Maxi San Miguel

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.

Homophily, Cultural Drift and the Co-Evolution of Cultural Groups

Studies of cultural differentiation have shown that social mechanisms that normally lead to cultural convergence—homophily and influence—can also explain how distinct cultural groups can form. However, this emergent cultural diversity has proven to be unstable in the face of cultural drift—small errors or innovations that allow cultures to change from within. The authors develop a model of cultural differentiation that combines the traditional mechanisms of homophily and influence with a third mechanism of network homophily, in which network structure co-evolves with cultural interaction. Results show that in certain regions of the parameter space, these co-evolutionary dynamics can lead to patterns of cultural diversity that are stable in the presence of cultural drift. The authors address the implications of these findings for understanding the stability of cultural diversity in the face of increasing technological trends toward globalization.

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.

Voter model dynamics in complex networks: Role of dimensionality, disorder, and degree distribution

We analyze the ordering dynamics of the voter model in different classes of complex networks. We observe that whether the voter dynamics orders the system depends on the effective dimensionality of the interaction networks. We also find that when there is no ordering in the system, the average survival time of metastable states in finite networks decreases with network disorder and degree heterogeneity. The existence of hubs, i.e., highly connected nodes, in the network modifies the linear system size scaling law of the survival time. The size of an ordered domain is sensitive to the network disorder and the average degree, decreasing with both; however, it seems not to depend on network size and on the heterogeneity of the degree distribution.

Generic absorbing transition in coevolution dynamics

We study a coevolution voter model on a complex network. A mean-field approximation reveals an absorbing transition from an active to a frozen phase at a critical value [see text for formula] that only depends on the average degree micro of the network. In finite-size systems, the active and frozen phases correspond to a connected and a fragmented network, respectively. The transition can be seen as the sudden change in the trajectory of an equivalent random walk at the critical point, resulting in an approach to the final frozen state whose time scale diverges as tau approximately |p(c) - p|(-)} near p(c).

Global culture: A noise-induced transition in finite systems

We analyze the effect of cultural drift, modeled as noise, in Axelrods model for the dissemination of culture. The disordered multicultural frozen configurations are found not to be stable. This general result is proven rigorously in d=1, where the dynamics is described in terms of a Lyapunov potential. In d=2, the dynamics is governed by the average relaxation time T of perturbations. Noise at a rate r <or=T(-1) induces monocultural configurations, whereas r > or =T(-1) sustains disorder. In the thermodynamic limit, the relaxation time diverges and global polarization persists in spite of a dynamics of local convergence.

Nonequilibrium transitions in complex networks: a model of social interaction

We analyze the nonequilibrium order-disorder transition of Axelrods model of social interaction in several complex networks. In a small-world network, we find a transition between an ordered homogeneous state and a disordered state. The transition point is shifted by the degree of spatial disorder of the underlying network, the network disorder favoring ordered configurations. In random scale-free networks the transition is only observed for finite size systems, showing system size scaling, while in the thermodynamic limit only ordered configurations are always obtained. Thus, in the thermodynamic limit the transition disappears. However, in structured scale-free networks, the phase transition between an ordered and a disordered phase is restored.

STOCHASTIC EFFECTS IN PHYSICAL SYSTEMS

The study of the effects of noise and fluctuations is a well established subject in several different disciplines ranging from pure mathematics (stochastic processes) to physics (fluctuations) and electrical engineering (noise and radiophysics). In traditional statistical physics, fluctuations are of thermal origin giving rise to small departures from a mean value. They tend to zero as one approaches the thermodynamic limit in which different statistical descriptions (different ensembles) become equivalent. Likewise, in more applied contexts fluctuations or noise are usually regarded as small corrections to a deterministic (noise free) behavior that degrades a signal-to-noise ratio or can cause transmission errors. In such framework fluctuations are a correction that can be usually dealt with through some sort of linearization of dynamics around a mean or noise free dynamics. A different point of view about fluctuations emerges, for example, in the study of critical phenomena in the 1970’s. The statistical physics description of these phenomena requires a formulation appropriate for a system dominated by fluctuations and nonlinear it ies. A linear theory only identifies the existence of a critical point by a divergence of fluctuations.

A measure of individual role in collective dynamics

Identifying key players in collective dynamics remains a challenge in several research fields, from the efficient dissemination of ideas to drug target discovery in biomedical problems. The difficulty lies at several levels: how to single out the role of individual elements in such intermingled systems, or which is the best way to quantify their importance. Centrality measures describe a nodes importance by its position in a network. The key issue obviated is that the contribution of a node to the collective behavior is not uniquely determined by the structure of the system but it is a result of the interplay between dynamics and network structure. We show that dynamical influence measures explicitly how strongly a nodes dynamical state affects collective behavior. For critical spreading, dynamical influence targets nodes according to their spreading capabilities. For diffusive processes it quantifies how efficiently real systems may be controlled by manipulating a single node.

Ordering dynamics with two non-excluding options: bilingualism in language competition

We consider an extension of the voter model in which a set of interacting elements (agents) can be in either of two equivalent states (A or B )o r in a third additional mixed (AB) state. The model is motivated by studies of language competition dynamics, where the AB state is associated with bilingualism. We study the ordering process and associated interface and coarsening dynamics in regular lattices and small world networks. Agents in the AB state define the interfaces, changing the interfacial noise driven coarsening of the voter model to curvature driven coarsening. This change in the coarsening mechanism is also shown to originate for a class of perturbations of the voter model dynamics. When interaction is through a small world network the AB agents restore coarsening, eliminating the metastable states of the voter model. The characteristic time to reach the absorbing state scales with system size as τ ∼ lnN to be compared with the result τ ∼ N for the voter model in a small world network.