Numerical models indicate that collective animal behavior may emerge from simple local rules of interaction among the individuals. However, very little is known about the nature of such interaction, so that models and theories mostly rely on aprioristic assumptions. By reconstructing the three-dimensional positions of individual birds in airborne flocks of a few thousand members, we show that the interaction does not depend on the metric distance, as most current models and theories assume, but rather on the topological distance. In fact, we discovered that each bird interacts on average with a fixed number of neighbors (six to seven), rather than with all neighbors within a fixed metric distance. We argue that a topological interaction is indispensable to maintain a flock's cohesion against the large density changes caused by external perturbations, typically predation. We support this hypothesis by numerical simulations, showing that a topological interaction grants significantly higher cohesion of the aggregation compared with a standard metric one.animal groups ͉ behavioral rules ͉ flocking ͉ self-organization C ollective behavior of large aggregations of animals is a truly fascinating natural phenomenon (1). Particularly interesting is the case when aggregations self-organize into complex patterns with no need of an external stimulus (2). Prominent examples of such behavior are bird flocks (3), fish schools (4) and mammal herds (5). Apart from its obvious relevance in ethology and evolutionary biology, collective behavior is a key concept in many other fields of science, including control theory (6), economics (7), and social sciences (8).How does collective behavior emerge? Numerical models of self-organized motion, inspired both by biology (9-15) and physics (16)(17)(18)(19)(20), support the idea that simple rules of interaction among the individuals are sufficient to produce collective behavior. Unfortunately, we have very scarce empirical information about the precise nature of such rules. The main theoretical assumptions (attraction among the individuals, short range repulsion, and alignment of the velocities) are reasonable, but generic, and there are as many different models as different ways to implement these assumptions. Without decisive experimental feedback it is difficult to select what is the ''right'' model and, therefore, to understand what are the underlying fundamental rules of animal collective behavior.The main goal of the interaction among individuals is to maintain cohesion of the group. This cohesion is a very strong biological requirement, shaped by the evolutionary pressure for survival: Stragglers and small groups are significantly more prone to predation than animals belonging to large and highly cohesive aggregations (4). Consider a flock of starlings under attack by a peregrine falcon: The flock contracts, expands, and even splits, continuously changing its density and structure. Yet, no bird remains isolated, and soon the flock reforms as whole. The question we want to answer i...

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From bird flocks to fish schools, animal groups often seem to react to environmental perturbations as if of one mind. Most studies in collective animal behavior have aimed to understand how a globally ordered state may emerge from simple behavioral rules. Less effort has been devoted to understanding the origin of collective response, namely the way the group as a whole reacts to its environment. Yet, in the presence of strong predatory pressure on the group, collective response may yield a significant adaptive advantage. Here we suggest that collective response in animal groups may be achieved through scale-free behavioral correlations. By reconstructing the 3D position and velocity of individual birds in large flocks of starlings, we measured to what extent the velocity fluctuations of different birds are correlated to each other. We found that the range of such spatial correlation does not have a constant value, but it scales with the linear size of the flock. This result indicates that behavioral correlations are scale free: The change in the behavioral state of one animal affects and is affected by that of all other animals in the group, no matter how large the group is. Scale-free correlations provide each animal with an effective perception range much larger than the direct interindividual interaction range, thus enhancing global response to perturbations. Our results suggest that flocks behave as critical systems, poised to respond maximally to environmental perturbations. O f all distinctive traits of collective animal behavior the most conspicuous is the emergence of global order, namely the fact that all individuals within the group synchronize to some extent their behavioral state (1-3). In many cases global ordering amounts to an alignment of the individual directions of motion, as in bird flocks, fish schools, mammal herds, and in some insect swarms (4-6). Yet, global ordering can affect also other behavioral states, as it happens with the synchronous flashing of tropical fireflies (7) or the synchronous clapping in human crowds (8).The presence of order within an animal group is easy to detect. However, order may have radically different origins, and discovering what is the underlying coordination mechanism is not straightforward. Order can be the effect of a top-down centralized control mechanism (for example, due to the presence of one or more leaders), or it can be a bottom-up self-organized feature emerging from local behavioral rules (9). In reality, the lines are often blurred and hierarchical and distributed control may combine together (10). However, even in the two extreme cases, discriminating between the two types of global ordering is not trivial. In fact, the prominent difference between the centralized and the self-organized paradigm is not order, but response.Collective response is the way a group as a whole reacts to its environment. It is often crucial for a group, or for subsets of it, to respond coherently to perturbations. For gregarious animals under strong predatory pressure,...

Flocking is a typical example of emergent collective behavior, where interactions between individuals produce collective patterns on the large scale. Here we show how a quantitative microscopic theory for directional ordering in a flock can be derived directly from field data. We construct the minimally structured (maximum entropy) model consistent with experimental correlations in large flocks of starlings. The maximum entropy model shows that local, pairwise interactions between birds are sufficient to correctly predict the propagation of order throughout entire flocks of starlings, with no free parameters. We also find that the number of interacting neighbors is independent of flock density, confirming that interactions are ruled by topological rather than metric distance. Finally, by comparing flocks of different sizes, the model correctly accounts for the observed scale invariance of long-range correlations among the fluctuations in flight direction.animal groups | statistical inference T he collective behavior of large groups of animals is an imposing natural phenomenon, very hard to cast into a systematic theory (1). Physicists have long hoped that such collective behaviors in biological systems could be understood in the same way as we understand collective behavior in physics, where statistical mechanics provides a bridge between microscopic rules and macroscopic phenomena (2, 3). A natural test case for this approach is the emergence of order in a flock of birds: Out of a network of distributed interactions among the individuals, the entire flock spontaneously chooses a unique direction in which to fly (4), much as local interactions among individual spins in a ferromagnet lead to a spontaneous magnetization of the system as a whole (5). Despite detailed development of these ideas (6-9), there still is a gap between theory and experiment. Here we show how to bridge this gap by constructing a maximum entropy model (10) based on field data of large flocks of starlings (11-13). We use this framework to show that the effective interactions among birds are local and that the number of interacting neighbors is independent of flock density, confirming that interactions are ruled by topological rather than metric distance (14). The statistical mechanics models that we derive in this way provide an essentially complete, parameter-free theory for the propagation of directional order throughout the flock.We consider flocks of European starlings, Sturnus vulgaris, as in Fig. 1A. At any given instant of time, following refs. 11-13, we can attach to each bird i a vector velocity~v i and define the normalized velocity~s i ¼~v i ∕j~v i j (Fig. 1B). On the hypothesis that flocks have statistically stationary states, we can think of all these normalized velocities as being drawn (jointly) from a probability distribution Pðf~s i gÞ. It is not possible to infer this full distribution directly from experiments, because the space of states specified by f~s i g is too large. However, what we can measure from field data is th...