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Evolutionary dynamics of group interactions on structured populations: a review

Perc, Matjaž, Gómez-Gardeñes, Jesús, Szolnoki, Attila, Floría, Luis M., Moreno, Yamir·2013·Journal of The Royal Society Interface
DOI 10.1098/rsif.2012.0997PMC PMC3565747
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Abstract

Interactions among living organisms, from bacteria colonies to human societies, are inherently more complex than interactions among particles and non-living matter. Group interactions are a particularly important and widespread class, representative of which is the public goods game. In addition, methods of statistical physics have proved valuable for studying pattern formation, equilibrium selection and self-organization in evolutionary games. Here, we review recent advances in the study of evolutionary dynamics of group interactions on top of structured populations, including lattices, complex networks and coevolutionary models. We also compare these results with those obtained on well-mixed populations. The review particularly highlights that the study of the dynamics of group interactions, like several other important equilibrium and non-equilibrium dynamical processes in biological, economical and social sciences, benefits from the synergy between statistical physics, network science and evolutionary game theory.

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Figure 1. Schematic of N = 3 microbes, where fractions are cooperators (producers) and are defectors (free-riders). For the most popular choice of benefit and cost functions, β(ρC) = rρC (r > 1) and α(ρC) = 1, respectively, individual pay-offs are and . An explicit computation of PC (ρC) () and PD(ρC) () reveals that they cannot be generated by means of pairwise interactions, thus illustrating the inherent irreducibility of group interactions.
Figure 1. Schematic of N = 3 microbes, where fractions are cooperators (producers) and are defectors (free-riders). For the most popular choice of benefit and cost functions, β(ρC) = rρC (r > 1) and α(ρC) = 1, respectively, individual pay-offs are and . An explicit computation of PC (ρC) () and PD(ρC) () reveals that they cannot be generated by means of pairwise interactions, thus illustrating the inherent irreducibility of group interactions.CC BY · PubMed Central
Figure 2. Schematic of different types of lattices. On the (a) square lattice, each player has four immediate neighbours, thus forming groups of size G = 5, whereas on the (b) honeycomb lattice, it has three, thus G = 4. In both cases, the clustering coefficient 𝒞 is zero. Yet, the membership of unconnected players in the same groups introduces effective links between them, which may evoke behaviour that is characteristic for lattices with closed triplets [28]. On the other hand, (c) the kagomé and (d) the triangular lattice both feature percolating overlapping triangles, which makes them less susceptible to effects introduced by group interactions. The kagomé lattice has G = 5 and , whereas the triangular lattice has and G = 7.
Figure 2. Schematic of different types of lattices. On the (a) square lattice, each player has four immediate neighbours, thus forming groups of size G = 5, whereas on the (b) honeycomb lattice, it has three, thus G = 4. In both cases, the clustering coefficient 𝒞 is zero. Yet, the membership of unconnected players in the same groups introduces effective links between them, which may evoke behaviour that is characteristic for lattices with closed triplets [28]. On the other hand, (c) the kagomé and (d) the triangular lattice both feature percolating overlapping triangles, which makes them less susceptible to effects introduced by group interactions. The kagomé lattice has G = 5 and , whereas the triangular lattice has and G = 7.CC BY · PubMed Central
Figure 3. Borders between the mixed C+D and the pure D phase in dependence on the normalized uncertainty by strategy adoptions K/G, as obtained on different lattices for (a) pairwise and (b) group interactions. Vertical axis depicts the defection temptation rate, i.e. the higher its value the smaller the value of r that still allows the survival of at least some cooperators. By pairwise interactions (G = 2), the absence of overlapping triangles is crucial (square and honeycomb lattices), as then there exists an intermediate value of K at which the evolution of cooperation is optimally promoted. If triangles do percolate (triangular lattice), the K → 0 limit is optimal. This behaviour is characteristic for all social dilemmas that are based on pairwise games, the most famous examples being the prisoner's dilemma and the snowdrift game (see figs 3 and 5 in Szabó et al. [30]). Conversely, when group interactions are considered (see figure 2 for G values) the topological differences between the lattices become void. Accordingly, the deterministic K → 0 limit is optimal, regardless of the topology of the host lattice [28].
Figure 3. Borders between the mixed C+D and the pure D phase in dependence on the normalized uncertainty by strategy adoptions K/G, as obtained on different lattices for (a) pairwise and (b) group interactions. Vertical axis depicts the defection temptation rate, i.e. the higher its value the smaller the value of r that still allows the survival of at least some cooperators. By pairwise interactions (G = 2), the absence of overlapping triangles is crucial (square and honeycomb lattices), as then there exists an intermediate value of K at which the evolution of cooperation is optimally promoted. If triangles do percolate (triangular lattice), the K → 0 limit is optimal. This behaviour is characteristic for all social dilemmas that are based on pairwise games, the most famous examples being the prisoner's dilemma and the snowdrift game (see figs 3 and 5 in Szabó et al. [30]). Conversely, when group interactions are considered (see figure 2 for G values) the topological differences between the lattices become void. Accordingly, the deterministic K → 0 limit is optimal, regardless of the topology of the host lattice [28].CC BY · PubMed Central
Figure 4. Characteristic snapshot of the evolutionary process for (a) small (G = 5) and (b) large (G = 301) groups. Cooperators are depicted by blue, whereas defectors are depicted by red. For small groups, the evolution of strategies proceeds with the characteristic propagation of the fronts of the more successful strategy (in this case D) until eventually the maladaptive strategy C goes extinct. For large groups, however, the cooperative clusters are strong and can outperform the defectors, even if r is very small. Still, as the density of defectors decreases, their pay-off suddenly becomes very competitive, and thus they can invade the seemingly invincible cooperative clusters. Such an alternating time evolution is completely atypical and was previously associated only with cooperators.
Figure 4. Characteristic snapshot of the evolutionary process for (a) small (G = 5) and (b) large (G = 301) groups. Cooperators are depicted by blue, whereas defectors are depicted by red. For small groups, the evolution of strategies proceeds with the characteristic propagation of the fronts of the more successful strategy (in this case D) until eventually the maladaptive strategy C goes extinct. For large groups, however, the cooperative clusters are strong and can outperform the defectors, even if r is very small. Still, as the density of defectors decreases, their pay-off suddenly becomes very competitive, and thus they can invade the seemingly invincible cooperative clusters. Such an alternating time evolution is completely atypical and was previously associated only with cooperators.CC BY · PubMed Central
Figure 5. Different realizations of the public benefit function B(S) = 1/(1 + exp[−β(Si−T)]), where T represents the threshold value and β is the steepness of the function [45]. For β = 0, the benefit function is a constant equalling 0.5, in which case the produced public goods are insensitive to the efforts of group members. Conversely, for β = +∞, the benefit function becomes step-like so that group members can enjoy the benefits of collaborative efforts via r only if the total amount of contributions in the group S exceeds a threshold. Otherwise, they obtain nothing. The depicted curves were obtained for T = 2.5 and β = 0.1 (dotted red), 1 (dashed green) and 10 (solid blue).
Figure 5. Different realizations of the public benefit function B(S) = 1/(1 + exp[−β(Si−T)]), where T represents the threshold value and β is the steepness of the function [45]. For β = 0, the benefit function is a constant equalling 0.5, in which case the produced public goods are insensitive to the efforts of group members. Conversely, for β = +∞, the benefit function becomes step-like so that group members can enjoy the benefits of collaborative efforts via r only if the total amount of contributions in the group S exceeds a threshold. Otherwise, they obtain nothing. The depicted curves were obtained for T = 2.5 and β = 0.1 (dotted red), 1 (dashed green) and 10 (solid blue).CC BY · PubMed Central
Figure 6. Time evolution of strategies on a square lattice having G = 25, for the critical mass (a) M = 2 and (b) M = 17 at r/G = 0.6 [48]. Defectors are marked by red, whereas cooperators are depicted by blue if their initial contributions are exalted or white if they go to waste. Accordingly, cooperators can be designated as being either ‘active’ or ‘inactive’. When M is low all cooperators are active, yet they do not have a strong incentive to aggregate because an increase in their density will not elevate their fitness. Hence, only a moderate fraction of cooperators coexists with the prevailing defectors in the stationary state. If the critical mass is neither small nor large, the status of cooperators varies depending on their location on the lattice: there are places where their local density exceeds the threshold and they can prevail against defectors. There are also places where the cooperators are inactive because their density is locally insufficient and loose against defectors. The surviving domains of active cooperators start spreading, ultimately rising to near dominance.
Figure 6. Time evolution of strategies on a square lattice having G = 25, for the critical mass (a) M = 2 and (b) M = 17 at r/G = 0.6 [48]. Defectors are marked by red, whereas cooperators are depicted by blue if their initial contributions are exalted or white if they go to waste. Accordingly, cooperators can be designated as being either ‘active’ or ‘inactive’. When M is low all cooperators are active, yet they do not have a strong incentive to aggregate because an increase in their density will not elevate their fitness. Hence, only a moderate fraction of cooperators coexists with the prevailing defectors in the stationary state. If the critical mass is neither small nor large, the status of cooperators varies depending on their location on the lattice: there are places where their local density exceeds the threshold and they can prevail against defectors. There are also places where the cooperators are inactive because their density is locally insufficient and loose against defectors. The surviving domains of active cooperators start spreading, ultimately rising to near dominance.CC BY · PubMed Central
Figure 7. Indirect territorial battle between (a) pure cooperators (blue) and peer-punishers (green), and between (b) pure cooperators (blue) and rewarding cooperators (light grey). In (a), pure cooperators and peer-punishers form isolated clusters that compete against defectors (red) for space on the square lattice. Because peer-punishers are more successful in competing against defectors than pure cooperators (also frequently referred to as second-order free-riders [66]), eventually the latter die out to a leave a mixed two-strategy phase (peer-punishers and defectors) as a stationary state (see Helbing et al. [59] for further details). In (b), defectors are quick to claim supremacy on the lattice, yet pure and rewarding cooperators both form isolated compact clusters to try and prevent this. While rewarding cooperators can outperform defectors, pure cooperators cannot. Accordingly, the latter die out, leaving qualitatively the same outcome as depicted in (a) (see Szolnoki & Perc [65] for further details).
Figure 7. Indirect territorial battle between (a) pure cooperators (blue) and peer-punishers (green), and between (b) pure cooperators (blue) and rewarding cooperators (light grey). In (a), pure cooperators and peer-punishers form isolated clusters that compete against defectors (red) for space on the square lattice. Because peer-punishers are more successful in competing against defectors than pure cooperators (also frequently referred to as second-order free-riders [66]), eventually the latter die out to a leave a mixed two-strategy phase (peer-punishers and defectors) as a stationary state (see Helbing et al. [59] for further details). In (b), defectors are quick to claim supremacy on the lattice, yet pure and rewarding cooperators both form isolated compact clusters to try and prevent this. While rewarding cooperators can outperform defectors, pure cooperators cannot. Accordingly, the latter die out, leaving qualitatively the same outcome as depicted in (a) (see Szolnoki & Perc [65] for further details).CC BY · PubMed Central
Figure 8. When the public goods game is staged on a complex network, cooperators can either bear a fixed cost per game, z (a), or this cost can be normalized with the number of interactions, i.e. z/(ki+1), where ki is the number of neighbours of each particular cooperator i. In the latter case, one effectively recovers a fixed cost per individual (b). This distinction has significant consequences for the evolution of public cooperation on complex interaction networks, as originally reported by Santos et al. [90]. Only if the cost is normalized with the number of neighbours does social heterogeneity significantly promote the evolution of public cooperation.
Figure 8. When the public goods game is staged on a complex network, cooperators can either bear a fixed cost per game, z (a), or this cost can be normalized with the number of interactions, i.e. z/(ki+1), where ki is the number of neighbours of each particular cooperator i. In the latter case, one effectively recovers a fixed cost per individual (b). This distinction has significant consequences for the evolution of public cooperation on complex interaction networks, as originally reported by Santos et al. [90]. Only if the cost is normalized with the number of neighbours does social heterogeneity significantly promote the evolution of public cooperation.CC BY · PubMed Central
Figure 9. Schematic display of the two different forms of encoding collaboration data. In the central plot (b), several collaborating groups represent the original data. The interactions among players can be translated into a projected complex network (a). However, if one aims at preserving all the information about the group structure, a representation as a bipartite graph (c) is more appropriate. Figure adapted from Gómez-Gardeñes et al. [103].
Figure 9. Schematic display of the two different forms of encoding collaboration data. In the central plot (b), several collaborating groups represent the original data. The interactions among players can be translated into a projected complex network (a). However, if one aims at preserving all the information about the group structure, a representation as a bipartite graph (c) is more appropriate. Figure adapted from Gómez-Gardeñes et al. [103].CC BY · PubMed Central
Figure 10. Cooperation level 〈C〉 as a function of the multiplication factor r for the public goods game played on the one-mode (projected) collaboration network and the bipartite graph preserving the original group structure. The two plots are for the public goods game played in (a) the fixed cost per game and (b) the fixed cost per individual mode. The strategy updating makes use of the Fermi function (see equation (2.2)). This figure is adapted from Gómez-Gardeñes et al. [103]. Filled squares, one mode; filled circles, bipartite.
Figure 10. Cooperation level 〈C〉 as a function of the multiplication factor r for the public goods game played on the one-mode (projected) collaboration network and the bipartite graph preserving the original group structure. The two plots are for the public goods game played in (a) the fixed cost per game and (b) the fixed cost per individual mode. The strategy updating makes use of the Fermi function (see equation (2.2)). This figure is adapted from Gómez-Gardeñes et al. [103]. Filled squares, one mode; filled circles, bipartite.CC BY · PubMed Central
Figure 11. (a) The multilevel hierarchical structure introduced in Wang et al. [107], where groups are hierarchically ordered as modules of a network in which the public goods game is played. (b) Two interdependent lattices as studied in Wang et al. [108]. Players can adopt different strategies within each layer, but coupling between the pay-offs obtained in each of the two layers (see equation (3.3)) makes their evolutionary dynamics interdependent.
Figure 11. (a) The multilevel hierarchical structure introduced in Wang et al. [107], where groups are hierarchically ordered as modules of a network in which the public goods game is played. (b) Two interdependent lattices as studied in Wang et al. [108]. Players can adopt different strategies within each layer, but coupling between the pay-offs obtained in each of the two layers (see equation (3.3)) makes their evolutionary dynamics interdependent.CC BY · PubMed Central
Figure 12. Coevolution of strategy and structure leads to high levels of public cooperation. Networks depict snapshots in time at (a) 0, (b) 2000 and (c) 20 000 iterations, whereby green links connect defector–cooperator pairs, blue links connect two defectors, while red links connect two cooperators. Accordingly, cooperators are depicted by red and defectors are depicted by blue. This figure was adapted from Wu et al. [127], where further details with respect to the simulation set-up can also be found.
Figure 12. Coevolution of strategy and structure leads to high levels of public cooperation. Networks depict snapshots in time at (a) 0, (b) 2000 and (c) 20 000 iterations, whereby green links connect defector–cooperator pairs, blue links connect two defectors, while red links connect two cooperators. Accordingly, cooperators are depicted by red and defectors are depicted by blue. This figure was adapted from Wu et al. [127], where further details with respect to the simulation set-up can also be found.CC BY · PubMed Central
Figure 13. Rough interfaces enable defectors (red) to have an effective exploitation of cooperators (blue), thus hindering spatial reciprocity. Upon the introduction of adaptive punishment (green, where darker (lighter) shades imply stronger (weaker) punishing activity), interfaces become smoother, which in turn invigorates spatial reciprocity and prevents defectors from being able to exploit the public goods. A prepared initial state, corresponding to a rough interface, is used to reveal the workings of this mechanism. Interestingly, here the stationary state is a pure C phase, while under the same conditions peer-punishment without coevolution yields a pure D phase. We refer to Perc & Szolnoki [69] for further details.
Figure 13. Rough interfaces enable defectors (red) to have an effective exploitation of cooperators (blue), thus hindering spatial reciprocity. Upon the introduction of adaptive punishment (green, where darker (lighter) shades imply stronger (weaker) punishing activity), interfaces become smoother, which in turn invigorates spatial reciprocity and prevents defectors from being able to exploit the public goods. A prepared initial state, corresponding to a rough interface, is used to reveal the workings of this mechanism. Interestingly, here the stationary state is a pure C phase, while under the same conditions peer-punishment without coevolution yields a pure D phase. We refer to Perc & Szolnoki [69] for further details.CC BY · PubMed Central

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Metadata

Authors
Perc, M., Gómez-Gardeñes, J., Szolnoki, A., Floría, L. M., & Moreno, Y.
Affiliations
  • 1
  • 2
  • 5
  • 3
Venue
Journal of The Royal Society Interface
Publisher
The Royal Society
Published
2013-03-06
Volume
10
Issue
80
ISSN
  • 1742-5689
  • 1742-5662
DOI
10.1098/rsif.2012.0997
PMC
PMC3565747
Cited by
1,340
References
12
Language
en
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