Julia Poncela

Complex Cooperative Networks from Evolutionary Preferential Attachment

In spite of its relevance to the origin of complex networks, the interplay between form and function and its role during network formation remains largely unexplored. While recent studies introduce dynamics by considering rewiring processes of a pre-existent network, we study network growth and formation by proposing an evolutionary preferential attachment model, its main feature being that the capacity of a node to attract new links depends on a dynamical variable governed in turn by the node interactions. As a specific example, we focus on the problem of the emergence of cooperation by analyzing the formation of a social network with interactions given by the Prisoners Dilemma. The resulting networks show many features of real systems, such as scale-free degree distributions, cooperative behavior and hierarchical clustering. Interestingly, results such as the cooperators being located mostly on nodes of intermediate degree are very different from the observations of cooperative behavior on static networks. The evolutionary preferential attachment mechanism points to an evolutionary origin of scale-free networks and may help understand similar feedback problems in the dynamics of complex networks by appropriately choosing the game describing the interaction of nodes.

Robustness of cooperation in the evolutionary prisoner's dilemma on complex networks

Recent studies on the evolutionary dynamics of the prisoners dilemma game in scale-free networks have demonstrated that the heterogeneity of the network interconnections enhances the evolutionary success of cooperation. In this paper we address the issue of how the characterization of the asymptotic states of the evolutionary dynamics depends on the initial concentration of cooperators. We find that the measure and the connectedness properties of the set of nodes where cooperation reaches fixation is largely independent of initial conditions, in contrast with the behaviour of both the set of nodes where defection is fixed, and the fluctuating nodes. We also check for the robustness of these results when varying the degree heterogeneity along a one-parametric family of networks interpolating between the class of Erdős–Renyi graphs and the Barabasi–Albert networks.

Cooperative scale-free networks despite the presence of defector hubs

Recent results have shown that heterogeneous populations are better suited to support cooperation than homogeneous settings when the Prisoners Dilemma drives the evolutionary dynamics of the system. The same occurs when the network growth is coevolving together with the evolutionary dynamics, which also gives rise to highly cooperative scale-free networks. In the latter case, however, the organization of cooperation is radically different with respect to the case in which the underlying network is static. In this paper we study the structure of cooperation in static networks grown together with evolutionary dynamics and show that the general belief that hubs can only be occupied by cooperators does not hold. Moreover, these scale-free networks support high levels of cooperation despite having defector hubs. Our results have several important implications for the explanation of cooperative behavior in scale-free networks and highlight the importance that the formation of complex systems have on its function.

Natural selection of cooperation and degree hierarchy in heterogeneous populations

One of the current theoretical challenges to the explanatory powers of Evolutionary Theory is the understanding of the observed evolutionary survival of cooperative behavior when selfish actions provide higher fitness (reproductive success). In unstructured populations natural selection drives cooperation to extinction. However, when individuals are allowed to interact only with their neighbors, specified by a graph of social contacts, cooperation-promoting mechanisms (known as lattice reciprocity) offer to cooperation the opportunity of evolutionary survival. Recent numerical works on the evolution of Prisoners Dilemma in complex network settings have revealed that graph heterogeneity dramatically enhances the lattice reciprocity. Here we show that in highly heterogeneous populations, under the graph analog of replicator dynamics, the fixation of a strategy in the whole population is in general an impossible event, for there is an asymptotic partition of the population in three subsets, two in which fixation of cooperation or defection has been reached and a third one which experiences cycles of invasion by the competing strategies. We show how the dynamical partition correlates with connectivity classes and characterize the temporal fluctuations of the fluctuating set, unveiling the mechanisms stabilizing cooperation in macroscopic scale-free structures.

Dynamical organization towards consensus in the Axelrod model on complex networks.

We analyze the dynamics toward cultural consensus in the Axelrod model on scale-free networks. By looking at the microscopic dynamics of the model, we are able to show how culture traits spread across different cultural features. We compare the diffusion at the level of cultural features to the growth of cultural consensus at the global level, finding important differences between these two processes. In particular, we show that even when most of the cultural features have reached macroscopic consensus, there are still no signals of globalization. Finally, we analyze the topology of consensus clusters both for global culture and at the feature level of representation.

COOPERATION IN THE PRISONER'S DILEMMA GAME IN RANDOM SCALE-FREE GRAPHS

In this paper we study the cooperative behavior of agents playing the Prisoners Dilemma game in random scale-free networks. We show that the survival of cooperation is enhanced with respect to random homogeneous graphs but, on the other hand, decreases when compared to that found in Barabasi–Albert scale-free networks. We show that the latter decrease is related to the structure of cooperation. Additionally, we present a mean field approximation for studying evolutionary dynamics in networks with no degree-degree correlations and with arbitrary degree distribution. The mean field approach is similar to the one used for describing the disease spreading in complex networks, making a further compartmentalization of the strategists partition into degree-classes. We show that this kind of approximation is suitable to describe the behavior of the system for a particular set of initial conditions, such as the placement of cooperators in the higher-degree classes, while it fails to reproduce the level of cooperation observed in the numerical simulations for arbitrary initial configurations.

Coordination and growth: the Stag Hunt game on evolutionary networks

Recently, the study of evolutionary games on networks has attracted great interest, focused mainly on the problem of the emergence of cooperation. A well studied framework for this problem is the Prisoners Dilemma game on fixed, evolving or growing networks. In this paper we present a complete picture of the behavior of another important social dilemma, the Stag Hunt game, under an evolutionary preferential attachment model, in which the network grows according to the dynamical states of the elements of the system. We observe the emergence of a scale-free and hierarchical organization of the strategies according to connectivity classes as a by-product of the diffusion of cooperation in the network. Depending on the parametrization of the game dynamics, we find a smooth transition from cooperation to defection and a polymorphic state with simultaneous presence of cooperator and defector hubs, which is very unusual in coordination games.

Growing Networks Driven by the Evolutionary Prisoner’s Dilemma Game

In this chapter, we present a model of growing networks in which the attachment of nodes is driven by the dynamical state of the evolving network. In particular, we study the interplay between form and function during network formation by considering that the capacity of a node to attract new links from newcomers depends on a dynamical variable: its evolutionary fitness. The fitness of nodes are governed in turn by the payoff obtained when playing a weak Prisoner’s Dilemma game with their nearest neighbors. Thus, we couple the structural evolution of the system with its evolutionary dynamics. On the one hand, we study both the levels of cooperation observed during network evolution and the structural outcome of the model. Our results point out that scale-free networks arise naturally in this setting and that they present non-trivial topological attributes such as degree-degree correlations and hierarchical clustering. On the other hand, we also look at the long-term survival of the cooperation on top of these networks, once the growth has finished. This mechanism points to an evolutionary origin of real complex networks and can be straightforwardly applied to other kinds of dynamical networks problems.