r→1 dr dr 2(m − 1)
Theorem 2.5. Suppose that there is a bi-level EG ρ2 (n, m, r) as above. If m1 < m2 n, and r∈(0,1), then ρ2 (n, m1 , r) < ρ2 (n, m2 , r). Proof. Since for r∈(0,1) and m ≥ 2, ∂ρ2 (n, m, r) −mr−m−1 (1 − r −1 ) = ∂m (1 − r −m )2 (n + 1) =
−mr m−2 (r − 1) > 0. (r m − 1)2 (n + 1)
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Thus, for m ≥ 2, ρ2 (n, m, r) is a monotone increase function in m. Therefore, for m1 < m2 and r∈(0,1), we have ρ2 (n, m1 , r) < ρ2 (n, m2 , r). Theorem 2.5 discusses the fixation probabilities of the bi-level EGs increase when numbers of the leaders increase. In the following, we will discuss the change of the bi-level EGs when the numbers of the followers increase. Theorem 2.6. Suppose that there is a bi-level EG ρ2 (n, m, r) as above. If 1 ≤ m n1 < n2 and r∈(0,1), we then have ρ2 (n1 , m, r) > ρ2 (n2 , m, r). Proof. Since for r∈(0,1), ∂ρ2 (n, m, r) (1 − r −1 ) < 0. =− ∂n (1 − r −m )(n + 1)2 Thus, ρ2 (n, m, r) is a monotone decrease function in n. Therefore, the conclusion is immediately obtained. Theorem 2.6 illustrates that the fixation probabilities of the bi-level EGs decrease when the number of the followers increase. The fixation probability of a bi-level EG is closely related to the fitness of the mutant. The following theorem will study the change of the fixation probabilities of the bi-level EGs with respect to the fitness of the mutants. Theorem 2.7. Suppose that there is a bi-level EG ρ2 (n, m, r) as above. If 0 < r1 < r2 <1, 2 ≤ m n, then ρ2 (n, m, r1 ) < ρ2 (n, m, r2 ). Proof. This conclusion can be obtained directly from (2) and Lemma 2.3. Theorems 2.2 and 2.4 inform us that the fixation probabilities of the bi-level EGs are lower than those of the one-rooted, either of the identical numbers of followers or of the same size. Since the fixation probability of a dynamics graph is opposite to the stability of the graph, the bi-level EGs are more stable. Theorems 2.5, 2.6 and 2.7 show that the fixation probabilities of the bi-level EGs increase with the increase of the numbers of the leaders, the decrease of the numbers of the followers and the increase of the fitness of the mutants. Thus, under the same other conditions, the fixation probability of a bi-level EG with just two leader is the least such that it is the most stable. Combining the structured populations, the above results illustrate that the fixation probability of a structured population is lower than that of an undivided population, such that they are more stable. There are different species in nature that often form mutualistic associations. For example, aphids are associated with
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ants. The association is based on what is generally assumed to be mutual beneficial services. Aphids provide the ants with honeydew and the ants defend the aphids from predators. These associations will increase the fitness of both ants and aphids. In other words, the fitness of the invaders will be relatively smaller. Thus, the fixation probabilities of the invaders will be smaller or the structured population is more stable. 3. Concluding remarks We have compared the bi-level EGs with the onerooted EGs, and some formal properties of the bi-level EGs have been obtained in the previous section. In this final section, we present some biologically oriented remarks. It seemed the one-rooted EG is a bi-level EG with just one leader, the fixation probability should be lower. In fact, this is not true. The reason is that a one-rooted EG has not the bi-level structure. The bi-level EGs come from structured population. According to the results of bi-level EGs, we conclude that the fixation probability of a structured population is lower than that of an undivided population. It is well known that the mutant in the medulla causes some diseases, such as myeloproliferative disorders, leukemia or multiple myeloma. But the bi-level structure of the cells in the bone depresses the occurrence of the diseases. Symbiosis plays an important role in the biological fields. Mutualism is a positive reciprocal relationship between two individuals of different species. Mycobiont and photobiont cannot growth well in some hard environment solely, but Lichens can grow well. One important reason is their symbiotic relations, which is of bi-level structure. Commensalism is also of bi-level structure. The clownfishes are protected from predators. Meanwhile, the clownfishes may attract other fishes for the anemone. It is known that the fixation probability of the bi-level EG is lower. Therefore the clownfishes and the sea anemones can fit the environment better. Acknowledgements Sincere thanks to the anonymous reviewer for his helpful suggestions. This work is partially supported by National Natural Science Foundation of China (No. 10372036, No. 10501019). References Bala, V., Goyal, S., 2000. A noncooperative model of network formation. Econometrica 68, 1181–1230.
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Giotopoulos, G., McCormick, C., Cole, C., 2006. DNA methylation during mouse hemopoietic differentiation and radiation-induced leukemia. Exp. Hematol. 34, 1462–1470. Li, K.P., 2006. Scale-free properties in traffic system. Commun. Theor. Phys. 46, 374–380. Lieberman, E., Hauert, C., Nowak, M.A., 2005. Evolutionary dynamics on graphs. Nature 433, 312–316. Michael, M.A., 2003. Fixation probability and time in subdivided populations. Genetics 164, 767–779. Nie, P.Y., 2007. Evolutionary graphs on two levels, Ars Combinatoria, in press. Ohtsuki, H., Hauert, C., Lieberman, E., Nowak, M.A., 2006. A simple rule for the evolution of cooperation on graphs and social networks. Nature 441, 502–505.
Santos, F.C., Pacheco, J.M., 2006. A new route to the evolution of cooperation. J. Evol. Biol. 19, 726–733. Santos, F.C., Pacheco, J.M., Lenaerts, T., 2006. Evolutionary dynamics of social dilemmas in structured heterogeneous populations. Proc. Natl. Acad. Sci. U.S.A. 103, 3490–3494. Taylor, C., Nowak, M.A., 2006. Evolutionary game dynamics with non-uniform interaction rates. Theor. Popul. Biol. 69, 243– 252. Traulsen, A., Sengupta, A.M., Nowak, M.A., 2005. Evolutionary game on two levels. J. Theor. Biol. 235, 393–401. Zhou, W.X., Sornette, D., Hill, R.A., et al., 2005. Discrete hierarchical organization of social group sizes. Proc. R. Soc. B-Biol. Sci. 272, 439–444.