Limits...
Spontaneous symmetry breaking in interdependent networked game.

Jin Q, Wang L, Xia CY, Wang Z - Sci Rep (2014)

Bottom Line: Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks.With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity.Moreover, our results can be well predicted by the strategy-couple pair approximation method.

View Article: PubMed Central - PubMed

Affiliation: 1] Center for Complex Network Research and Department of Physics, Northeastern University, Boston, MA 02115, USA [2] School of Physics, Nankai University, Tianjin 300071, China.

ABSTRACT
Spatial evolution game has traditionally assumed that players interact with direct neighbors on a single network, which is isolated and not influenced by other systems. However, this is not fully consistent with recent research identification that interactions between networks play a crucial rule for the outcome of evolutionary games taking place on them. In this work, we introduce the simple game model into the interdependent networks composed of two networks. By means of imitation dynamics, we display that when the interdependent factor α is smaller than a threshold value α(C), the symmetry of cooperation can be guaranteed. Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks. With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity. Moreover, our results can be well predicted by the strategy-couple pair approximation method.

No MeSH data available.


Related in: MedlinePlus

Typical cross sections of phase diagrams for α = 0.4 (a) and α = 0.9 (b) on the interdependent regular lattices (green squares) and small-world (SW) networks (red triangles) with fraction of rewired links equalling 0.05. The colored regions represent different sections: yellow is pure cooperators section (PC), cyan is mixed strategies section (MS), gray is pure defectors section (PD) and purple is symmetry breaking section (SB). Additionally, the lines in (c) denote the results of strategy-couple pair approximation (SCPA) for α = 0.9, which is qualitatively similar to the case of (b). The dashed line is an unstable solution of SCPA approach (note that the range of figures is different, since the analysis method is only for predicting the evolution trend).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
getmorefigures.php?uid=PMC3924213&req=5

f3: Typical cross sections of phase diagrams for α = 0.4 (a) and α = 0.9 (b) on the interdependent regular lattices (green squares) and small-world (SW) networks (red triangles) with fraction of rewired links equalling 0.05. The colored regions represent different sections: yellow is pure cooperators section (PC), cyan is mixed strategies section (MS), gray is pure defectors section (PD) and purple is symmetry breaking section (SB). Additionally, the lines in (c) denote the results of strategy-couple pair approximation (SCPA) for α = 0.9, which is qualitatively similar to the case of (b). The dashed line is an unstable solution of SCPA approach (note that the range of figures is different, since the analysis method is only for predicting the evolution trend).

Mentions: In order to study the phase transition in the system, now we turn to some typical cross sections of phase diagrams under different cases. Fig. 3(a) is the case of α = 0.4, which displays the existence of three sections: pure cooperators section (PC), mixed strategies section (MS) and pure defectors section (PD), which is resonant with previous report of PDG study232847. However, in Fig. 3(b), where α is set as 0.9 (larger than αC), a novel section emerges: the symmetry breaking section (SB). In this section, usually with one of the networks showing a pure-cooperation behavior, two networks do not share the same fraction of cooperators. Notably, this interesting discovery (the coexistence of pure-cooperation and quasi-cooperation in both systems) can also be interpreted using the realistic instances. Take the maintenance of biological species as an example. Collective (pure) cooperation behavior is greatly beneficial for resisting the invasion of predators and further expanding the populations. On the other hand, individual survival is also faced with the temptation of obtaining high benefit yet no contribution, which thus leads to the existence of free-riders123. We need to argue that if no additional rule is introduced, this pure-cooperation phenomenon goes beyond what can be supported by the traditional spatial reciprocity5. Moreover, it will be instructive to check the universality of this interesting behavior on other networks. From the presented results in Fig. 3(a) and Fig. 3(b), we find that the interdependent regular lattices and small-world (SW) networks actually share the same phase transition, implying that this behavior is robust to different coupled networks. It is worth mentioning that when the fraction of rewired links is very small, the transition details of cooperation behavior on the small-world network are identical with those of regular lattice. However, with the fraction of rewired links increasing, the extinction value of cooperators will also boost.


Spontaneous symmetry breaking in interdependent networked game.

Jin Q, Wang L, Xia CY, Wang Z - Sci Rep (2014)

Typical cross sections of phase diagrams for α = 0.4 (a) and α = 0.9 (b) on the interdependent regular lattices (green squares) and small-world (SW) networks (red triangles) with fraction of rewired links equalling 0.05. The colored regions represent different sections: yellow is pure cooperators section (PC), cyan is mixed strategies section (MS), gray is pure defectors section (PD) and purple is symmetry breaking section (SB). Additionally, the lines in (c) denote the results of strategy-couple pair approximation (SCPA) for α = 0.9, which is qualitatively similar to the case of (b). The dashed line is an unstable solution of SCPA approach (note that the range of figures is different, since the analysis method is only for predicting the evolution trend).
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC3924213&req=5

f3: Typical cross sections of phase diagrams for α = 0.4 (a) and α = 0.9 (b) on the interdependent regular lattices (green squares) and small-world (SW) networks (red triangles) with fraction of rewired links equalling 0.05. The colored regions represent different sections: yellow is pure cooperators section (PC), cyan is mixed strategies section (MS), gray is pure defectors section (PD) and purple is symmetry breaking section (SB). Additionally, the lines in (c) denote the results of strategy-couple pair approximation (SCPA) for α = 0.9, which is qualitatively similar to the case of (b). The dashed line is an unstable solution of SCPA approach (note that the range of figures is different, since the analysis method is only for predicting the evolution trend).
Mentions: In order to study the phase transition in the system, now we turn to some typical cross sections of phase diagrams under different cases. Fig. 3(a) is the case of α = 0.4, which displays the existence of three sections: pure cooperators section (PC), mixed strategies section (MS) and pure defectors section (PD), which is resonant with previous report of PDG study232847. However, in Fig. 3(b), where α is set as 0.9 (larger than αC), a novel section emerges: the symmetry breaking section (SB). In this section, usually with one of the networks showing a pure-cooperation behavior, two networks do not share the same fraction of cooperators. Notably, this interesting discovery (the coexistence of pure-cooperation and quasi-cooperation in both systems) can also be interpreted using the realistic instances. Take the maintenance of biological species as an example. Collective (pure) cooperation behavior is greatly beneficial for resisting the invasion of predators and further expanding the populations. On the other hand, individual survival is also faced with the temptation of obtaining high benefit yet no contribution, which thus leads to the existence of free-riders123. We need to argue that if no additional rule is introduced, this pure-cooperation phenomenon goes beyond what can be supported by the traditional spatial reciprocity5. Moreover, it will be instructive to check the universality of this interesting behavior on other networks. From the presented results in Fig. 3(a) and Fig. 3(b), we find that the interdependent regular lattices and small-world (SW) networks actually share the same phase transition, implying that this behavior is robust to different coupled networks. It is worth mentioning that when the fraction of rewired links is very small, the transition details of cooperation behavior on the small-world network are identical with those of regular lattice. However, with the fraction of rewired links increasing, the extinction value of cooperators will also boost.

Bottom Line: Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks.With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity.Moreover, our results can be well predicted by the strategy-couple pair approximation method.

View Article: PubMed Central - PubMed

Affiliation: 1] Center for Complex Network Research and Department of Physics, Northeastern University, Boston, MA 02115, USA [2] School of Physics, Nankai University, Tianjin 300071, China.

ABSTRACT
Spatial evolution game has traditionally assumed that players interact with direct neighbors on a single network, which is isolated and not influenced by other systems. However, this is not fully consistent with recent research identification that interactions between networks play a crucial rule for the outcome of evolutionary games taking place on them. In this work, we introduce the simple game model into the interdependent networks composed of two networks. By means of imitation dynamics, we display that when the interdependent factor α is smaller than a threshold value α(C), the symmetry of cooperation can be guaranteed. Interestingly, as interdependent factor exceeds α(C), spontaneous symmetry breaking of fraction of cooperators presents itself between different networks. With respect to the breakage of symmetry, it is induced by asynchronous expansion between heterogeneous strategy couples of both networks, which further enriches the content of spatial reciprocity. Moreover, our results can be well predicted by the strategy-couple pair approximation method.

No MeSH data available.


Related in: MedlinePlus