Network reciprocity is an example of how cooperation can be promoted by an appropriate interaction topology. Such an interaction topology may be achieved dynamically if the agents can choose their interaction partners according to their strategies, that is if the
agents interact in an adaptive network. Moreover, network adaptivity can lead to a number of less obvious mechanisms allowing for the evolution and persistence of cooperation (a list of various co-evolutionary mechanisms including network adaptivity has been assembled by Perc and Szolnoki, 2010). In this section, I give an overview on several different mechanisms by which cooperation can be promoted in adaptive networks.
Local effects
In adaptive networks, a node is not constrained to remain within the static interaction topology of its—possibly unfavourable—neighbourhood. It can rather try to increase its fitness, which is determined by the payoff it extracts from this neighbourhood, by discontinuing unprofitable links and seeking to establish new advantageous connections, for instance.
It is generally advantageous for cooperators to connect to other cooperators and form tightly connected clusters, as these produce high payoff and cannot be invaded easily by defectors (network reciprocity). In adaptive networks, cooperator clusters can be created actively by appropriate link dynamics directly or indirectly favouring assortative interactions between cooperating agents. Indeed, ‘cooperation prevails when individuals adjust their social ties,’ i.e. prefer links to cooperators over links to defectors (Santos et al., 2006a). Such link dynamics can be quite counter-intuitive. In (Zimmermann et al., 2000, 2001, 2004; Eguíluz et al., 2005), for instance, cooperators cannot alter their connections, but defectors are allowed to rewire, seeking new cooperators to exploit. Nevertheless, the population reaches a highly cooperative stationary state that cannot be invaded by defectors.
Structure
Not only local clustering of cooperators but also heterogeneity in the number of contacts per agent can significantly promote cooperation. Notably, the positive effect of degree heterogeneity has been observed in a somewhat unifying way for different games on scale-free networks (Santos and Pacheco, 2005; Santos et al., 2006b). Santos and Pacheco (2005) argued that cooperators typically occupy the high-degree hub nodes in static scale-free networks, acting as leaders in the population.
In adaptive networks, such leadership emerges naturally in the hierarchical topolo- gies that can self-organize due to the interplay between the strategy dynamics and the topological dynamics. Starting from some initial configuration, the network may thus be reshaped into a more favourable structure enabling cooperators to survive. This was first observed in (Zimmermann et al., 2000, 2001, 2004; Eguíluz et al., 2005), where an initially random network evolves into a stationary hierarchy of well-connected influential cooperators (leaders), which are imitated by a large number of followers. Consequently, a substantial fraction of the population cooperates.
model by Ebel and Bornholdt (2002), which is based on an iterated version of thePD game. There, in addition to promoting cooperation, the topological dynamics also lead to increased clustering and assortative mixing, features observed in many real-world networks.
Mixing
Although the topological change in adaptive networks may lead to a high level of hetero- geneity, it can also reduce the influence of the network structure on the node dynamics by providing an effective mixing of the population. When the agents are allowed to choose their connections in a sufficiently random way, they effectively sample interactions from the whole network. If this proceeds on a fast time-scale, one expects similar dynamics as in well-mixed populations.
When the linking dynamics proceed totally random, the topological evolution effec- tively decouples from the strategy dynamics, simply mixing the underlying network structure. In this limit, the network is not an adaptive network any more. This was observed, for instance, in aPDmodel with random, state-independent rewiring of the adjacent links after each node update (Kun and Scheuring, 2009). Due to this effective mixing, the probability for cooperators to survive and spread is very low, similar to the case of a well-mixed population.
Active linking, on the other hand, may result in a rather different behaviour. In this case, links are formed and disconnected at state-dependent rates. If these processes are fast compared to the node updates, the situation is transformed into an effective game in a well-mixed population with a rescaled payoff matrix, which depends on the stationary regime of the linking processes (Santos et al., 2006b; Pacheco et al., 2006a,b). ThePD game can thus be mapped on an effective coordination game in a well-mixed population, where cooperation is greatly enhanced. Similarly, theSDgame is transformed into an effective harmony game, which even allows for full cooperation of every single agent. The number of cooperators surviving in the stationary state is thus a function of the ratio between the time scales of the strategy dynamics and active linking.
Growth
So far, we have considered only adaptive networks of a fixed size, i.e. a given number of agents. The complex dynamics during network formation and growth, however, also play an important role for the evolution of cooperation. When the interaction network grows adaptively, i.e. new agents connect to existing ones depending on the strategy dynamics, cooperation can be favoured by a novel mechanism rooted in this adaptive growth process.
Two prominent examples are the growing network models of Ren et al. (2006) and Poncela et al. (2009b). In these models, new agents are added continuously and connect to existing ones according to a preferential attachment rule. In contrast to theBAalgorithm
for scale-free networks, here the preference is for fitness rather than degree. Thus, the agents receive new connections with a probability proportional to their fitness. With this rule, highly hierarchical networks emerge where the most successful players in terms of payoff are also the most connected ones. For the imitation processes, a probabilistic update rule is used in synchronous updates of all agents, so that they occasionally adopt the worse strategy. Notably, a high level of cooperation is achieved in (Poncela et al., 2009b), which decays once the network stops growing. Hence, a genuinely novel mechanism is at work in this model: It is the preferential growth process itself that facilitates cooperation in this system rather than the resulting heterogeneous hierarchical topology.
In a stricter variant of the model where the agents never adopt a worse strategy, coope- ration increases even further after the final system size has been reached (Poncela et al., 2008). Thus, only rather rational agents (never adopting a worse strategy) can achieve and sustain a cooperative state when the network stops growing. Still, the mechanism at work is different from the way cooperation is promoted in static scale-free networks (Santos and Pacheco, 2005). In contrast to the latter, cooperators do not necessarily occupy the most connected hub nodes. In fact, a high level of cooperation is reached ‘despite the presence of defector hubs’ (Poncela et al., 2009a).
Dynamics
The central question in evolutionary game theory is how cooperation can evolve and survive in a population of selfish individuals. For this reason, most studies in this field focus on average and therefore quasi-static quantities such as the stationary fraction of cooperators in the population or the probability that an initially rare strategy can spread and fixate in the population (fixation probability). Cooperation on adaptive networks is, however, an inherently dynamic process exhibiting a number of interesting phenomena apart from the fact that it can be promoted substantially by network adaptivity. We have already seen that cooperation may be promoted dynamically as long as the network keeps growing. Furthermore, oscillations in the number of cooperators have been observed in adaptive networks (Hanaki et al., 2007; Suzuki et al., 2008; Szolnoki et al., 2008; Szolnoki and Perc, 2009a,b). In (Zimmermann et al., 2000, 2001, 2004; Zimmermann and Eguíluz, 2005; Eguíluz et al., 2005), a perturbation of the highly cooperative stationary state results in large reorganization avalanches until a new stationary regime of high cooperation is reached.
In the remaining part of this chapter, I discuss how cooperation can be promoted dynamically in a specific model where the agents have access to non-local information about the average performance of the strategies. In this model, fast topological change can even achieve asymptotic full cooperation in infinite populations.