A Model of Scale-Free Proportion Based on Mutual Anticipation

Recently, new empirical research of flocking behavior has been accumulated. Scale-free proportion has revealed how a flock can appear to behave as if it has one mind and body. The notion of scale-free proportion implies that the correlated domain within a flock is not constant size, but is proportional to flock size. Scale-free proportion can be explained by previous models, such as BOIDS based on the fixed radius neighborhood where an agent interacts with others if the critical valued parameter and a huge neighborhood are given. However, it is hard to explain under the normal neighborhood condition. The authors propose a new computational model that, although also based on BOIDS, incorporates mutual anticipation, which is implemented by modeling the resonance between the potential transitions available to each agent, allowing overlap between them. Via mutual anticipation, this model implements interactions not only among individuals but also between individuals and the field. The authors show that this model reveals the dynamic and robust structure of a flock or swarm, as well as scale-free proportion over a wide range of the flock sizes, comparing previous models, and that its predictions correlate well with empirical field data. DOI: 10.4018/jalr.2012010104 International Journal of Artificial Life Research, 3(1), 34-44, January-March 2012 35 Copyright © 2012, IGI Global. Copying or distributing in print or electronic forms without written permission of IGI Global is prohibited. & 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. Here, we present a new theoretical model that allows for ambiguity in the definition of neighborhood in a two-dimensional lattice space. In our model, each individual has its own principal vector, and a number of potential transitions are derived from the principal vector with the maximal angle. Updating of our model is asynchronously implemented with three steps. First, Individuals move to the popular site at which potential transitions among individuals is concentrated. These transitions implement using mutual anticipation. Second, if the mutual anticipation does not provide a transition, individuals follow their antecessors. Third, If Individuals could not move at above two steps, they move freely by choosing a direction from a list of potential transitions. We show that our model can demonstrate scale-free proportion, comparing previous models. (Recently, we also showed that scale-free proportion could be explained in our flock model by a switch, dependent on a particular condition, from a topological neighborhood to metric one) (Niizato & Gunji, 2010).