We explore the coupled dynamics of the internal states of a set of interacting elements and the network of interactions among them. Interactions are modeled by a spatial game and the network of interaction links evolves adapting to the outcome of the game. As an example we consider a model of cooperation, where the adaptation is shown to facilitate the formation of a hierarchical interaction network that sustains a highly cooperative stationary state. The resulting network has the characteristics of a small world network when a mechanism of local neighbor selection is introduced in the adaptive network dynamics. The highly connected nodes in the hierarchical structure of the network play a leading role in the stability of the network. Perturbations acting on the state of these special nodes trigger global avalanches leading to complete network reorganization.

We investigate the relationship between the nested organization of mutualistic systems and their robustness against the extinction of species. We establish that a nested pattern of contacts is the best possible one as far as robustness is concerned, but only when the least linked species have the greater probability of becoming extinct. We introduce a coefficient that provides a quantitative measure of the robustness of a mutualistic system. 2

We propose a model for stochastic formation of opinion clusters, modeled by an evolving network, and herd behavior to account for the observed fat-tail distribution in returns of financial-price data. The only parameter of the model is h, the rate of information dispersion per trade, which is a measure of herding behavior. For h below a critical h ‫ء‬ the system displays a power-law distribution of the returns with exponential cutoff. However, for h . h ‫ء‬ an increase in the probability of large returns is found and may be associated with the occurrence of large crashes. Recently, there has been significant interest in applications of physical methods in social and economical sciences [1]. In particular, the analysis of financial stock market prices has been found to exhibit some universal characteristics similar to those observed in physical systems with a large number of interacting units, and several microscopic models have been developed to study them [2][3][4]. For example, the distribution of the so-called returns, i.e., the logarithmic change of the market price, has been observed to present pronounced tails larger than in a Gaussian distribution [2,3,[5][6][7]. Several models have been put forward which phenomenologically show the fat-tail distributions. Among the more sophisticated approaches are dynamic multiagent models [4,8] based on the interaction of two distinct agent populations ("noisy" and "fundamentalists" traders), which reproduces the desired distributions, but fails to account for the origin of the universal behavior. An alternative approach, explored in this Letter, is that herd behavior [9,10] may be sufficient to induce the desired distributions. Herding assumes some degree of coordination between a group of agents. This coordination may arise in different ways, either because agents share the same information or they follow the same rumor. This approach has been recently formalized by Cont and Bouchaud [11] as a static percolation model. We present a model for opinion cluster formation and information dispersal by agents in a network. As a first approach to model this complicated social behavior we consider (i) a random dispersion of information, (ii) agents sharing the same information form a group that makes decisions as a whole (herding), and (iii) whenever a group performs an action, the network necessarily adapts to this change. We then apply the model to study the price dynamics in a financial market. Our results show that when the information dispersion is much faster than trading activity, the distribution of the number of agents sharing the same information behaves as a power law. Using a linear relationship for the price update in terms of the order size [11,12], the price returns also exhibit this universal feature (with a different exponent). On the other hand when the dispersion of information becomes slower, a smooth transition to truncated exponential tails is observed, with a portion of the distribution remaining close to the power law. In our approach the average conne...

By means of extensive computer simulations, the authors consider the entangled coevolution of actions and social structure in a new version of a spatial Prisoners Dilemma model that naturally gives way to a process of social differentiation. Diverse social roles emerge from the dynamics of the system: leaders are individuals getting a large payoff who are imitated by a considerable fraction of the population, conformists are unsatisfied cooperative agents that keep cooperating, and exploiters are defectors with a payoff larger than the average one obtained by cooperators. The dynamics generate a social network that can have the topology of a small world network. The network has a strong hierarchical structure in which the leaders play an essential role in sustaining a highly cooperative stable regime. But disruptions affecting leaders produce social crises described as dynamical cascades that propagate through the network.

Cooperative behavior among a group of agents is studied assuming adaptive interactions. Each agent plays a Prisoner's Dilemma game with its local neighbors, collects an aggregate payoff, and imitates the strategy of its best neighbor. Agents may punish or reward their neighbors by removing or sustaining the interactions, according to their satisfaction level and strategy played. An agent may dismiss an interaction, and the corresponding neighbor is replaced by another randomly chosen agent, introducing diversity and evolution to the network structure. We perform an extensive numerical and analytical study, extending results in M. G. Zimmermann, V. M. Eguíluz, and M. San Miguel, Phys. Rev. E 69, 065102(R) (2004). We show that the system typically reaches either a full-defective state or a highly cooperative steady state. The latter equilibrium solution is composed mostly by cooperative agents, with a minor population of defectors that exploit the cooperators. It is shown how the network adaptation dynamics favors the emergence of cooperators with the highest payoff. These "leaders" are shown to sustain the global cooperative steady state. Also we find that the average payoff of defectors is larger than the average payoff of cooperators. Whenever "leaders" are perturbed (e.g., by addition of noise), an unstable situation arises and global cascades with oscillations between the nearly full defection network and the fully cooperative outcome are observed.

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We introduce and describe periodic coherent structures of the CGLE, called Modulated Amplitude Waves (MAWs). MAWs of various period P occur naturally in phase chaotic states. A bifurcation study of the MAWs reveals that for sufficiently large period, pairs of MAWs cease to exist via a saddle-node bifurcation. For periods beyond this bifurcation, incoherent near-MAW structures occur which evolve toward defects. This leads to our main result: the transition from phase to defect chaos takes place when the periods of MAWs in phase chaos are driven beyond their saddle-node bifurcation.PACS numbers: 47.52.+j, 03.40.Kf, 05.45.+b, 47.54.+r Spatially extended systems can exhibit, when driven away from equilibrium, irregular behavior in space and time: this phenomenon is commonly referred to as spatio-temporal chaos [1]. The one-dimensional complex Ginzburg-Landau equation (CGLE):describes pattern formation near a Hopf bifurcation and has become a popular model to study spatiotemporal chaos [1][2][3][4][5][6][7][8][9][10][11][12][13]. As a function of c 1 and c 3 , the CGLE exhibits two qualitatively different spatiotemporal chaotic states known as phase chaos (when A is bounded away from zero) and defect chaos (when the phase of A displays singularities where A = 0). The transition from phase to defect chaos can either be hysteretic or continuous; in the former case, it is referred to as L 3 , in the latter as L 1 (Fig. 1). Despite intensive studies [5][6][7][8][9][10][11][12][13], the phenomenology of the CGLE and in particular its "phase"-diagram [5,7] are far from being understood. Moreover, it is under dispute whether the L 1 transition is sharp, and whether a pure phase-chaotic (i.e. defect-free) state can exist in the thermodynamic limit [9]. It is the purpose of this paper to elucidate these issues by presenting the mechanism which creates defects in transient phase chaotic states. Our analysis consists of four parts: (i) We describe a family of Modulated Amplitude Waves (MAWs), i.e., pulse-like coherent structures with a characteristic spatial period P . (ii) A bifurcation analysis of these MAWs reveals that their range of existence is limited by a saddle-node (SN) bifurcation. For all c 1 , c 3 within a certain range, we define P SN as the period of the MAW for which this bifurcation occurs. (iii) We show that for P > P SN , i.e., beyond the SN bifurcation, near-MAW structures display a nonlinear evolution to defects. It is found that, in phase chaos, near-MAWs with various P 's are created and annihilated perpetually. L1, L2 and L3 transitions (after [7]). Between the L2 and L3 curves, there is the hysteretic regime where either phase or defect chaos can occur; in the latter case, defects persist up to the L2 transition. Notice how the L1 and L3 transitions to defect chaos lie above our lower (P → ∞) bounds. Also shown are the SN locations for P = 20, 50.The transition to def...

Abstract. Delay games are two-player games of infinite duration in which one player may delay her moves to obtain a lookahead on her opponent's moves. For ω-regular winning conditions it is known that such games can be solved in doubly-exponential time and that doubly-exponential lookahead is sufficient.We improve upon both results by giving an exponential time algorithm and an exponential upper bound on the necessary lookahead. This is complemented by showing ExpTimehardness of the solution problem and tight exponential lower bounds on the lookahead. Both lower bounds already hold for safety conditions. Furthermore, solving delay games with reachability conditions is shown to be PSpace-complete.

The transition from phase chaos to defect chaos in the complex Ginzburg-Landau equation (CGLE) is related to saddle-node bifurcations of modulated amplitude waves (MAWs). First, the spatial period P of MAWs is shown to be limited by a maximum P SN which depends on the CGLE coefficients; MAW-like structures with period larger than P SN evolve to defects. Second, slowly evolving near-MAWs with average phase gradients ν ≈ 0 and various periods occur naturally in phase chaotic states of the CGLE. As a measure for these periods, we study the distributions of spacings p between neighboring peaks of the phase gradient. A systematic comparison of p and P SN as a function of coefficients of the CGLE shows that defects are generated at locations where p becomes larger than P SN . In other words, MAWs with period P SN represent "critical nuclei" for the formation of defects in phase chaos and may trigger the transition to defect chaos. Since rare events where p becomes sufficiently large to lead to defect formation may only occur after a long transient, the coefficients where the transition to defect chaos seems to occur depend on system size and integration time. We conjecture that in the regime where the maximum period P SN has diverged, phase chaos persists in the thermodynamic limit.