^{1}

^{2}

^{3}

^{3}

^{1}

^{*}

The authors have declared that no competing interests exist.

Conceived and designed the experiments: GFY LS NEL. Performed the experiments: GFY. Analyzed the data: GFY AC IG NEL. Contributed reagents/materials/analysis tools: GFY LS AC IG NEL. Wrote the paper: GFY NEL.

Flocks of starlings exhibit a remarkable ability to maintain cohesion as a group in highly uncertain environments and with limited, noisy information. Recent work demonstrated that individual starlings within large flocks respond to a fixed number of nearest neighbors, but until now it was not understood why this number is seven. We analyze robustness to uncertainty of consensus in empirical data from multiple starling flocks and show that the flock interaction networks with six or seven neighbors optimize the trade-off between group cohesion and individual effort. We can distinguish these numbers of neighbors from fewer or greater numbers using our systems-theoretic approach to measuring robustness of interaction networks as a function of the network structure, i.e., who is sensing whom. The metric quantifies the disagreement within the network due to disturbances and noise during consensus behavior and can be evaluated over a parameterized family of hypothesized sensing strategies (here the parameter is number of neighbors). We use this approach to further show that for the range of flocks studied the optimal number of neighbors does not depend on the number of birds within a flock; rather, it depends on the shape, notably the thickness, of the flock. The results suggest that robustness to uncertainty may have been a factor in the evolution of flocking for starlings. More generally, our results elucidate the role of the interaction network on uncertainty management in collective behavior, and motivate the application of our approach to other biological networks.

Starling flocks move in beautiful ways that both captivate and intrigue the observer. Previous work has shown that starlings pay attention to their seven closest neighbors, but until now it was not understood why this number is seven. Our paper explains the mystery: when uncertainty in sensing is present, interacting with six or seven neighbors optimizes the balance between group cohesiveness and individual effort. To prove this result we develop a new systems-theoretic approach for understanding noisy consensus dynamics. The approach allows the evaluation of robustness over a family of hypothesized sensing strategies using observations of the spatial positions of birds within the flock. We apply this approach to experimental data from wild starling flocks, and find that six or seven neighbors yield maximal robustness. The implication that robustness of cohesion may have been a factor in the evolution of flocking has significant consequences for evolutionary biology. In addition, the results and the versatility of the approach have implications for uncertainty management in social and technological networks.

Flocks of birds and schools of fish exhibit striking and robust collective behaviors despite the challenging environments in which they live

Recent analysis of position

Here, we address the question of what is the connection between the number of neighbors used by each bird for social information and the robustness of the flock as a whole. We evaluate robustness for starling flocks using three-dimensional positions of birds studied in

Our systems-theoretic approach makes it possible to evaluate robustness to uncertainty over a parameterized family of hypothesized individual sensing strategies given observations of the group. For the starling flocks we evaluate the set of strategies corresponding to each individual sensing and responding to a fixed number of closest neighbors. Since the interaction structure of each starling flock network is determined by the measured spatial distribution of the birds and the strategy that each bird uses to determine which neighbors it senses, we can apply our metric to the starling flock data to distinguish which strategy (i.e., which number of neighbors), among a parameterized family of strategies (i.e., the family parameterized by number of neighbors), minimizes the influence of uncertainty on how close the birds come to consensus.

Assuming that every bird in a flock responds to a fixed number of neighbors (

By analyzing variations between different flocks, we show further that for the range of flocks observed the optimal number of neighbors (

Most models of flocking are based on consensus behavior

Recent work has shown that

By defining an

In this setting, consensus corresponds to

We measure the robustness of consensus to noise by the expected steady-state disagreement when every agent has a unit-intensity i.i.d. source of white noise (

The disagreement in the system is the length of

The metric depends on

Our robustness metric is most suitable to our purposes: since the metric only depends on the sensing graph, we can evaluate robustness for different sensing strategies (e.g., choice of

Previous analysis of the observed positions of starlings within large flocks (440 to 2600 birds) has shown that the birds interact with seven nearest neighbors, irrespective of flock density

It is possible that the birds weight the information from different neighbors differently, for example, depending on their distance or how well they are sensed. To be conservative and consistent, we consider that each individual uses an unweighted average of the information from its

The cost for an individual starling to sense the behavior of each neighbor comes from sensory and neurological requirements as well as time lost for watching for predators or searching for a roosting site, etc. It is known that birds have a limited and thus costly capability for tracking multiple objects

We computed robustness per neighbor (

For small values of

For all ten flocks studied in

For each flock the curve shown is the average of all snapshots taken of that flock, with error bars showing the standard deviation. The overall average, shown as the blue curve, is an average of the twelve flocks, with error bars showing the standard deviation. If, instead, an average is taken of every snapshot (394 in total), the resulting curve and standard deviations are almost identical (see

We further investigated observed variation in the value of

In a fully random group, the number of neighbors required for connectivity, and hence

Different snapshots from the same flock have different numbers of birds due to occlusions.

Instead we observed a strong dependence of both

Flock thickness is defined as the ratio of smallest to largest dimension of an ellipsoid having the same principal moments of inertia as the flock.

To further understand the dependence on thickness, we generated random flocks of varying thickness (with 1200 individuals). For uniformly distributed flocks,

Our analysis shows that the size (seven) of each starling's neighborhood

The trade-off seen here between robustness and sensing cost is not observed for performance metrics related to responsiveness, such as the speed of convergence to consensus (

Although we observed variability in our computed values of

Although here we have focussed on the sensing strategy of interacting with

The nonlinear dependence on thickness observed in the random flocks suggests that a transition between “2-d” and “3-d” behavior takes place as thickness increases, with a flock behaving as fully 3-d when its thickness is above about 0.4. There appear to be aerodynamic reasons why starling flocks should be thin and sheet-like

More generally, our work demonstrates the significant role of who is interacting with whom in the ability of a network to efficiently manage uncertainty when seeking to maintain consensus. This suggests possibilities for understanding and evaluating uncertainty management in other social and technological networks. Our systems-theoretic approach to evaluating robustness to uncertainty in consensus can be applied to interaction networks in these other contexts; distinguishing interaction strategies that yield networks that optimize robustness can be useful both for better understanding observed group behavior and, when control is available, for designing high performing groups.

(EPS)

(EPS)

(EPS)

(EPS)

(EPS)

(EPS)

(EPS)

(EPS)

(EPS)

The authors thank three anonymous reviewers for helpful comments and suggestions. A.C. and I.G. thank the Lewis-Sigler Institute for Integrative Genomics and the Physics Department at Princeton University for hospitality at the beginning of this work. Our collaboration was facilitated by the Initiative for the Theoretical Science at the Graduate Center of the City University of New York.