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Evolutionary game dynamics in a growing structured population
J. Poncela, J. Gomez-Gardenes, A. Traulsen, Y. Moreno
TL;DR
Analytical descriptions of evolutionary dynamics in structured populations are difficult and limited to special cases. The paper models evolutionary play during network growth with payoff-preferential attachment, and reports that grown networks themselves do not significantly promote cooperation.
Problem
Analytical descriptions of evolutionary dynamics in structured populations are tedious and possible only in special cases.
Source: §1 · p. 3Method
The model plays an evolutionary game as the network grows, with network growth based on payoff-preferential attachment.
Source: §1 · p. 3, §4 · p. 14Results
The authors report that the grown networks themselves do not significantly promote cooperation.
Source: §4 · p. 13Takeaways & Limitations
The paper distinguishes cooperation on grown networks from the growth phase as a context relevant to cooperation.
Source: §4 · p. 13The authors identify further study of the model as future work.
Source: §6 · p. 15
Abstract
from arXiv · showhide
We discuss a model for evolutionary game dynamics in a growing, network-structured population. In our model, new players can either make connections to random preexisting players or preferentially attach to those that have been successful in the past. The latter depends on the dynamics of strategies in the game, which we implement following the so-called Fermi rule such that the limits of weak and strong strategy selection can be explored. Our framework allows to address general evolutionary games. With only two parameters describing the preferential attachment and the intensity of selection, we describe a wide range of network structures and evolutionary scenarios. Our results show that even for moderate payoff preferential attachment, over represented hubs arise. Interestingly, we find that while the networks are growing, high levels of cooperation are attained, but the same network structure does not promote cooperation as a static network. Therefore, the mechanism of payoff preferential attachment is different to those usually invoked to explain the promotion of cooperation in static, already-grown networks.
1. In tro du tion
Analytical descriptions of evolutionary dynamics in structured populations are difficult and possible only in special cases, with few general statements available. The paper studies games as the network grows, examining how growth and evolutionary dynamics interact and separating effects of topology from effects of network growth.
1. In tro du tion
- The model evolves the network while the evolutionary game is played, rather than growing a network first and simulating strategies on it afterward.This setup lets the authors examine the interplay between growth and game dynamics and disentangle effects based on topology from those based on network growth.Source: §1 · p. 3
2. Gro wing stru tured p opulations
New individuals join a growing network, play an evolutionary game, and connect to existing players according to a payoff-preference rule. The section describes how attachment settings shape network structure and whether strategy dynamics affect that structure.
2. Gro wing stru tured p opulations
- The model begins with a small complete network and can address games with different payoff rules and any number of strategies.When τ is large, game dynamics can bring the system close to equilibrium before another node is added.Source: §2 · pp. 4, 6
- At each time step, players synchronously choose between their old strategy and that of a randomly selected neighbor, adopting the neighbor’s strategy probabilistically; β is selection intensity, and selection is weak for β ≪ 1.Source: §2 · p. 4
- New individuals arrive with random strategies and establish m links to existing nodes, preferentially selecting nodes by their previous-step performance.At small α, attachment is approximately linear in payoff; as α approaches infinity, newcomers attach to the m most successful players.Source: §2 · p. 4
- At α = 0, random attachment yields an exponentially decaying degree distribution, and topology remains independent of selection intensity β.The topology is independent of strategies even when individuals play different strategies and receive different payoffs.Source: §2 · p. 5
- For α > 0, topology and strategy dynamics interact, with node degree, strategy, and neighbors influencing attachment probability.The interaction depends on strategies producing different payoffs.Source: §2 · p. 5
- For large α, the initial complete-network nodes become super-hubs connected to almost all nodes added during growth.Their emergence hinges on the nonlinear form of the attachment rule.Source: §2 · p. 5
3. Pla ying ev olutionary games during gro wth
The section examines evolutionary games during network growth, where strategy dynamics and topology interact. Payoff-preferential attachment promotes cooperation most when network growth outpaces strategy updates.
3. Pla ying ev olutionary games during gro wth
- The framework can address general games, while this section uses the Prisoner's Dilemma as an example and considers games with repeated interactions or many strategies.Strategy updates follow dynamics in which individuals can sometimes adopt a worse strategy.Source: §3 · p. 6
- Payoff-preferential attachment significantly raises cooperation versus random attachment; the effect persists but is less pronounced under weak selection.Cooperation increases most at small τ, when many nodes are added before players update strategies.Source: §3 · p. 6
- With α=0, degree distributions decay exponentially; at α=0.1 hubs emerge and the distribution begins to resemble a power law, while at α=1 over 99.9% of nodes have degree two.For α=1, the initial m=2 nodes connect to almost all other nodes.Source: §3 · p. 7
- For τ above a threshold, cooperation becomes independent of τ, indicating that incorporating a fixed number of new players is enough to reach dynamical equilibrium.At large τ, strategies have equilibrated at least locally before the next random-strategy entrant arrives.Source: §3 · pp. 6, 8
- Game dynamics have a relatively weak influence on degree distributions at small degrees, with clear differences only for large α and small b/c.The relatively few highly connected nodes are more sensitive to b/c or β; stronger selection tends to increase low-degree nodes and decrease highly connected ones.Source: §3 · pp. 8–9
4. Promotion of o op eration in gro wing net w orks
Cooperation is strongest during payoff-preferential network growth; the resulting fixed topology alone does not sustain comparable cooperation. The outcomes also depend on attachment strength, selection intensity, and benefit-to-cost ratio.
4. Promotion of o op eration in gro wing net w orks
- At small b/c, cooperation decreases as attachment preference β increases, while at large b/c it increases with selection intensity.The passage distinguishes the effects of β by benefit-to-cost regime.Source: §4 · p. 10
- For large b/c, cooperators can dominate at the very beginning of network growth, but this threshold argument is quantitative only then.A larger initial cooperative cluster and more nodes added before strategy updates make cooperation easier to spread initially.Source: §4 · p. 10
- With α = 0.1, cooperation exceeds 50% only at very high b/c; with α = 1.0, cooperators are more abundant, even at neutral selection.For α = 1.0, payoff-preferential attachment can yield high cooperation when β = 0.Source: §4 · pp. 10–11
- For small α, fixation is more likely within 10^4 steps under weaker selection, and becomes less likely as b/c increases.The authors report that fixation times increase with b/c in this regime.Source: §4 · p. 11
- For large α, strong selection speeds fixation across a wide parameter range, except when b/c is very high.For α = 1.0, the network is more heterogeneous; the passage identifies very high b/c as the exception to faster fixation under strong selection.Source: §4 · pp. 11–12
- Near the b/c values where cooperation crosses 50%, neither strategy is clearly favored initially, and fixation takes longer.When both strategies initially spread at roughly constant abundance, one is harder to eliminate later.Source: §4 · p. 12
- Growth produces comparatively high cooperation, whereas the resulting fixed topology alone does not significantly promote it.The authors report that cooperation declines once growth stops; the growth phase, rather than the grown network by itself, is crucial.Source: §4 · pp. 13–14
6. Dis ussion
The discussion finds that cooperation can reach high levels while the network grows, but the resulting structure alone does not promote it in a static network. Payoff-based attachment creates hubs and makes cooperation depend on the growth process and initial conditions.
6. Dis ussion
- Even moderate payoff-preferential attachment, such as α = 0.1, produces super-hubs that attract many new links.The resulting networks resemble those produced by nonlinear preferential attachment.Source: §6 · pp. 14–15
- High cooperation is possible during growth, but the network structure alone does not produce similarly high cooperation when static.The authors argue that the growth phase matters and that super-hubs alone are insufficient to sustain cooperation.Source: §6 · p. 15
- The growth phase distinguishes payoff-preferential attachment from mechanisms that promote cooperation on static networks.The authors emphasize that the way a network structure is obtained matters, not only the interaction network's structure.Source: §6 · pp. 15–16
- When β and b/c are sufficiently large, cooperation and defection can coexist for a long time despite eventual fixation under irregular strategy updates.Individuals sometimes adopt a strategy that performs worse, which can eventually lead one strategy to fixation.Source: §6 · p. 15
- Cooperation in these grown networks is more sensitive to initial conditions: the starting number of cooperators near super-hubs helps determine its fate.The discussion contrasts this with the weak initial-condition dependence reported for static scale-free networks.Source: §6 · p. 15
- The authors identify studying other 2×2 games as future work; whether game dynamics generally have little effect on network structure remains open.They note that game dynamics appear to have a weak impact on the resulting network structure in the model discussed.Source: §6 · p. 15
A kno wledgemen ts
The authors acknowledge colleagues and institutions and identify support from COST, the DFG, and MICINN.
A kno wledgemen ts
- The authors thank J.M. Pacheco and acknowledge institutional hospitality and funding from COST, the DFG, and MICINN.Y.M. also acknowledges support through the Ramón y Cajal Program.Source: A kno wledgemen ts · p. 16
Referen es
The references include work on evolutionary games, cooperation, and network structure and dynamics.
Referen es
- The bibliography cites research on evolutionary games and the dynamics of cooperation in structured and complex networks.Listed titles include work on evolutionary games on graphs, scale-free networks, and cooperative networks.Source: Referen es · pp. 16–17