Martin G. Zimmermann

Coevolution of dynamical states and interactions in dynamic networks.

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 in which 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.

Transmission of information and herd behavior: An application to financial markets

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(*) the system displays a power-law distribution of the returns with exponential cutoff. However, for h>h(*) an increase in the probability of large returns is found and may be associated with the occurrence of large crashes.

Cooperation and the Emergence of Role Differentiation in the Dynamics of Social Networks

By means of extensive computer simulations, the authors consider the entangled coevolution of actions and social structure in a new version of a spatial Prisoner’s 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.

Why nestedness in mutualistic networks

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.

Pulse bifurcation and transition to spatiotemporal chaos in an excitable reaction-diffusion model

We address the stability of solitary pulses as well as some other traveling structures near the onset of spatiotemporal chaos in a two-species reaction-diffusion model describing the oxidation of C ...

Modulated amplitude waves and defect formation in the one-dimensional complex Ginzburg-Landau equation

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 PSN which depends on the CGLE coefficients; MAW-like structures with period larger than PSN 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 neighbouring peaks of the phase gradient. A systematic comparison of p and PSN as a function of coefficients of the CGLE shows that defects are generated at locations where p becomes larger than PSN. In other words, MAWs with period PSN 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 PSN has diverged, phase chaos persists in the thermodynamic limit.

Modulated Amplitude Waves and the Transition from Phase to Defect Chaos

The mechanism for transitions from phase to defect chaos in the one-dimensional complex Ginzburg-Landau equation (CGLE) is presented. We describe periodic coherent structures of the CGLE, called modulated amplitude waves (MAWs). MAWs of various periods P occur 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 evolve towards 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.

Analysis and assembling of network structure in mutualistic systems.

It has been observed that mutualistic bipartite networks have a nested structure of interactions. In addition, the degree distributions associated with the two guilds involved in such networks (e.g., plants and pollinators or plants and seed dispersers) approximately follow a truncated power law (TPL). We show that nestedness and TPL distributions are intimately linked, and that any biological reasons for such truncation are superimposed to finite size effects. We further explore the internal organization of bipartite networks by developing a self-organizing network model (SNM) that reproduces empirical observations of pollination systems of widely different sizes. Since the only inputs to the SNM are numbers of plant and animal species, and their interactions (i.e., no data on local abundance of the interacting species are needed), we suggest that the well-known association between species frequency of interaction and species degree is a consequence rather than a cause, of the observed network structure.

Cooperation, Adaptation and the Emergence of Leadership

A generic property of biological, social and economical networks is their ability to evolve in time, creating and suppressing interactions. We approach this issue within the framework of an adaptive network of agents playing a Prisoner’s Dilemma game, where each agent plays with its local neighbors, collects an aggregate payoff and imitates the strategy of its best neighbor. We allow the agents to adapt their local neighborhood according to their satisfaction level and the strategy played. We show that a steady state is reached, where the strategy and network configurations remain stationary. While the fraction of cooperative agents is high in these states, their average payoff is lower than the one attained by the defectors. The system self-organizes in such a way that the structure of links in the network is quite inhomogeneous, revealing the occurrence of cooperator “leaders” with a very high connectivity, which guarantee that global cooperation can be sustained in the whole network. Perturbing the leaders produces drastic changes of the network, leading toglobal dynamical cascades.These cascades induce a transient oscillation in the population of agents between the nearly all-defectors state and the all-cooperators outcome, before setting again in a state of high global cooperation.

Cooperation in an Adaptive Network

We study the dynamics of a set of agents distributed in the nodes of an adaptive network. Each agent plays with all its neighbors a weak prisoners dilemma collecting a total payoff. We study the case where the network adapts locally depending on the total payoff of the agents. In the parameter regime considered, a steady state is always reached (strategies and network configuration remain stationary), where co-operation is highly enhanced. However, when the adaptability of the network and the incentive for defection are high enough, we show that a slight perturbation of the steady state induces large oscillations (with cascades) in behavior between the nearly all-defectors state and the all-cooperators outcome.