No abstract

A large-scale system consisting of self-propelled particles, moving under the directional alignment rule ͑DAR͒, can often self-organize to an ordered state that emerges from an initially rotationally symmetric configuration. It is commonly accepted that the DAR, which leads to effective long-range interactions, is the underlying mechanism contributing to the collective motion. However, in this paper, we demonstrate that a swarm under the DAR has unperceived and inherent singularities. Furthermore, we show that the compelled symmetry-breaking effects at or near the singularities, as well as the topological connectivity of the swarm in the evolution process, contribute fundamentally to the emergence of the collective behavior; and the elimination or weakening of singularities in the DAR will induce an unexpected sharp transition from coherent movement to isotropic dispersion. These results provide some insights into the fundamental issue of collective dynamics: What is the underlying mechanism causing the spontaneous symmetry breaking and leading to eventual coherent motion?

-An important natural phenomenon surfaces that ultrafast consensus can be achieved by introducing predictive mechanisms. By predicting the dynamics of a network several steps ahead and using this information in the consensus protocol, it is shown that, without changing the topology of the network, drastic improvements can be achieved in terms of the speed of convergence towards consensus and of the feasible range of sampling periods, compared with the routine consensus protocol. In natural science, this study provides an evidence for the idea that some predictive mechanisms exist in widely-spread biological swarms, flocks, and schools. From the industrial engineering point of view, inclusion of an efficient predictive mechanism allows for a significant increase in the consensus speed and a reduction of the required communication energy. . The collective dynamics of networks of interconnected agents is currently a subject of intensive research that has potential applications in biology, physics and engineering. In this area, one of the most general and attractive topic is the consensus problem [6][7][8], where groups of agents agree upon certain quantities such as attitude, position, temperature and voltage. Furthermore, solving consensus problem using distributed computational methods has direct implications on sensor network data fusion, load balancing, swarm control, unmanned air vehicles (UAVs), attitude alignment of satellite clusters, congestion control of communication networks, multi-agent formation control, etc. [9][10][11].One of the central problems in the study of collective dynamics is the design of consensus protocols allowing to reach consensus as quickly as possible using the least amount of communication energy. Recently, by using parallel and distributed computing approaches [12] or (a) E-mail: mc274@le.ac.uk numerical statistical analysis, some methods aiming at enhancing the consensus speed have been proposed, such as convex optimization [13], introduction of long-range interactions [14], stability analysis [15], the relation between communication intensity and convergence speed [16], design of the heterogeneous influences [17], and use of adaptive velocity [18]. Most of the previous studies on the consensus of interconnected agents has been based on the implicit assumption that an agent's state at the next discrete-time step is solely determined by the current (network) state. However, in natural bio-groups, individuals typically have some higher-level intelligence, namely predictive intelligence, which is the ability of predicting the future position of some group members based on their past and current positions. Microscopically, recent experiments on the bio-eyesight systems have revealed the predictive functions of the optical and acoustical apparatuses, especially the retina and cortex [19][20][21]. Macroscopically, experiments on bee groups [22,23] have provided evidences for the existence of predictive mechanisms in bee swarm formation and foraging. Unfortunately, predictive m...