Information transfer in finite flocks with topological interactions

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Highlights

  • Long term information flow in the topological Vicsek model behaves similarly to the standard, metric Vicsek model.

  • Short term information flow in the topological Vicsek model converges to the long term behaviour in smaller observation window sizes than in the metric model.

  • Global Transfer Entropy stays maximal above the phase transition.

Abstract

The Vicsek model is a flocking model comprising simple point particles originally proposed with metric interactions: particles align to neighbours within a radius. Later, topological interactions were introduced such that particles align with their closest k neighbours. We simulate the Vicsek model utilising topological neighbour interactions and estimate information theoretic quantities as a function of noise, the variability in the extent to which each particle aligns with its neighbours, and the flock direction. These quantities have been shown to be important in characterising phases transitions, such as that exhibited by the Vicsek model. We show that these quantities, mutual information and global transfer entropy, are in fact dependent on observation time, and in comparison to the canonical Vicsek model which utilises range-based interactions, the topological variant converges to the long-term limiting behaviour with smaller observation windows. Finally, we show that in contrast to the metric model, which exhibits maximal information transfer for the ordered regime, the topological model maintains this maximal information transfer dependent on noise and velocity, rather than the current phase.

Introduction

The scale and grace of bird flocks are nature’s most impressive displays, arising from seemingly chaotic flight paths of individual birds. Flocking behaviour is not restricted to just birds, either, with many other species displaying similar movements, from schools of fish [1] to colonies of bacteria [2]. The phenomenon of collective motion, and its constituent components, are subject to much research [3].

While new stereographic camera techniques and equipment allow research into large real-world flocks, much of the literature continues to use abstract models, such as the Standard Vicsek Model (SVM) [4] or the Inertial Spin Model [5]. These models approximate real-world systems as point particles whose decision making processes are encapsulated by local neighbour interaction rules and errors (noise). The SVM is perhaps the most minimal such model in that it contains only a single rule obeyed by all particles: assume the average direction of the local neighbourhood of radius r with some random perturbation added [4], where the magnitude of the noise determines whether the flock is coherent or not. The alternate model, the Topological Vicsek Model, behaves likewise, with the local neighbourhood instead defined as the k closest neighbours.

An implication of local interaction rules is that observing one flock member can reveal information about its nearby flock mates—for instance, we can better guess, or be more certain of, neighbouring headings. Information Theory allows this reduction in uncertainty to be quantified. Mutual Information (MI) [6] defines information sharing, the symmetric and instantaneous reduction in uncertainty about the heading of one flock member when any other member is observed. Information transfer instead quantifies the temporal reduction in uncertainty between flock members, that is, how knowledge of the current heading affects estimation of a future heading, which we measure here with Global Transfer Entropy (GTE) [7].

Information sharing among flock members is crucial to flock formation and stability, yet in the SVM it behaves unexpectedly in these finite flocks [8], [9]: MI is shown to diverge as flocks become increasingly ordered, while information transfer converges to a finite, non-zero value, where both are expected to vanish in a highly ordered flock.

The unexpected behaviour in the SVM is due to continuous symmetry in the system. Fluctuations in a highly ordered flock allow the flock orientation to proceed on a random walk without affecting overall order. When observed over small periods of time, an ordered flock is confined to a small region of phase space – that is, the system is restricted to only a small number of all possible states or flock configurations – producing more expected results—namely, MI and GTE tend to zero as flocks become more ordered, as seen in [10] for MI. However as observation time is increased, flock orientation gradually drifts on a random walk, eventually becoming uniform over 2π in the long-term (infinite) limit (See Fig. 1), exploring the entire phase space along the way. When the system is confined in phase space, it is said to have broken ergodicity; the assumption that behaviour averaged over time is the same when averaged over phase space. Thus over short observation periods, the SVM breaks ergodicity, while in the long-term observation limit, it is restored, leading to continuously-broken ergodicity [11] and the phenomena of diverging MI and non-zero GTE. The central question of this letter is whether the same holds for the TVM.

The SVM flock members interact with neighbours within a fixed radius, i.e., metric interactions. However, it is becoming increasingly apparent that this is not the case for many real-world flocks. Ballerini et al. [12] show via detailed 3D recordings of starlings that birds in real-world flocks instead interact in a topological manner. That is, a bird will interact with its closest six to seven neighbours, regardless of distance. Similar work has shown that 3D schools of fish, and (effectively) 2D herds of sheep and deer also utilise topological interactions [13], [14]. Niizato and Gunji [15] further shows that in “imperfect” biological organisms there might not be a strict delineation between the two, as both interaction methods can be seen as “uncertain” approximations of each other.

Here we apply the information theoretic metrics, MI and GTE, to a topological variant of the Vicsek Model (TVM) in which flock members take the average direction of their k closest neighbours rather than those within r units.

By noting a fundamental instability in the TVM over long time scales [16] we show here that the information theoretic behaviours of TVM are not only comparable to those of its metric counterpart, but in fact converge to the long-term limiting behaviours with shorter observation windows.

Section snippets

The topological Vicsek model

The two dimensional TVM comprises a set of N point particles (labelled i=1,,N) moving on a plane of linear extent, L, with periodic boundary conditions. Each particle moves with constant speed, v, and interacts only with the closest k neighbours. Positions, xi(t), and headings, θi(t), are updated synchronously at discrete time intervals Δt=1 according to xi(t+Δt)=xi(t)+vi(t)Δt,θi(t+Δt)=φi(t)+ωi(t), respectively, where vi(t) is constructed from θi(t) and the constant v, φi(t) is the

Methodology

We start in Section 3.1 with definitions of the information theoretic quantities we employ in this study, along with the dimensionally reduced forms outlined in [8], [9]. Following this, we discuss experimental simulation details in Section 3.2.

Results

Fig. 2 shows the long-term MI, IpwLT, estimated in simulation according to Eq. (5), which behaves similarly to the metric case, particularly, the absence of any peak at the phase transition, and divergence to as η0.

The long-term GTE, TglLT, on the other hand behaves completely differently to the metric case around the phase transition. In the metric case, the noise value, η, at which TglLT reached the convergence value scaled with both the velocity and the peak in χ, where higher velocities

Conclusion

Analysis of the topological Vicsek model over short observation windows reveals that information transfer more rapidly approaches the long term-limit behaviour in the TVM than its metric counterpart. This phenomenon arises due to an inherent instability in the topological Vicsek model which enables much faster exploration of the total phase volume. Future work would measure the information theoretic quantities of more stable models, perhaps even those measured from real-world flocks.

While the

CRediT authorship contribution statement

Joshua M. Brown: Software, Investigation, Methodology, Visualization, Writing - original draft. Terry Bossomaier: Conceptualization, Project administration, Writing - review & editing. Lionel Barnett: Conceptualization, Writing - review & editing.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

Joshua Brown would like to acknowledge the support of his Ph.D. program and this work from the Australian Government Research Training Program Scholarship.

The National Computing Infrastructure (NCI) facility provided computing time for the simulations under project e004, with part funding under Intersect and the Australian Research Council Linkage Infrastructure scheme.

Joshua Brown received his Ph.D. in Computer Science in 2019 from Charles Sturt University, Bathurst, Australia. His research interests are computational modelling, optimisation, and information theory.

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    Joshua Brown received his Ph.D. in Computer Science in 2019 from Charles Sturt University, Bathurst, Australia. His research interests are computational modelling, optimisation, and information theory.

    Terry Bossomaier is professor of computer systems at Charles Sturt University. His research interests range from complex systems to computer games. He has published numerous research articles in journals and conferences and several books, the most recent, on information theory appearing in December 2016.

    Lionel Barnett is a member of the Sackler Centre of Conciousness Science at the University of Sussex, United Kingdom. He has a number of publications with particular interest in evolutionary computation and Granger causality. He is also co-author with Terry Bossomaier of a Transfer Entropy book published in 2016.

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