New tools for characterizing swarming systems: A comparison of minimal models

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Abstract

We compare three simple models that reproduce qualitatively the emergent swarming behavior of bird flocks, fish schools, and other groups of self-propelled agents by using a new set of diagnosis tools related to the agents’ spatial distribution. Two of these correspond in fact to different implementations of the same model, which had been previously confused in the literature. All models appear to undergo a very similar order-to-disorder phase transition as the noise level is increased if we only compare the standard order parameter, which measures the degree of agent alignment. When considering our novel quantities, however, their properties are clearly distinguished, unveiling previously unreported qualitative characteristics that help determine which model best captures the main features of realistic swarms. Additionally, we analyze the agent clustering in space, finding that the distribution of cluster sizes is typically exponential at high noise, and approaches a power-law as the noise level is reduced. This trend is sometimes reversed at noise levels close to the phase transition, suggesting a non-trivial critical behavior that could be verified experimentally. Finally, we study a bi-stable regime that develops under certain conditions in large systems. By computing the probability distributions of our new quantities, we distinguish the properties of each of the coexisting metastable states. Our study suggests new experimental analyses that could be carried out to characterize real biological swarms.

Introduction

For more than two decades, there has been a continuing interest in finding simple models that can describe the collective motion of groups of self-propelled agents [1], [2], [3], [4], [5], [6], [7], [8]. In nature, these systems typically correspond to swarms of living organisms, such as bird flocks, fish schools, or herds of quadrupeds [9], [10], [11], [12], [13]. While their specific biological details can be very different and quite intricate, in all of these examples the agents are able to achieve a surprising amount of coordination in their collective displacements, in spite of having no apparent leader or long-range communication mechanism. At the origin of this swarming behavior is the ability that groups of agents have to, under certain circumstances, align and start heading in a common direction. Various models have been introduced to study this phenomenon [1], [4], [7], [8], [14], [15], [16]. Among the simplest is the one originally described by Vicsek et al. in Ref. [4]. It was in this model that numerical computations first showed the emergence of a dynamical phase transition from a disordered state (in which agents move in random directions) to an ordered one (where they head in approximately the same direction) as the noise level is decreased or the mean density is increased. More recently, other simple models with qualitatively similar agent dynamics have been introduced [7], [14], [15], [16]. Typically, they all describe groups of agents moving with an imposed non-zero speed and that tend to align with their short-range neighbors. They also focus on similar order parameters to measure the amount of self-organization in the system, namely: the degree of alignment of the agents’ headings. When noise is introduced, all these models display a transition from a disordered to an ordered state, analogous to the one observed in the original model by Vicsek et al. The similarity of these results presents a challenge, since we are left with no easy way to discriminate which minimal description is best at capturing the basic swarming mechanisms. Indeed, while a transition to a state where agents move in a common direction is an essential part of any credible swarming description, this appears to be the generic result of all algorithms that mimic a reasonable swarm-like behavior.

In this work we introduce a set of global quantities that allow us to better characterize the collective states resulting from different swarming dynamics. We compare three commonly used simple swarming models and find that, while the standard order parameters (that measure alignment) behave equivalently in all cases, our newly introduced quantities provide a more detailed description that can clearly distinguish their properties. These quantities can also be measured experimentally, allowing further comparison to real swarms. It is important to point out, however, that there are currently very few experiments that can measure the relative position of all agents in a swarm, which is required by our averaged quantities. Fortunately, the recent implementation of the experiments described in Refs. [30], [32], [33], [34], together with a growing interest in swarming systems by communities in Biology, Physics, and Engineering, makes us expect more sets of swarming data to be obtained in the near future. It is unclear at this point, though, how these data sets will connect to our numerical calculations. The data sets recently obtained in Ref. [30], for example, seem to indicate that the interactions are not based on which agents are within a certain range (as assumed by our algorithms), but instead on a fixed number of neighboring agents that can be processed by the Starling’s nervous system. For this situation, most of the results obtained here are therefore not applicable. However, Ref. [30] corresponds to experiments in a very specific system and there is no reason to expect the many other examples of swarming biological agents to behave equivalently. In summary, as more experiments become available, the analysis tools that we introduce will have to be applied in a case by case basis. In some situations, the quantities that we study will only be accessible through indirect measurements or require additional adjustable parameters. In other situations, the experiments may unveil dynamic rules that are very different from the ones assumed in our simulations, rendering our tools ineffective or meaningless. Given the variety of swarms found in Nature and our limited understanding of their mechanisms, however, we expect that our work can provide a first approach for the analysis of many of them. In some cases it will have to be adapted to the possibilities and limitations of each particular experiment, but this link to specific systems is beyond the scope of the current paper. The paper is organized as follows. In Section 2 we describe the models and order parameters considered. Here we attempt to clarify a long-standing confusion by distinguishing between two different ways of implementing the model by Vicsek et al. Section 3 presents our main results, comparing various systems by using the standard and new order parameters. Section 4 uses the cluster size distribution to further investigate the global states of the system. In Section 5, we apply our analysis techniques to shed some light on a current controversy regarding the order of the phase transition in the Vicsek model. Finally, Section 6 is our conclusion.

Section snippets

Models

We will consider in this paper three similar models that were designed to be minimal, i.e., to capture the essence of a swarming behavior while including only its basic components. They are all comprised of N agents defined as point particles that (for simplicity) are forced to move at the same constant speed s in a two-dimensional periodic square box of side L. At every time-step, each agent interacts with all agents within a neighborhood of radius R, computes its new direction of motion, adds

Order parameter analysis

We begin by computing the standard order parameter ψ defined in Eq. (5) as a function of the amount of noise for the OVA, the SVA, and the GCA. The length- and time-scales of the computation are fixed by defining the interaction range as R=1 and using Δt=1. All our data was obtained by integrating the dynamics for over 107 time-steps and averaging every 104 steps after an initial relaxation time.

Fig. 1 shows three typical snapshots of moderately ordered states computed using the OVA, SVA, and

Cluster size distribution analysis

In this section, we will further analyze the OVA, SVA, and GCA swarming simulations by computing their distribution of cluster sizes. In previous work, we had shown that the SVA typically produces clusters of all sizes that are continuously forming and breaking apart, and that the distribution of cluster sizes resembles a power-law in its ordered phase [20]. We found no characteristic cluster size but that instead the probability of having larger clusters decays smoothly with size. We will

Bi-stability analysis of a large system

In Refs. [14], [29] Grégoire and Chaté claim that all three minimal models (the OVA, SVA, and GCA) present a first-order phase transition in the thermodynamic limit. They argue that this transition may only appear to be of second order in some finite-size systems (such as in the top panels of Fig. 2, Fig. 3) when the size L of the simulation box is smaller than a certain crossover length-scale L(ρ,s). The L>L condition would be required to develop density bands that become metastable to a

Conclusion

We have examined various global statistical quantities that characterize in new ways the collective behavior of swarming systems. In contrast to previous analyses, we concentrate not only on the agents’ orientation but also on their spatial distribution. We focused in this paper on three well-known minimal models that display similar features when analyzed with standard tools. By computing the new quantities, we were able to better characterize and clearly distinguish the dynamics of the three

Acknowledgements

The work of C. Huepe was supported by the National Science Foundation under Grant No. DMS-0507745. M. Aldana acknowledges CONACyT and PAPIIT-UNAM for partial support under grants P47836-F and IN112407-3, respectively.

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