Flocking in complex environments -- attention trade-offs in collective information processing

TL;DR

This paper models collective behavior in complex environments, revealing that limiting individual cognitive capacity can enhance large-scale coordination by reducing distraction, highlighting a fundamental trade-off between coordination and responsiveness.

Contribution

It introduces a model showing how attentional bottlenecks can improve collective coordination in complex environments, providing new insights into biological and artificial swarm design.

Findings

Large-scale coordination is maximized by limiting individual cognitive capacity.

A trade-off exists between collective coordination and environmental responsiveness.

Self-organized dynamics enable the collective to isolate from distracting cues.

Abstract

The ability of biological and artificial collectives to outperform solitary individuals in a wide variety of tasks depends crucially on the efficient processing of social and environmental information at the level of the collective. Here, we model collective behavior in complex environments with many potentially distracting cues. Counter-intuitively, large-scale coordination in such environments can be maximized by strongly limiting the cognitive capacity of individuals, where due to self-organized dynamics the collective self-isolates from disrupting information. We observe a fundamental trade-off between coordination and collective responsiveness to environmental cues. Our results offer important insights into possible evolutionary trade-offs in collective behavior in biology and suggests novel principles for design of artificial swarms exploiting attentional bottlenecks.

Equations

$$C=⟨\frac{1}{N}\sum _j{\hat{u}}_j\cdot {\hat{u}}_p⟩$$

$$\frac{d{\vec{x}}_i}{dt}$$

$$\frac{d{\phi }_i}{dt}$$

$$g_i(t)=\{\begin{array}{l}1\text{for}l\in \text{kNO}\text{and}{d}_{il}<r\\ 0\text{else}.\end{array}$$

$$F(C,A)=C+\delta (A-1),$$

$$\tilde{C}=\frac{1}{N}⟨\sum _{i=1}^N{\hat{u}}_i⟩,$$

$$\tilde{F}(C,A,k)=b_{c}C+b_a(A-1)$$

$$F(C,A)=C+\delta (A-1),$$

Full text

Introduction

Consensus formation, coordination and collective response to environmental cues are important aspects in collective behavior of many interacting agents in biology, physics, robotics and computational social science. The understanding of these processes is fundamental for a better comprehension of collective intelligence that confers groups the ability to solve problems collectively using strategies which are beyond the reach of single individuals Krause et al. (2010). Crucial for understanding benefits of collective behavior, is the understanding of the mechanisms underlying collective information processing: How new information is acquired, how information is shared and combined within the collective, and how the collective deals with conflicting information are among the most compelling and elusive questions on the self-organization of collectives. Generic flocking models (see e.g. Huth and Wissel (1992); Vicsek et al. (1995); Grünbaum (1998); Couzin et al. (2002)) allow to study the interplay between emergent collective behaviors and collective information processing in a dynamical system setting. For example, using flocking models it has been demonstrated, that only a small fraction of informed individuals is sufficient to accurately guide large collectives Couzin et al. (2005); Cucker and Huepe (2008); Guttal and Couzin (2010). It was also shown that groups are able to collectively track dynamic environmental gradients not detectable by individuals Grünbaum (1998); Berdahl et al. (2013), or that groups can make efficient consensus decisions in conflict situations without any implicit knowledge about the majority-minority relationships Couzin et al. (2005, 2011). Only recently, predictions of such models on fundamental decision bifurcations in spatial movement decisions have been confirmed in the collective migration of baboon groups Strandburg-Peshkin et al. (2015).

A fundamental aspect of the self-organization of collectives – from collective decision-making to consensus formation, including coordinated movements – is that individuals are limited in terms of perception and cognition. Without direct access to the state of the whole group they must rely on local information Moussaïd et al. (2009, 2011); Strandburg-Peshkin et al. (2013); Pearce et al. (2014). The emergent collective patterns in agent-based models have been shown to depend strongly on the field of view of individuals Strömbom (2011); Barberis and Peruani (2016), and more generally on what local information individuals pay attention to Romanczuk and Schimansky-Geier (2012); Romensky et al. (2014). Furthermore, even for a strongly limited field of view, the sensory input of individual agents may contain a large number of social and non-social cues. However, social interaction in animal groups appear to be restricted to a rather low number of neighbors Ballerini et al. (2008); Jiang et al. (2017). On the one hand, this suggests additional cognitive constraints on the processing of available sensory information, which is also in-line with a wide range of experimental results on limited capacity for visual tracking of multiple objects in animals and humans Intriligator and Cavanagh (2001); Todd and Marois (2004); Cavanagh and Alvarez (2005); Green and Bavelier (2006); Alvarez and Franconeri (2007). On the other hand, it has been shown in generic flocking models that the emergent, large scale collective behavior depends strongly on how many neighbors a given individual can pay attention to Ginelli and Chaté (2010); Chou et al. (2012); Shang and Bouffanais (2014). However, to our knowledge the explicit role of cognitive constraints on collective information processing has not been systematically explored.

Up-to-date most theoretical and empirical research focused on information sharing and collective decision making, in idealized, laboratory-like environments, including the works mentioned above, discussing cognitive and sensorial limitations (see e.g. Couzin et al. (2005, 2011); Lemasson et al. (2009); Ward et al. (2011)). However, collectives in realistic scenarios need to cope with complex environments with a large number of potentially informative or distracting environmental cues (see Fig. 1a and Strandburg-Peshkin et al. (2017)). Whereas recently it was shown using minimal flocking models that self-organized, collective behaviors are strongly affected by complex environments Chepizhko et al. (2013); Chepizhko and Peruani (2015), it remains open how complex environments impact collective information processing, which is the main question we focus on here. More specifically, we investigate the emergence of collective behaviors in a generic model of socially interacting agents in a complex environment containing many potentially dangerous sites. Individuals try to avoid these sites, while at the same time trying to coordinate with their neighbors. In addition, some informed individuals have also private information on a global preferred direction of migration, which may be in conflict with the local environmental cues. Our main aim is to explore how the cognitive, and/or sensory constraints of the individuals affect group-level coordination, information exchange and collective response to environmental cues.

Our results show that in heterogeneous environments, strongly limited attention capability of individual agents results in higher accuracy with respect to large-scale coordination, which is in stark contrast to previous results obtained in simple environments Chou et al. (2012); Shang and Bouffanais (2014). This is caused by a dynamical, spatial “echo chamber”-like effect, where individual attention becomes saturated by social information and non-social cues are largely ignored. However, if these non-social cues provide important information about potential environmental dangers, the emergent dynamical “echo chambers” become strongly detrimental to the ability of the collective to safely navigate the environment. Note that information exchange through social interactions is typically believed to be beneficial for the collective Sirot (2006); Berdahl et al. (2013). Here, our analysis shows that below a critical threshold in attention capacity, groups perform worse at acquiring new information about the environment than non-interacting agents. This is due to the emergent self-isolation from environmental cues, which is exactly what facilitates group coordination in complex environments. Our findings not only suggest a fundamental trade-off in collective behavior in natural systems, but also provide important insights for the design of communication in artificial, distributed systems, such as robotic swarms.

Results

Model. We consider a flocking model consisting of $N$ agents moving in a two-dimensional environment of size $L\times L$ with periodic boundary conditions. In addition, we assume the presence of $N_{\textnormal{inf}}$ informed individuals with private information about a preferred direction of motion $\hat{u}_{p}$. The informed fraction of the collective is $R_{\textnormal{inf}}=N_{\textnormal{inf}}/N$.

The environment contains non-social cues, which represent features of the environment that solitary agents in general try to avoid, as they, for example, signal potential dangers. We will refer to these disrupting cues as “danger sites” (or distraction sites) in short as DS. Each DS is surrounded by an effective repulsion zone of radius $r=1$. The environment contains $N_{DS}$ randomly distributed DSs at fixed positions. In particular at high DS densities $\rho_{DS}=N_{DS}/L^{2}$, the corresponding repulsion zones may overlap (see Fig. 2a, b). We note that the agents and the danger sites (DS) are assumed to be point-like, as in many agent-based models for collective movement, e.g. Vicsek-type models Vicsek et al. (1995). This corresponds to the scenario where the sensory ranges are large compared to the physical size of moving agents and DSs.

Based on experimental observations, it was suggested that animals interact with a limited number of conspecifics Ballerini et al. (2008); Strandburg-Peshkin et al. (2013); Jiang et al. (2017). Motivated by these findings, and with intention of explicitly studying the impact of limited attention in a generic flocking model, we assume that each agent can pay attention only to $k$ nearest objects ($kNO$) in its vicinity, irrespective whether it is another agent (social cue) or a DS (non-social cue). Thus, $k$ can be interpreted as the number of available attention slots for each agent. The parameter $k$ quantifies the individual attention capacity.

Each agent moves with a constant speed $v_{0}$ and reacts to neighbors and DSs through corresponding changes in its direction of motion $\hat{u}_{i}=(\cos\varphi_{i},\sin\varphi_{i})^{T}$ defined by a polar orientation angle $\varphi_{i}$. All agents - informed and uninformed - turn away from DSs when two conditions are met simultaneously: 1) the agent perceives the DS – that means that the DS in question, say $l$, is within its $k$ nearest objects – and 2) the distance between the agent and DSs ($d_{il}=|\vec{x}_{i}-\vec{x}_{l}|$) is smaller than the radius of its repulsion zone. In all other cases, individuals ignore the DSs and coordinate their motion with other individuals within their kNO by aligning their direction of motion with the average direction of their neighbors. Informed individuals exhibit an additional bias to orient towards the preferred direction of motion that we denote $\hat{u}_{p}$. Throughout this work, the preferred direction of motion of informed individuals will be along the $x$-axis: $\hat{u}_{p}=(1,0)^{T}$. We emphasize that the response to DSs, once detected, dominates all other behavioral responses of individuals, whether informed or uninformed. The specific formulation of the mathematical model in terms of stochastic differential equations, together with the parameters used, are given in Methods.

The finite attention capacity to $k$ nearest objects leads to a natural competition between social and non-social cues: If the $k$ nearest objects of the focal agent are other agents, it will be not capable to “sense” a DS $l$ even if $d_{il}<1$ (see Fig. 1b). Note that in the case of vanishing density of DSs $\rho_{DS}\to 0$ (homogeneous environment), the model reduces to a simple flocking model with so-called metric-free alignment interaction with k-nearest neighbors (see e.g. Ballerini et al. (2008); Peshkov et al. (2012); Chou et al. (2012)) with the additional feature of informed individuals. If instead of topological, metric interaction are used, then the model reduces in the limit of $\rho_{DS}\to 0$ and $R_{inf}=0$ to a Vicsek-type alignment model Vicsek et al. (1995), which has been extensively discussed in Vicsek and Zafeiris (2012); Marchetti et al. (2013). For $R_{inf}>0$ it is closely related to the model explored in Couzin et al. (2005), while for $R_{inf}=0$ and $\rho_{DS}>0$ it reduces to the model studied in Chepizhko et al. (2013); Chepizhko and Peruani (2015).

Interaction Networks & Collective Accuracy. In order to quantify the emergent collective dynamics and study the effect of varying attention capability, we have performed systematically numerical simulations of the above model for varying attention limit $k$, DS density $\rho_{DS}$, and the ratio of informed individuals $R_{\textnormal{inf}}$ (see Methods for details). By neglecting the directed nature of inter-individual links, the entire agent system can be viewed as a time-dependent, undirected interaction network. For all DS densities, we obtain the same qualitative picture in the stationary state: For low $k$, we observe strongly fragmented dynamical networks, which at given time $t$ are characterized by a large number of small, disconnected sub-groups (see Fig. 2a and Supp. Fig. S1). Each such cluster corresponds to an isolated connected component. These components are not static: We observe continuous fission-fusion of clusters over time due to randomness in individual motion and interactions with DSs (see Movie S1). By increasing the attention limit $k$, we observe a fast decrease in the number of disconnected clusters that results in an increase in average cluster size (see Fig. 2b and Supp. Fig. S2a). Eventually, by increasing $k$ above a critical value, we can obtain fully connected networks with a single connected component. Correspondingly, the average life-time of a connection between specific agents grows strongly with increasing attention limit $k$, whereas an increase in DS density $\rho_{DS}$ reduces the life-time of individual edges in the network (see Supp. Fig. S2b). In general, as one would intuitively expect, the network of social interactions becomes more tightly connected with increasing attention capacity $k$.

We measure the accuracy of the collective migration through the average agreement between the heading $\hat{u}_{j}$ of individuals and the preferred direction of motion $\hat{u}_{p}$:

[TABLE]

with $\langle\cdot\rangle$ indicating the temporal average in the stationary state. If all agents move perfectly along the preferred direction, which is available only to informed individuals, then we obtain $C=1$, whereas for disordered movement we observe $C\approx 0$.

For collective behavior in homogeneous environments it has been shown that increasing the connectivity of the interaction networks is beneficial for coordination Lemasson et al. (2009); Shang and Bouffanais (2014), which is also in line with general results on synchronization in (dynamic) networks of oscillators Belykh et al. (2004); Dörfler et al. (2013). In particular for few informed individuals, one intuitively expects that a strongly connected information network ensures that information about the preferred direction of motion diffuses more efficiently across large parts of the collective. Therefore, the natural prediction would be that collective accuracy increases with increasing attention capacity $k$. This is indeed the case in the limit of vanishing density of DSs $\rho_{DS}=0$ (homogeneous environment, see Fig. 3a), where we observe a monotonous increase in accuracy $C$ with the attention capacity $k$, for a fixed ratio of informed individuals: Whereas for $k=1$, in order to achieve an accuracy of $C>0.9$, we require the majority of the collective to be informed ($R_{\textnormal{inf}}\geq 0.6$), for $k=6$ it is already sufficient to have a small fraction $R_{\textnormal{inf}}=0.2$ of informed individuals to achieve the same level of collective accuracy.

The situation completely reverses for high DS densities (see Fig. 3b). For $\rho_{DS}=0.2$ we observe a monotonous decrease in the collective accuracy with increasing $k$ for all values $R_{\textnormal{inf}}>0$. In order to achieve a certain level of collective accuracy (e.g. $C=0.5$) for larger $k$ we need a larger fraction of the system to be informed. In other words, stronger connected flocks become more difficult to guide. Even more dramatically, for the largest attention capacity investigated ($k=24$) the maximal attainable average accuracy for a fully informed system ($R_{\textnormal{inf}}=1$) is $C\approx 0.62$. This is lower than the average collective accuracy of $C\approx 0.66$ for collectives at minimal attention capacity ($k=1$) with only a tiny fraction of informed individuals $R_{\textnormal{inf}}\approx 0.013$ ($1.3\%$ of the entire collective). In fact, it appears that for $k=1$, the collective accuracy $C$ versus $R_{\textnormal{inf}}$ depends only very weakly on the DS density, whereas the accuracy for high $k$ is massively decreased for all $R_{\textnormal{inf}}>0$.

Thus, in complex environments, strongly limited attention resulting in a very sparse, fragmented - yet dynamic - interaction networks, turns out to be highly beneficial for global consensus formation (see Fig. 3c). This counter-intuitive effect can be understood through an analysis of information flows and how environmental information is processed by the collective, for example through analysis of the (stationary) probability distribution of agents having a particular combination of in and out-degree. The out-degree $D_{out}$ quantifies how many individuals a focal individual pays attention to, whereas the in-degree $D_{in}$ is the number of others paying attention to the focal individual. Agents directly responding to a DS have a social out-degree $D_{out}=0$ - they ignore their neighbors. However, their neighbors can still pay attention to them so their social in-degree $D_{in}$ can be larger than zero (see Supp. Information and Fig. 2c,d), in this case they “broadcast” information on the DS through their evasion behavior to others. Therefore, environmental cues affect collective behavior directly through the individuals directly responding to a DS, as well as, indirectly through information transmitted to other agents via social interactions. High consensus in complex environments requires effective self-isolation of the collective from distracting environmental cues. It can emerge due to two mechanisms: 1) The number of direct responders remains small; 2) Their influence on others is weak due to a low $D_{in}$, in particular compared to the in-degree of non-responders. In our case both effects play a role: For low $k$, the fraction of agents directly responding to DSs remains low even at very high $\rho_{DS}$. At low $k$, aligned individuals move together in dense sub-groups, and even if one or more agents enter a repulsion zone, there is a high probability that there are $k$ neighbors closer than the DS, which prevents the detection of the latter. In addition, the in-degree of agents responding to DSs is on average significantly lower than that of agents not responding to the DSs (Fig. 2c,d). Agents evading a DS, have a high probability to move away from their neighbors, which in turn decreases the probability that they will be within the $kNO$ of others. Thus, in particular for low $k$, the indirect response to DSs, mediated through social interactions, is strongly inhibited. These emergent small, dense agent clusters at low $k$ permanently merge and split up over time, which leads to exchange of directional information across the collective on long time scales, eventually leading to global consensus and high coordination levels.

This situation changes with increasing attention capacity $k$. The chance of an agent to detect a DS increases strongly, as the distance to the $k$ nearest objects is an increasing function of $k$. As soon as this distance becomes larger than the distance to the next DS ($d_{il}\lesssim 1$), agents are capable to detect the environmental cues reliably, and react to them if they are within their repulsion zone. In addition, larger $k$ also increases the number of other agents paying attention to a direct-responder. The motion of the emergent sub-groups is still well coordinated at a local scale (see Supplementary Movie S2). However, these sub-groups interact now predominantly with the environment by changing their direction of motion and by complex fission-fusion dynamics directly triggered by the DSs. This results in quick loss of directional information across time and space, and in a vanishing impact of the directional information provided by the informed individuals, which yields a strong decrease in collective accuracy $C$.

The contribution of both effects to the emergent self-isolation for low $k$ is shown in Fig 4a: The fraction of direct responders $r_{d}$ is much lower for low $k$ and shows only a slow increase with increasing DS density $\rho_{DS}$. The same holds for the fraction of first-order indirect responders $r_{i}$, defined by agents paying attention to at least one direct responder. For $k=2$, $r_{i}<0.1$ for all DS densities studied, whereas for $k=24$ at high DS densities, it saturates at more than half of the entire collective ($r_{i}\approx 0.55$).

Collective Avoidance of Repulsion Zones. While limited attention is beneficial for consensus formation in complex environments, it may be very detrimental to the collective if the environmental cues (DSs) provide reliable informations about environmental dangers (e.g. predators).

In order to quantify the performance of the collective to respond to environmental cues, we introduce in the following a DS avoidance measure. First, we compute the average fraction of agents in “safe” areas, i.e. outside the DS repulsion zone. As a reference, let us estimate the same quantity in a control numerical experiment with the same number of agents but who are only interacting exclusively with the environment and not with other agents. Our DS avoidance measure $A$ is then defined as the ratio between these two quantities (see Methods for details). This measure $A$ allows us not only to compare the relative collective performance of the flock at different attention capacities $k$, but also to compare it with the performance of solitary agents without any social interactions. For $A>1$, social interactions provide a benefit with respect to solitary individuals. For $A<1$, a collective performs on average worse than solitary agents in avoiding the potentially dangerous areas. Fig. 4b shows clearly that for small values of $k$, the collective performs much worse than solitary individuals in avoiding the repulsion zones. Here, increasing $k$ results in a monotonous increase in $A$. However, we observe $A>1$ only for sufficiently large $k\gtrsim 12$. For lower $k$ it turns out that social interactions are detrimental with respect to DS avoidance. The observed behavior is directly linked to the emergent local echo-chamber effect at small $k$. Socially interacting individuals can only outperform solitary agents in responding to environmental cues, if sufficient amount of information is able to enter the interaction network and spread effectively through it.

We can quantify the emergent trade-off between migration accuracy (quantified by $C$) and DS avoidance (measured by $A$) through a global fitness function $F(C,A)$ (see Methods and Supplementary Information for details). While $F$ depends implicitly on the attention limit $k$ through $A$ and $C$, we introduce the parameter $\delta$ to quantify the relative benefits of avoidance versus accuracy. Whereas in safe environments, benefit of avoidance may be negligible ($\delta\approx 0$), in environments where local cues provide important information about potential dangers one can assume $\delta\gg 1$. Fig. 4c shows $F$ as a function of $k$ and $\delta$ for collectives in random DS fields. We note that $F=0$ corresponds to the average fitness expected for solitary individuals. For low $\delta$, we observe a single maximum of $F$ at low attention capacities $k\approx 2$. For increasing benefits of DS avoidance $\delta$, a second maximum emerges at large $k$, while at low $k$, the socially interacting collectives are on average outperformed by solitary individuals.

This trade-off between accuracy and avoidance, or “responsiveness”, becomes particularly prominent if we consider structured, inhomogeneous environments containing free paths and voids in a landscape otherwise filled with high density of DSs, instead of homogeneous, random environments. At low $k$ the agents completely ignore the environmental structure and their dynamics is dominated by social interactions with high directional accuracy. At high $k$, the situation reverses and the collective dynamics is dominated by the environment, where agents track the environmental structure, staying preferentially in areas with low DS density, while ignoring the preferred direction of migration (see Fig. 4d–f, see also Supp. Fig. S3 and Supp. Movie S3, S4).

Model Variations & Generality of Results. So far, we have considered social interactions which do not pay special attention to neighbors responding to DSs. This is motivated by the idea of social interactions being based on observations of behavior of others but absence of direct communication about the cause of their particular behavior. In order to test the robustness of our results, we explored an extension of the basic model, by introducing active signaling about potential danger by agents interacting with a DS. In this case, all neighboring agents connected to the signaler put their full attention on the signaling agent and respond only to it, while ignoring other non-signaling agents. As expected, the additional signaling improves the collective DS avoidance due to increased saliency of the corresponding cues within the network. However, the general results - in particular the coordination and responsiveness trade-off - remain unchanged (see Supp. Fig. S4).

Furthermore, the general trade-off between consensus and responsiveness to DSs does not depend on the presence of informed individuals. For $R_{\textnormal{inf}}=0$ and $\rho_{DS}=0$, our system reduces to a Vicsek model with topological interactions in homogeneous environments Ginelli and Chaté (2010); Peshkov et al. (2012); Chou et al. (2012). With $R_{\textnormal{inf}}=0$ and $\rho_{DS}>0$, i.e. without the bias provided by informed individuals, the model becomes a topological Vicsek-like model in heterogeneous environments. All the phenomena discussed above hold also in this case, if we replace the collective accuracy $C$ by the (normalized) average velocity $\tilde{C}=(\sum_{i}\hat{u}_{i})/N$, which quantifies the overall degree of (orientational) order in the system (see Supp. Information and Supp. Fig. S5). It is worth stressing that there exist fundamental differences with metric Vicsek-like models in heterogeneous environments (cf. Chepizhko et al. (2013); Chepizhko and Peruani (2015)), where agents are not subject to any cognitive limitation and display exclusively a limited perception capacity. Finally, we note that variations of the topological interaction mechanism do not affect the reported results. Specifically, we study the behavior of a model where the $k$-nearest objects are selected from the Voronoi neighborhoods (Supp. Fig. S6). This version of the model resembles the spatially balanced topological flocking algorithm proposed in Camperi et al. (2012) and represents a better approximation of visual networks Strandburg-Peshkin et al. (2013). All this suggests that the fundamental coordination-responsiveness trade-off discussed here is independent on the specific choice of the social interaction model.

Discussion

Using a generic flocking model we have demonstrated the importance of finite attention capacity of individuals for collective information processing in complex environments. In our model, agents dynamically allocate their limited attention to process social and environmental stimuli, whereby the saliency of different stimuli is governed by their spatial vicinity. We demonstrated, that contrary to the general intuition, large-scale coordination and, as a result, the accuracy of collective migration in complex environments is maximized for strongly limited individual attention capacity. High levels of accuracy for agents which can pay attention only to few stimuli at a time, are a direct consequence of a self-isolation from distracting environmental cues through social interactions. On the other hand, the increased ability of agents to respond collectively to environmental cues for high attentional capacity leads to a breakdown of collective accuracy for a minority of informed individuals as the strong information inflow through local distractions overrides the information on the global preferred direction of motion available only to a minority of informed individuals. This demonstrates a fundamental trade-off between large-scale coordination and collective accuracy on the one hand, and the dynamical response to local environmental cues in complex environments, on the other hand. We would like to emphasize that our general finding of weaker connected networks achieving higher global accuracy in complex environments is diametrically opposed to widely accepted and intuitive knowledge in network science that more connections lead more effective information exchange and thus higher levels of synchronization (see e.g. Lemasson et al. (2009); Shang and Bouffanais (2014); Belykh et al. (2004); Dörfler et al. (2013)). We recover this intuitive result for flocking in empty environments, which demonstrates how taking into account environmental disturbances may dramatically change the collective behavior of self-organizing systems.

Our results suggest a specific link between cognitive and sensory capabilities of flocking animals and the ecological context. For example, for migrating animals, with high fitness benefits associated with coordination and information sharing on a preferred migration direction, with no (or very low) fitness costs of ignoring local environmental cues, strongly limited attention appears to be beneficial. However, if collective response to environmental cues is highly relevant for individual fitness but global coordination is not, as for example in foraging reef fish Hein et al. (2018), then being able to pay attention to many stimuli simultaneously becomes important. This yields testable hypotheses, on how the attention capability of different species exhibiting grouping behavior should co-vary with ecological niche, or how individuals within the same species should modulate their attention capabilities across contexts. Here, we note that a recent analysis of collective behavior in Hemigrammus rhodostomus, a strongly schooling fish species, suggests that an individual fish appears to pay attention only to one or two neighbors at a time Jiang et al. (2017).

Interestingly, being social offers an advantage over solitary behavior with respect to response to DSs only above a critical attention capacity. This poses some fundamental questions regarding the co-evolution of social behavior and individual attention capacity, especially taking into account potential developmental costs of higher attention capacity. Overall, our results point towards a complex interrelation between pre-existing attention capability, evolution of grouping behavior and the ecological niche.

We note that in our minimal model we varied only a single dimension of cognitive capabilities, namely the total number of objects an individual can pay attention to. There are other more complex cognitive processes, which affect the individual processing of a large number of social and non-social stimuli. For example object recognition and classification, may enable individuals to dynamically vary the relative saliency of social and non-social stimuli, which is not considered in this work but could allow individuals to adapt to different behavioral contexts. In general, the strength of the observed trade-off will depend on model choice and model details. For example, making the agents more likely to detect danger sites, will make the collectives more responsive to the environmental cues. However, if the overall attention capacity has an upper bound - as assumed here - it must come at the cost of decreasing social interactions. Hence, the existence of the general effects discussed here, in particular the surprising increase in collective accuracy with decreasing attention capabilities, is not restricted to particular model choice. The qualitative results should hold as long as the following three conditions are met: 1) the attention capacity is limited, 2) the salience of social cues has some distance-based component, and 3) the structure of the interaction emerges naturally from spatial self-organization. Overall, this work demonstrates the fundamental importance of potential constraints in sensory and cognitive abilities of individuals on emergence and function of collective behavior.

We considered explicitly the case of spatial flocking behavior, however the coordination-responsiveness trade-off as a general principle should be observable in different collective behaviors. Only recently it was shown that Guinea baboon exhibit stronger response to known social cues than to novel ones. It was hypothesised that this unexpected behavior can be linked to the complex social environments in which Guinea baboon groups live, and the corresponding necessity to filter out irrelevant or distracting information (“social noise”) Faraut and Fischer (2019).

Our findings have also implications for the design of interaction networks in artificial distributed systems, such as robotic swarms that operate in complex environments. Instead of continuously increasing the sensory and computational abilities of individual agents in order to cope with the consensus problems in complex environments, it may be promising to think about constraining the “cognition” of swarming robots. By generating specifically tailored attentional bottlenecks, resulting in emergent self-isolation as observed here, one can facilitate coordination and exchange of relevant information in complex environments. Attentional bottlenecks based on static features, e.g. colors, which can be easily distinguished from the background, are widely used in swarm robotics Brambilla et al. (2013). Here, we demonstrated that dynamical features (as opposed to static ones), like relative distance, or relative speed Romanczuk and Schimansky-Geier (2012), could provide effective means of coordination in complex environments, where “filtering” based on static features is difficult or not feasible.

Last but not least, our work yields potentially interesting implication for social sciences, where “echo-chambers” have received considerable attention recently. In human social networks, this effect is typically linked to homophily and confirmation bias Dandekar et al. (2013); Del Vicario et al. (2016). Our results show collective self-isolation from conflicting external information as agents moving in the same direction form tightly interacting social groups which ignore environmental cues. This can be viewed as an “echo-chamber”-like effect, which emerges naturally even in the absence of such explicit biases and self-sorting mechanisms. On the one hand, this suggests that these self-isolation tendencies may be much more prevalent and easier to obtain for agents with limited attention capacity. On the other hand, our results provide support for the evolution of proximate, socio-psychological mechanism facilitating the formation of echo chambers, such as homophily, by demonstrating how an emergent “echo chambers”-like effect strongly increase intra-group synchronization for a collective in a complex environment.

Methods

Agent-based Model. We consider a system of $N$ self-propelled agents and $N_{DS}$ danger sites DSs in a two dimensional domain of size $L\times L$ with periodic boundary conditions (torus). The agent and DS densities are thus $\rho=N/L^{2}$ and $\rho_{DS}=N_{DS}/L^{2}$. The agents move with a fixed speed $v_{0}=0.5$ and respond to other agents and DSs by changing their direction of motion $\hat{u}_{i}=(\cos\varphi_{i}(t),\sin\varphi_{i}(t))^{T}$. The behavior of each agent is mathematically described by following stochastic equations of motion, whereby $d\varphi_{i}/dt$ corresponds to the turning rate of agent $i$:

[TABLE]

The first term in the turning response equation 5, is the alignment interaction. A focal individual aligns with the strength $\gamma_{s}=1$ with neighbors $j$, which are part of its kNO-set. In addition, informed individuals have a (weak) tendency $\gamma_{p}=0.1$ to move in a preferred direction, here $+\hat{x}=(1,0)^{T}$ ($\gamma_{p}=0$ for non-informed individuals). The turning away (repulsion) from the DS $l$, which are in the kNO-set is given by the third term ($\gamma_{l}=1$). Here, $\alpha_{i,l}$ is the spatial position angle of the focal agent relative to the DS $l$. Both interactions are normalized by the respective number of agents and DSs, respectively ($n_{y}=\sum_{y\in\text{kNO}}1$). The function $g_{i}(t)$ determines whether the agent responds to DSs, or whether it aligns with its neighbors, and, in the case of informed individuals, biases its motion towards the preferred direction of motion. It is defined as

[TABLE]

Note that an agent responding to a non-social cue ($g(t)=1$), will not interact socially with other agents until its interaction with the DS is terminated, either because it leaves the repulsion zone or the site falls out of its $k$-nearest object set. The last term in equation 5 accounts for the stochasticity in the motion of individuals, with $\eta$ being the angular noise strength and $\xi_{i}(t)$ a normally distributed Gaussian white noise with standard deviation $1$. In all the simulations discussed, $\eta=0.25$ and the average density of particles is fixed at $1$ in a box of linear size $L=25$ ($N=625$). We have confirmed, that a detailed choice of $v_{0}$, $\eta$, and $L$ does not change the qualitative observations discussed here.

Avoidance Parameter. We quantify avoidance of repulsion zones through $\tilde{A}=1-\langle N_{\textnormal{rz}}(t)/N\rangle$. Here $N_{\textnormal{rz}}(t)$ is the number of agents, which are within at least one repulsion zone at time $t$, and $\langle\cdot\rangle$ represents temporal average in the stationary state. $\tilde{A}$ will always decrease with DS density, as more and more space is occupied by repulsion zones. In order to control for this trivial effect, we rescale $\tilde{A}$ by the corresponding value for non-interacting agents $A=\tilde{A}/\tilde{A}_{\textnormal{ni}}$. Thus, $A=1$ indicates same average performance of the flock as solitary agents without any social interactions. Here, solitary individuals are always responding to a DS, once they are within the corresponding repulsion zone.

Fitness Function. We can quantify the emergent trade-off between collective accuracy and responsiveness, through the following global fitness function depending on the collective accuracy C and DS avoidance:

[TABLE]

with $\delta$ being the relative benefits of DS avoidance with respect to migration accuracy. The above function was defined in a way so that a value $F=0$ corresponds to the behavior of solitary individuals, where the average accuracy vanishes ($C=0$) and the DS avoidance is $A=1$, according to the definition above. Thus, only for $F>0$ the collectives perform better than single individuals (see Supp. Information for more details).

Acknowledgements.

P. Romanczuk & P. Rahmani acknowledge support by the German Science Foundation (DFG) through RO 4766/2-1 and the German Academic Exchange Service (DAAD). P. Romanczuk acknowledges funding by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2002/1 “Science of Intelligence” – project number 390523135. P. Rahmani acknowledges support from the Ministry of Science, Research and Technology of Iran. F. Peruani was supported by the Agence Nationale de la Recherche via project BactPhys, Grant No. ANR-15-CE30-0002-01. We thank Ana Sofía Peruani for the artwork in Fig.1a.

Supplementary Information

Stationary distribution of in and out-degrees

From our numerical simulations, we can extract the stationary probability distribution $f(D_{out},D_{in})$ for an agent having a particular combination of out-degree $D_{out}$ and in-degree $D_{in}$.

In homogeneous environments, $\rho_{DS}\to 0$, the out-degree of each agent is directly set by the attention capacity $k$. Thus, the distribution of individual in and out-degrees $f(D_{\textnormal{out}},D_{\textnormal{in}})$ is sharply peaked at $D_{\textnormal{out}}=k$. The in-degree is not fixed: Whereas a focal individual $i$ pays attention only to $k$ nearest neighbors, the number of other agents $m$ paying attention to it may be lower or higher. Therefore, we observe a spreading out of $f(D_{\textnormal{out}},D_{\textnormal{in}})$ along the in-degree axis. This pattern holds also for finite number of DSs, however, now in addition, there is also a finite probability that an agent interacting with a DS ignores its neighbors (see Fig. 2c,d). In this case, its out-degree with respect to social interactions is zero: $D_{\textnormal{out}}=0$, while the in-degree can assume many possible values depending on how many other agents pay attention to it at a given time. In general, for a finite DS density, the combined distribution $f(D_{\textnormal{out}},D_{\textnormal{in}})$ shows a bimodal distribution with two maxima at $D_{\textnormal{out}}=0$ and $D_{\textnormal{out}}=k$.

For low $k$, the fraction of agents responding to DSs remains low even at very high $\rho_{DS}$. The average out-degree as well as the average in-degree increases as expected with increasong $k$. However, the expected in-degree of direct responders with $D_{out}=0$ is always lower then the expected in-degree of other agents with $D_{out}>0$. As a results, for low $k<3$, DS responders become often completely isolated from other agents by having $D_{in}=D_{out}=0$.

Emergence of global order in the absence of informed individuals

In the absence of informed individuals $R_{inf}=0$, there is no preferred direction of motion. Here, instead of collective accuracy, coordination can be quantified using average polarization of the flock

[TABLE]

which is equivalent to the ferromagnetic order parameter in physics. By replacing $C$ by $\tilde{C}$, we observe the same fundamental coordination-responsiveness trade-off as discussed in the main text (Fig. S5).

Quantifying the Coordination-Responsiveness Trade-Off – Global Fitness Function

We can quantify the emergent trade-off between coordination and responsiveness, by introducing the following global fitness function depending on the collective accuracy $C$ and DS avoidance $A$:

[TABLE]

with $b_{c}$ being the benefits per unit of directional consensus, and $b_{a}$ the benefits per unit of DS avoidance. Without loss of generality, the equation can be rescaled, by $b_{c}$ to

[TABLE]

where $\delta=b_{a}/b_{c}$ represents the relative benefits of DS avoidance versus coordination. Please note that $F$ is not fitness in a strict evolutionary sense (selection at the level of individuals), as it is a function collective variables $A$ and $C$. Here, “fitness” refers rather to a collective utility function, where (local) maxima correspond to (local) optimal collective strategies for homogeneous collectives. For a given set of parameters, $A$ and $C$ are calculated in a stationary regime of a stochastic system. Thus both variables reflect the average ability of individuals to avoid DSs and to coordinate with their neighbors in homogeneous groups.

For $\delta=1$ both behaviors yield equal benefits, whereas for $\delta>1$ ($\delta<1$) DS avoidance is more (less) beneficial than coordination. For a wide range of DS densities and angular noise, we observe two distinct maxima of $F$ in the $\delta,k$-plane. For $\delta\ll 1$ coordination is much more beneficial than DS avoidance and the maximum of $F$ is located at small $k$ where coordination is highest. With increasing $\delta$ this maximum decreases and we eventually see a rise of a second maximum at large $k$, leading to two local maxima at intermediate $\delta$. Eventually, for large $\delta$, where DS avoidance becomes far more important than coordination, we observe a single global maximum at large $k$.

Numerical Implementation and Experiments

The mathematical model was implemented in C/C++ with the k nearest object interaction implemented using the kd-tree implementation in the CGAL library cgal (1). The kNN interaction implementation in periodic boundary conditions was combining the standard kd-tree based algorithm with generation of mirror images of the environment to account for the interactions on a torus.

The stochastic equations of motion were numerically integrated using a standard Euler-Maruyama scheme manella (2). The numerical time step was set $dt=0.1$, whereby also smaller time steps were tested to show that the general results do not depend on the time step.

All the data were obtained by averaging over 20 realizations, each including $5\times 10^{4}$ relaxation time steps and $10^{5}$ stationary time steps.