cohesion in flocking models Spatially balanced topological interaction grants optimal

Dipartimento di Fisica, Universita` di Roma 3, Roma, ItalyModels of self-propelled particles (SPPs) are an indispensable tool to investigate collectiveanimal behaviour. Originally, SPP models were proposed with metric interactions, whereeach individual coordinates with neighbours within a fixed metric radius. However, recentexperiments on bird flocks indicate that interactions are topological: each individual interactswith a fixed number of neighbours, irrespective of their distance. It has been argued thattopological interactions are more robust than metric ones against external perturbations, asignificant evolutionary advantage for systems under constant predatory pressure. Here, wetest this hypothesis by comparing the stability of metric versus topological SPP models inthree dimensions. We show that topological models are more stable than metric ones. Wealso show that a significantly better stability is achieved when neighbours are selected accord-ing to a spatially balanced topological rule, namely when interacting neighbours are evenlydistributed in angle around the focal individual. Finally, we find that the minimal numberof interacting neighbours needed to achieve fully stable cohesion in a spatially balancedmodel is compatible with the value observed in field experiments on starling flocks.1. INTRODUCTIONOne of the prominent features of collective animal be-haviour is the way animal groups manage to staytogether in spite of predatory attacks and environ-mental perturbations [1]. During aerial display ofstarlings, for example, flocks fly for almost an hourabove the roost, in the presence of falcons and seagullsexerting continuous disturbances. In this respect, flocksexhibit a very efficient response, with a large degree ofcoordination and very robust cohesion. Flocks are anemblematic case of self-organized collective behaviour,where global patterns emerge from local interactionbetween individuals [2,3]. In this respect, a crucial ques-tion is: what kind of interactions are able to grant thegroup the robustness to perturbations that we observe?Experimental results on flocks of starlings (Sturnusvulgaris)[4,5] gave some insight into the nature of theinteraction between birds. In the study of Balleriniet al. [4], it was discovered that interactions are topolo-gical, each individual coordinating with a fixed number(approx. 7) of closest neighbours, irrespective of theirdistances. This result contrasted to what assumed bymost models of self-organized collective motion, wheremetric interactions were used [6–12].In the study of Ballerini et al. [4], it was argued thattopological interactions grant more robust cohesion thanmetric ones, and are therefore more effective from ananti-predatory point of view. In the present work, wetest this hypothesis. To this end, we resort to numericalmodels of self-propelled particles (SPPs) [9]whichhavebeen extensively used in the last 20 years to studythe emergence of order in polarized systems. Most of thepast literature on flocking models was devoted either tocharacterizetheonsetofordering[9,13–18],ortodescribethe features of the ordered phase [6–8,10,19–23]. Lessattentionwasgiventoresponseandrobustnesstoexternalperturbations, and to understand what determines at amicroscopic level specific traits of the global behaviour.Besides, the greatest part of numerical analysis has beenperformed in two dimensions, dealing either with smallfinite groups or with fluids of SPPs.However, to really understand what happens in realaggregations, we need to consider three-dimensionalmodels and look at large finite groups of individuals.A few very recent works [24–28] implemented topologi-cal rules both in two- and three-dimensional models,but did not consider the question we want to addressin this paper: what are the features of the microscopicinteractions that grant robust cohesion to the group?