The macroscopic behavior of the system depends on the interplay between the microscopic dynamics through connections and the pattern of interactions. Life form has evolved through evolutionary processes over billions of years.
Introduction
For example, the correlation length in the paramagnetic system follows the power law at the critical temperature. An individual from the population can be represented as a node and interaction patterns between any two individuals can be represented as a link.
Definitions and concepts
Compared to the classical random networks whose degree distribution follows the Poisson distribution introduced by Eldos and Renyi, the average degree cannot explain the scale-free networks well. The probabilistic interpretation of the degree distribution, P(k), is the probability that a node of degree k is chosen by random selection.
Structural characteristics of complex networks
Each network is representative of the degree-homogeneous (ER) and the degree-heterogeneous (BA) structure. Alternative way to show the degree mixing is to measure the average neighbor degree as a function of
Generation of networks
Static model. For a configuration model, each of the Nnodes is assigned an indexn N} and the corresponding weight n−α, where α∈[0,1). We consider a variety of dynamic processes taking place in a general population and attempt to find the correlations between the dynamic processes and the underlying population structure.
Introduction
For the population of fixed size, say, x+y=1, and nowx, and represent the relative abundance of each type. where,φ=αx+βyensuresx+y=1.φ is the average fitness of the population. Therefore, the population will be taken over by the mutant or the mutant will eventually die out. Note that the fixation probability for a single wild-type individual is equal to the probability that N−1 of the failed mutant take over the population set,ρM,N−1, hence, the wild-type fixation probability,ρW ,1 is given by ,.
If the product less than 1,ρW<ρM, which means a single one, the mutant strain is more likely to take over the wild-type population compared to the opposite case. In general, the ratio between typeA and typeB, ρA/ρBi, is indicative of the time spent that the population is regulated by typeA or typeB[92]. In this case the relative fitness of the mutant, r=1, fixation occurs with probability 1/N.
This is called the neutral drift, which means that the probability of the population becoming a descendant of a particular individual is one over the size of the population. The rate at which a mutation occurs is Nu, and the probability that the mutant takes over the population is ρ. If introduced mutation is neutral, the fixation reduces to ρ=1/N, and the rate of neutral evolution is given by,R=u, where the rate of neutral evolution is simply equal to the mutation rate, independent of population size.
Evolutionary processes on graphs
The isothermal graph has the property that the probability of fixing the evolutionary dynamics in it is the same as the probability of fixing the Moran process of the same size [58]. If the ratio,γm=Tm−/Tm+in a given evolutionary graph is identical to the ratio from the Moran process, γm=1/r, the probability of fixation of the mutant with the relative fitness of the graph will be the Moran fixation. In the case of the star plot, most of the seed mutant appears one from the leaf node.
Because the death of the seed mutant requires two events at the same time, namely the birth of the wild type at the center and chosen to be replaced. Therefore, the seed mutant survives with high probability and when it chooses to give birth, it sends its offspring to the center. The spread of the mutant to another leaf node happens fortunately when the middle node is chosen for birth with the mutant type.
As the number of mutants among the leaves increases, the probability that the center has the mutant type increases. Therefore, the seed mutant survives with high probability and sends its offspring to the center when the opportunity arises. From (a) to (d) there is the 'frozen' node that will never be replaced by others in terms of temperature.
Interplay between evolutionary dynamics and population structure
In the case of the generalized structures with the degree heterogeneity and degree mixing, the selection under the DB is also suppressed as shown in Fig. The effect of the degree heterogeneity under the DB appeared as the more heterogeneous, the more suppressed. Interestingly, the effect of the degree mixing is not significant, but under BD, degree-mixed structures negatively enhance mutant selection, while under DB the effect is opposite.
Why under death-birth dynamics is selection repressed in structured populations and why is the effect of scale mixing inconsistent? In the case of the star, the fixation time is too long under BD and too short under DB. In the case of the star, the central node is the hot spot under BD, which means it is influential under DB.
Under the BD, the fixation probability is inversely proportional to the degree of seed,ρM(k)BD∝1/kwhile under the DB,ρM(k)BD∝k. In contrast, under the DB the evolutionary success of mutant proportional to the degree of seed. In the case of the star chart, the fixation time is longest under the BD and shortest under the DB.
Summary
Introduction
For the simplest two-strategy game, we have four combinations of strategies, so the payoff matrix is a 2×2 matrix. Connect the fitness function to the Eq. 4.5) The behavior of the system can be classified into five categories depending on the order relationship of the elements in the payoff matrix. A dominates B: If a>c andb>d, the average fitness of A always exceeds that of B for any composition of the population.
The fate of the system depends on the initial condition, ifx(0)
In the case of repeated Prisoner's Dilemma games, it had been asked "what is the best strategy?" by Axelrod [15]. Therefore, the payoff for cooperators using TFT isb−c and for defectors is −konly for the first game, and 0 for both. In this framework, we have defectors and collaborators who work together based on the individual's reputation to indicate a defector.
Evolutionary game dynamics on graphs
Further studies on the degree of heterogeneous networks discussed that the heterogeneity actually suppresses the choice of cooperation. Interestingly, based on synchronous updating, the scale-free networks were much better than other types in maintaining the high level of cooperation under the high temptation to defect. The high level of cooperation in the scale-free population was maintained by high degree nodes, the hubs.
Once the hub has become a cooperative partner, its yield will be kept increased by neighbors who mimic the cooperation from the hub and therefore the cooperation is maintained. When the population structure is highly mixed, neither assortative nor disassortative mixing shows higher levels of cooperation compared to neutral mixing (Fig. 4.3). In the case of assortative mixing, a sudden drop in the cooperative fraction was observed as the temptation to defect increased.
Clustering has no critical impact, but it is of little use in maintaining a higher level of cooperation due to tightly clustered mutual ties. Despite much discussion about the effect of structured population on maintaining the level of cooperation, it is still unclear how scale heterogeneity and scale-to-scale correlation influence the evolution of cooperation. It is an indicator to show that a certain network is facilitating or not for the evolution of cooperation.
Emergence of cooperation on complex population structures
The importance of taking control of the centers for the evolutionary success of the cooperators was manifested in their apparent advantage of dispersal ability. Furthermore, in the evolutionary game the individual's fitness is proportional to the number of cooperators in the neighborhood, the influence of the cooperative center will be increased by its evolutionary success. As we discussed earlier from the probability of arrival there is a certain threshold fraction of cooperators that ensure the occurrence of cooperation.
4.11 (a) The probability of arrival for a given fraction of cooperators from a randomly placed single cooperator, (b) the conditional fixation probability of a given fraction of cooperators on the structured population with γ=2.5 atb/c=8.0 . The evolutionary success of collaboration partners for a hub in this phase determines whether its further expansion takes place. Note that the aligned increase in local cooperators around hubs in diverse admixture increases the total share of cooperators in the entire population.
In addition to the interference through the structural equivalence, the utilization of defector hubs by collaborators also depends on the level of structural similarity. Consequently, this different level of local share of collaborators around hubs leads to a completely opposite effect on the choice of collaboration. In disassortative mixing and the high level of local share of collaborators around C-hub leads a large influence of C-hub.
Summary
Second, we treated the population structural preference for cooperative selection based on evolutionary game theory. Emergence of cooperation in the complex structured population containing degree-heterogeneity and degree-to-degree correlation achieved along the following stages. Survival: the survival of the seed's associate and its first evolutionary success depends on the degree of the seed and the harmonic mean of the degree of its neighbors.
Below the BD, center transitions are much more likely to occur rather than leaf transitions. Under the BD, the center transition (Tm,h:W+ →M,Tm,h:M→W− ) is much more likely to occur than leaf transitions (Tm,h=M+ ,Tm,h=W−. BD, the evolutionary success from seed to the center is easy to achieve (m=1 → m=2), but the spread of mutants from the center to the other leaf depends on the relative fitness.
Meanwhile, under DB, the evolutionary success of the mutant is extremely dependent on the center. From the conditional probability of fixation, we concluded that the marginal fraction of cooperators, fC∗, which ensures further evolutionary success until fixation, depends on the degree of degree heterogeneity and degree of mixing. Since the fitness of individuals in the dynamics of an evolutionary game depends on their payoffs, the relative fitness of participants increases as the local stake increases.
The overall conditional fixation probability summarizes the effect of given structural characteristics of the population on the evolution of cooperation. Seed cooperators' survival and up to a few evolutionary successes depend on the degree of seediness, kseed.
