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Top1. Introduction
Collective behavior, from animals to microorganisms, is a fascinating natural phenomenon (Gueron, Levin, & Rubenstein, 1996; Emlen, 1952; Schaller, Weber, Semmrich, Frey, & Bausch, 2010; Allison & Hughes, 1991; Rauch, Millonas, & Chialvo, 1995), and of particular interest is the manner in which a flock or a swarm which moves very quickly can appear to behave as if it were a single organism (Couzin, 2007). Regardless of the kind of animal that is considered, the similarities in the behavior of different groups of animals suggest that a simple and local rule may underlie collective motion. The classical model of collective behavior BOIDS is proposed to use in Computer Graphics (Reynolds, 1987). Agent in BOIDS has neighborhood with certain radius (called the metric distance) in which they separate from neighbors if they are too close each other (collision avoidance), align their direction (velocity matching) and approach to neighbors if they separate each other (flock centering). The model specialized in “velocity matching” is SPP (self-propelled particle) (Szabo, Nagy, & Vicsek, 2009; Vicsek, Czirok, Ben-Jacob, & Shochet, 1995). In this model, each agent interacts with their neighbors in a neighborhood with a fixed radius by averaging the directions of motion of its neighbors (i.e., the velocity matching), coupled with external random noise. The SPP model shows a phase transition with respect to the average of the entire alignment plotted against the external noise. Based on experimental studies of actual animals, it has been proposed that this transition depends on density. Couzin and others showed that, in aggregating locusts, a one-dimensional phase transition from disordered to ordered is dependent on this density and that this behavior can be mimicked by SPP (Buhl, Sumpter, Couzin, Hale, Despland, Miller, & Simpson, 2006).
The model based on a neighborhood with metric distance has been regarded as a powerful tool to explain animal collective behavior in general, although discrepancies between the metric distance and empirical data have also been reported (Gregoire, Chate, & Tu, 2003; Gregoire & Chate, 2004). Use of a neighborhood with an indefinite boundary described by fuzzy logic or with a variable boundary has also been introduced as a variation on BOIDS or SPP (Bajec, Zimic, & Mraz, 2005).
Ballerini and others, on the basis of their empirical studies, have shown that the interaction between individuals in a flock does not rely on metric distance, but rather on topological distance (Ballerini et al., 2008a, 2008b). Although the notion of metric distance means that a bird interacts with neighbors within a fixed distance, the notion of topological distance means that a bird interacts with a fixed number of neighbors (e.g., six or seven). In their simulation with predator’s attack, once individuals leave their metric distance neighborhood, they can no longer interact with other members and cannot return to the flock. Agents in a topological flock, however, can rejoin their group if they stray, because their interactions are based on topological distance. Although the notion of topological distance models some aspects of flocking behavior well, like metric distance, it has some problems. First, if each agent interacts with only seven neighbors, information propagation through an entire flock is very slow when size of the flock is very large. Second, it is not clear how a single bird would be able to monitor its distant flockmates. Another important discovery about animal collective behavior is scale-free proportion in a bird flock (Cavagnaa et al., 2010). The distribution of fluctuation vectors in a flock indicates that it contains a conspicuous sub-domain in which fluctuation vectors are aligned and synchronized. Here, fluctuation vector means that the difference of the velocity of the individual from the mean in a flock. It has also been shown that the scale of this sub-domain is proportional to the size of the flock.