Groups of animals coordinate remarkable, coherent, movement patterns during periods of collective motion. Such movement patterns include the toroidal mills seen in fish shoals, highly aligned parallel motion like that of flocks of migrating birds, and the swarming of insects. Since the 1970’s a wide range of collective motion models have been studied that prescribe rules of interaction between individuals, and that are capable of generating emergent patterns that are visually similar to those seen in real animal group. This does not necessarily mean that real animals apply exactly the same interactions as those prescribed in models. In more recent work, researchers have sought to infer the rules of interaction of real animals directly from tracking data, by using a number of techniques, including averaging methods. In one of the simplest formulations, the averaging methods determine the mean changes in the components of the velocity of an individual over time as a function of the relative coordinates of group mates. The averaging methods can also be used to estimate other closely related quantities including the mean relative direction of motion of group mates as a function of their relative coordinates. Since these methods for extracting interaction rules and related quantities from trajectory data are relatively new, the accuracy of these methods has had limited inspection. In this paper, we examine the ability of an averaging method to reveal prescribed rules of interaction from data generated by two individual based models for collective motion. Our work suggests that an averaging method can capture the qualitative features of underlying interactions from trajectory data alone, including repulsion and attraction effects evident in changes in speed and direction of motion, and the presence of a blind zone. However, our work also illustrates that the output from a simple averaging method can be affected by emergent group level patterns of movement, and the sizes of the regions over which repulsion and attraction effects are apparent can be distorted depending on how individuals combine interactions with multiple group mates.

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Emergent patterns of collective motion are thought to arise from local rules of interaction that govern how individuals adjust their velocity in response to the relative locations and velocities of near neighbours. Many models of collective motion apply rules of interaction over a metric scale, based on the distances to neighbouring group members. However, empirical work suggests that some species apply interactions over a topological scale, based on distance determined neighbour rank. Here, we modify an important metric model of collective motion (Couzin et al. in J Theor Biol 218(1): [1][2][3][4][5][6][7][8][9][10][11] 2002), so that interactions relating to orienting movements with neighbours and attraction towards more distant neighbours operate over topological scales. We examine the emergent group movement patterns generated by the model as the numbers of neighbours that contribute to orientation-and attraction-based velocity adjustments vary. Like the metric form of the model, simulated groups can fragment (when interactions are influenced by less than 10-15% of the group), swarm and move in parallel, but milling does not occur. The model also generates other cohesive group movements including cases where groups exhibit directed motion without strong overall alignment of individuals. Multiple emergent states are possible for the same set of underlying model parameters in some cases, suggesting sensitivity to initial conditions, and there is evidence that emergent states of the system depend on the history of the system. Groups that do not fragment tend to stay relatively compact in terms of neighbour distances. Even if a group does fragment, individuals remain relatively close to near neighbours, avoiding complete isolation.