Noise-induced breakdown of coherent collective motion in swarms

We consider swarms formed by populations of self-propelled particles with attractive long-range interactions. These swarms represent multistable dynamical systems and can be found either in coherent traveling states or in an incoherent oscillatory state where translational motion of the entire swarm is absent. Under increasing the noise intensity, the coherent traveling state of the swarms is destroyed and an abrupt transition to the oscillatory state takes place.Comment: 6 pages, 5 figures; to appear in Phys. Rev.

On the genealogy of a population of biparental individuals

If one goes backward in time, the number of ancestors of an individual doubles at each generation. This exponential growth very quickly exceeds the population size, when this size is finite. As a consequence, the ancestors of a given individual cannot be all different and most remote ancestors are repeated many times in any genealogical tree. The statistical properties of these repetitions in genealogical trees of individuals for a panmictic closed population of constant size N can be calculated. We show that the distribution of the repetitions of ancestors reaches a stationary shape after a small number Gc ~ log N of generations in the past, that only about 80% of the ancestral population belongs to the tree (due to coalescence of branches), and that two trees for individuals in the same population become identical after Gc generations have elapsed. Our analysis is easy to extend to the case of exponentially growing population.Comment: 14 pages, 7 figures, to appear in the Journal of Theoretical Biolog

Disturbing synchronization: Propagation of perturbations in networks of coupled oscillators

We study the response of an ensemble of synchronized phase oscillators to an external harmonic perturbation applied to one of the oscillators. Our main goal is to relate the propagation of the perturbation signal to the structure of the interaction network underlying the ensemble. The overall response of the system is resonant, exhibiting a maximum when the perturbation frequency coincides with the natural frequency of the phase oscillators. The individual response, on the other hand, can strongly depend on the distance to the place where the perturbation is applied. For small distances on a random network, the system behaves as a linear dissipative medium: the perturbation propagates at constant speed, while its amplitude decreases exponentially with the distance. For larger distances, the response saturates to an almost constant level. These different regimes can be analytically explained in terms of the length distribution of the paths that propagate the perturbation signal. We study the extension of these results to other interaction patterns, and show that essentially the same phenomena are observed in networks of chaotic oscillators.Comment: To appear in Eur. Phys. J.

Dynamical clustering in oscillator ensembles with time-dependent interactions

We consider an ensemble of coupled oscillators whose individual states, in addition to the phase, are characterized by an internal variable with autonomous evolution. The time scale of this evolution is different for each oscillator, so that the ensemble is inhomogeneous with respect to the internal variable. Interactions between oscillators depend on this variable and thus vary with time. We show that as the inhomogeneity of time scales in the internal evolution grows, the system undergoes a critical transition between ordered and incoherent states. This transition is mediated by a regime of dynamical clustering, where the ensemble recurrently splits into groups formed by varying subpopulations.Comment: 4 pages, 3 figure

Analysis of scale-free networks based on a threshold graph with intrinsic vertex weights

Many real networks are complex and have power-law vertex degree distribution, short diameter, and high clustering. We analyze the network model based on thresholding of the summed vertex weights, which belongs to the class of networks proposed by Caldarelli et al. (2002). Power-law degree distributions, particularly with the dynamically stable scaling exponent 2, realistic clustering, and short path lengths are produced for many types of weight distributions. Thresholding mechanisms can underlie a family of real complex networks that is characterized by cooperativeness and the baseline scaling exponent 2. It contrasts with the class of growth models with preferential attachment, which is marked by competitiveness and baseline scaling exponent 3.Comment: 5 figure

Global firing induced by network disorder in ensembles of activerotators

Abstract.: We study the influence of repulsive interactions on an ensemble of coupled excitable rotators. We find that a moderate fraction of repulsive interactions can trigger global firing of the ensemble. The regime of global firing, however, is suppressed in sufficiently large systems if the network of repulsive interactions is fully random, due to self-averaging in its degree distribution. We thus introduce a model of partially random networks with a broad degree distribution, where self-averaging due to size growth is absent. In this case, the regime of global firing persists for large sizes. Our results extend previous work on the constructive effects of diversity in the collective dynamics of complex system

Propagation of small perturbations in synchronized oscillator networks

We study the propagation of a harmonic perturbation of small amplitude on a network of coupled identical phase oscillators prepared in a state of full synchronization. The perturbation is externally applied to a single oscillator, and is transmitted to the other oscillators through coupling. Numerical results and an approximate analytical treatment, valid for random and ordered networks, show that the response of each oscillator is a rather well-defined function of its distance from the oscillator where the external perturbation is applied. For small distances, the system behaves as a dissipative linear medium: the perturbation amplitude decreases exponentially with the distance, while propagating at constant speed. We suggest that the pattern of interactions may be deduced from measurements of the response of individual oscillators to perturbations applied at different nodes of the network

Global firing induced by network disorder in ensembles of active rotators

We study the influence of repulsive interactions on an ensemble of coupled excitable rotators. We find that a moderate fraction of repulsive interactions can trigger global firing of the ensemble. The regime of global firing, however, is suppressed in sufficiently large systems if the network of repulsive interactions is fully random, due to self-averaging in its degree distribution. We thus introduce a model of partially random networks with a broad degree distribution, where self-averaging due to size growth is absent. In this case, the regime of global firing persists for large sizes. Our results extend previous work on the constructive effects of diversity in the collective dynamics of complex systems.Comment: 8 pages, 6 figure