key: cord-0557914-du4wmk4k
authors: Geiss, Daniel; Kroy, Klaus; Holubec, Viktor
title: Signal propagation and linear response in the delay Vicsek model
date: 2022-05-24
journal: nan
DOI: nan
sha: 8a3b1466e30623bad810cbf9423776ac563d1d43
doc_id: 557914
cord_uid: du4wmk4k

Retardation between sensation and action is an inherent biological trait. Here we study its effect in the Vicsek model, which is a paradigmatic swarm model. We find that: (i) a discrete time delay in the orientational interactions diminishes the ability of strongly aligned swarms to follow a leader and increases their stability against random orientation fluctuations; (ii) both longer delays and higher speeds favor ballistic over diffusive spreading of information (orientation) through the swarm; (iii) for short delays, the mean change in the total orientation (the order parameter) scales linearly in a small local orientation bias and inversely in the delay time, while its variance first increases and then saturates with increasing delays; (iv) the linear response breaks down when orientation conservation is broken by sizable swarm motility or heterogeneity.

Information exchange in animal groups has been suggested to foster survival by improving foraging [60] and predator avoidance [16, 43, 52, 54, 57] . The emerging collective behavior is associated with swarm intelligence [13, 32] and a collective mind [29] . Indeed, if short-range interactions between group members give rise to scale-free correlations [12, 20, 70] , this allows for an efficient information propagation through large groups [4, 6, 20] . And their sensory range is thereby increased way beyond that of its individual members. Inspired by a historical naval battle, where peer-to-peer signalling was employed to spy beyond the horizon, this is sometimes referred to as the Trafalgar effect [17, 69] . It is plausible that animal interactions were evolutionary optimized: too low mutual sensitivity might jeopardize a group's cohesion, too high sensitivity could magnify random perturbations into involuntary collective spasms. As examples for advanced collective strategies, previous work has identified leadership [7, 26, 56, 67] , collective escape waves [44, 63] , and hierarchical structures [7, 26, 56, 67] . From a physical perspective, studies of motile ensembles, ranging from bacterial colonies [11, 76] to flocks of birds [25, 63] , are often subsumed into the field of active matter [10, 40, 65] . The associated (active) manybody-theory perspective has already helped to uncover a variety of exotic non-equilibrium collective phenomena [9, 18, 34, 42, 68] and hinted at potential technological applications [14, 47, 55, 71] .

A key ingredient in the interactions of living agents is a time delay between perception and reaction. Signal propagation and processing, as well as actuation suffer from inevitable speed limitations. In general, the associated time delay in the mutual interactions between the agents may enrich the resulting collective dynamics with a hierarchy of instabilities, including oscillations, multistability, and chaos [36, 37, 73] . These have been the * viktor.holubec@mff.cuni.cz subject of intense study in the field of dynamical systems [2] . Conversely, delays have also be employed to stabilize unstable orbits in chaotic systems, e.g., via the Pyragas method [64] . In the context of information propagation, time delay has moreover been investigated in connection with sensory [74] , social [75] and financial (bitcoin) [31] networks, as well as in human traffic [5] . However, to the best of our knowledge, there are no studies of delay effects on the information propagation and response of motile active matter systems. For self-propelling particles with retarded pairwise attractions, time delays can trigger behavioral patterns [35, 53] reminiscent of dynamical regimes known from linear stochastic delay differential equations [37] . Small perception-reaction delays in a Vicsek model (VM) may foster clustering [61] and flocking [45] , similarly as in the Cucker-Smale model [33] . In contrast, long delays have the opposite effect [33, 45, 61] . From a more technological perspective, time delays naturally play an important role in feedback systems [47, 55] . Finally, it was recently suggested that retardation induced by inertia [50] or by discrete time lags could be the missing link between the VM and the behavior empirically observed in natural swarms [3, 4, 21, 45] .

In the following, we study retarded information spreading and linear response in the two-dimensional time delay VM, defined in Sec. II. First, we focus on the ability of an aligned system (flock) to perform collective turns (Sec. III), and its reaction to local perturbations [38] (Sec IV). For the former case, we study in Sec. V how the signal spreads in space and time and determine the corresponding dispersion relation. In Sec. VI, we consider the response of the system to an orientation bias applied to a subgroup of agents (so-called leaders) and in particular evaluate the conditions for linearity [28, 59] . Finally, Sec. VII summarizes our findings and gives an outlook to possible extensions of the present work.

The VM [27, 72] is a paradigmatic model for dry active matter [65] . Its original version [72] consists of N mobile spins moving with constant speed v 0 , and interacting via alignment interactions. We consider the two-dimensional version with time delayed metric interactions [45] , where each particle at time t + 1 assumes the average orientation of all neighbors at distance less than R at an earlier time t − τ < t, up to some noise. Velocities v i (t) and positions r i (t) of the individual particles obey the equations of motion

where the noise operator R η v realizes a uniform random rotation by an angle in the range [−η, η] around its normalized argument v/|v|. For the chosen metric interactions, the elements of the connectivity matrix n ij are given by n ij (t) = 1 if r ij (t) = |r i (t) − r j (t)| < R and n ij (t) = 0 otherwise. We take R as our length unit, i.e. we set R = 1, and the time step ∆t = 1 between two consecutive interactions as the time unit. In contrast to the closely related XY model [48] the connectivity matrix is time-dependent. In the VM with topological interactions, which seems more appropriate for natural bird flocks [8] , we expect qualitatively similar results for information spreading as described below [38] . Without a time delay, the model (1)-(2) is just the standard VM [72] , which exhibits overdamped orientational dynamics of v i (t) [21, 45] . At (imposed) high densities, a highly aligned phase emerges, reminiscent of flocking birds. The disordered state found at low densities resembles an insect swarm. The transition between these two phases is discontinuous ("first order") [39] . A finite delay time τ [45] or, alternatively, some finite inertia [24] in the dynamical equation for the velocity is known to bring the model predictions closer to the space-time correlations and scaling observed in natural swarms [21] .

To sustain their cohesion, flocks are able to perform collective maneuvers [23] . We now investigate the ability of individual members to initiate such maneuvers in the time delay VM. Specifically, we consider an initially (t = 0) fully polarized flock, with all particles moving in the same direction v init /v 0 = (1, 0, 0) , corresponding to the polarization Φ = ( i v i ) 2 /v 0 N = 1. The initial positions are distributed randomly inside a circle of diameter L F around the origin. The only exception is a single leader placed at the front of the flock, r L (0) = (L F , 0, 0) . To initialize the particle velocities, we assume that the whole flock had evolved along the x-axis according to Eqs. (1), (2) for t < 0, while, for t ≥ 0, the leader is constrained to make a deterministic turn The operator T ϕ rotates its argument by an angle ϕ. Its deterministic time evolution acts as a persistent source of information fed to the flock dynamics. For the sake of simplicity, we neglect the noise, η(t) = 0, for now. In Fig. 1 , we show typical turning events of a small flock of N = 30 agents for four different values of the time delay. For short delays, all the particles turn with the same radius [4] , in a manner that is loosely reminiscent of natural bird flocks [23, 62] . However, due to the accumulating mismatch between the navigation of the flock and its leader, the latter ultimately escapes its followers. Its cumulative impact on the overall flock orientation (the order parameter) is determined by its dwell time. Since the conductive spreading of orientation is diffusive [38] , the particles that spend most time in the vicinity of the leader follow it more closely than the more remote ones. This leads to a reshuffling of the flock formation and an associated defocusing that can eventually rip the flock apart. Interestingly, the effect becomes increasingly pronounced with growing delay, but fades again beyond a certain threshold time. This non-monotonicity is indicative of a trade-off between an increase in the time spent in the vicinity of the leader needed for successful alignment for longer delays, which favors flock break-up, and a correspondingly shortened dwell time of the leader inside the flock, which reduces its cumulative impact.

In the VM, flocks thus only follow slowly turning leaders. Due to the metric interaction rule, better coherence is naturally achieved by more compact flocks. However, no matter the conditions, the leader eventually always leaves a finite flock, as witnessed by the decreasing polarization in Fig. 2 . The generally poor take-up of the leader's direction by the VM flocks is a direct consequence of the averaging interactions, which disperse the information, preventing a perfect alignment. Recent investigations [23] show that in the VM augmented with orientational inertia (the so-called inertia spin model), the turning information can propagate much more efficiently. As a result, the inertia spin model allows for coherent collective turns where the leader stays inside the flock. Contrary to the inertia, the time delay thus makes the ability of the VM to perform coherent turns even worse. This striking difference between the effects of inertia and delay is interesting because other aspects of the two models, e.g., dynamical critical exponent and shape of time correlation functions, are similar [25, 45] .

In Sec. VI, the turning scenario is revisited from the perspective of the flock's linear response. Before that, we discuss another interpretation of the turning event, in which the poor "obedience" of the flock has a more advantageous connotation.

In contrast to an intended maneuver, a sudden turn of a leader in the scenario of the preceding section can be interpreted as a random perturbation in the flock. In stable systems, such perturbations should not create a strong response. This again reflects the necessity for tuning the information propagation in the system to an intermediate strength. And, again, delayed interactions can play a role in this respect. Efficient information propagation promotes cohesion and a coherent response of the flock to a stimulus but also tends to make the system overly sensitive to accidental local perturbations that do not qualify as reasonable cues for the whole flock.

To study the response of the time delay VM to such "accidental perturbations", we consider the same scenario as in the preceding section -with the important difference that the leader abruptly turns by an angle ϕ at time t = 0 but subsequently obeys the dynamic equations (1), (2), rather than following an externally prescribed route. In this case, the system receives information only at time t = 0 and relaxes freely, thereafter. The response of the flock in terms of its polarization is plotted in Fig. 3a , again for four different values of time delay. Three main effects of the time delay on the response can be discerned. First, as already gleaned from the above discussion, delayed interactions cause a lag of the collective response. This increases the relaxation time of the polarization. Which can, secondly, be interpreted as a form of swarm memory. The spatio-temporal spreading of the perturbation, which is already present in the standard VM is temporally stretched further, over τ time-steps, which might be called a reverberation. It effectively softens the overall impact onto the flock. In the figure, this is witnessed by a smaller overall decay of the polarization at late times for larger delays. Where there is reverberation, its more potent sister echo is not far. Indeed, thirdly, and most spectacularly, a sufficiently strong retardation of the interactions introduces an oscillatory response in the form of periodic spikes in the polarization dynamics. Their period is given by τ + 1 and their amplitude increases with growing τ and decays with growing lab time t. Oscillatory instabilities are well known to be a hallmark of delay dynamical systems [37, 45] . In the time delay VM, they can be explained as follows. After the leader has been perturbed, it will spread the perturbation to the rest of the flock but also tend to realign itself with its (former) neighbors. The leader may therefore already have realigned with the initial direction of its more distant neighbors once these neighbors are affected by the spreading impulse of the initial perturbation. Their delayed reaction may then echo back to the leader, after yet another delay time. This wave-like oscillatory spreading pattern clearly constitutes a pulse propagation. Apart from the crucial delay interaction, it is tied to the finite extent of the perturbing impulse and the subsequent free relaxation of the leader, so that it could not be observed in the turning scenario studied further above, where the leader's dynamics is at all times determined by Eq. (3), which provides a persistent information influx preventing the backlash.

So far, we concentrated on purely deterministic dynamics, for simplicity. But qualitatively similar behavior can also be observed for systems affected by noise. One may think of external noise as consisting of a large ensemble of "accidental perturbations" of the same type as just discussed. They permanently trigger random chains of similar orientational adaptation events as just described, reverberating across the whole flock. Let us now consider an initially fully polarized flock whose polarization relaxes (to zero, if the flock is infinitely large) due to strong noise. In Fig. 3b , we show that also in this case the time delay has a stabilizing effect on the system, as it in general increases its relaxation time. However, the relaxation time does not grow monotonically in τ . Instead, it exhibits a peak for intermediate delays, and then decreases to a finite asymptotic value. We conjecture that this peak results from a competition between the reduced alignment strength and the improved stability against perturbations, caused by the delay. Similar nonmonotonic effects in the time delay VM were previously reported in Refs. [45, 61] .

Recent studies have investigated the information spreading in the standard VM [15, 19] . In this vein, we now study how information about the leader's turn propagates through a flock, in the time delay VM. In our setting, it is not obvious how to employ the usual information theoretical notions such as entropy or channel capacity [30, 66] . Therefore, our analysis relies on the concept of maximal response [38] or, more generally, correlations [25] .

The change of state of a Vicsek flock is described by the accelerations a i (t) =v i (t) or, equivalently, by changeṡ

) of the individual particles. Following Ref. [38] , we take the turning rateθ(t) as a measure for the strength of the signal arriving at the particle i at time t. For our specific setup, this approach is equivalent to the accelerationcorrelation based approach applied in Refs. [22, 25] . Similarly as in Sec. III, we consider in this section an initially highly polarized flock with a leader performing a deterministic turn. However, in contrast to the above scenarios we assume that the leader changes its orientation instantaneously at time t = 0 by an angle ϕ that remains fixed thereafter. In this setting, the signal source is better localized in space and time than in the case of the continuous turn of Sec. III. To minimize boundary effects, we also initialize the leader in the center of the flock instead of its front. 

Let us first discuss the signal spreading through the flock of immobile (v 0 = 0) agents with a static interaction network. In this case, the distance to the source of the signal is a well defined quantity and thus we can take the corresponding results as a standard for further analysis. Besides, since the lattice is fixed, the information spreads homogeneously and solely by conduction [38] . In Fig. 4 , we show the resulting orientations, θ i (t), and their rates of changeθ i (t), as functions of the distance d i to the leader for τ = 0 and τ = 1. To avoid a dependence on the initial condition, the shown values are averaged with respect to N runs simulation runs as follows. We first collect all data

..,N from the individual simulations runs and sort the total data with respect to the distance d i . The so obtained ordered set of data is then smoothened by averaging with respect to the same number N runs of nearest neighbors. The corresponding average is denoted by the symbol • nn .

Panels a) and c) show that, for τ = 0, the information spreads in "steps" of the width of one interaction radius R = 1. After the first time step, the information is homogeneously distributed within the interaction radius R = 1 because the corresponding particles interacted equally with the information source. The rateṡ θ i (t) monotonically decrease at short distances but increase at large distances from the leader, because particles closer to the leader interact with more neighbors that have already changed their orientation. For time delay τ = 1, oscillations of period τ + 1 are induced in θ i (t) (panel d), for the same reasons as explained above. As a consequence, the rate of change is strongly damped and θ i (t) (panel b) increases in steps. Regardless of the time delay, the orientations θ i (t) exhibit a discontinuity for d i = R that increases with time, as a result of the continuous influx of information from the fixed leader.

For moving flocks with v 0 > 0, the distance traveled by the signal is no longer unambiguously defined. Even though it might seem inadequate, one possible choice of d i is still the distance to the leader at time t = 0 as it gives qualitatively the same results as other definitions, e.g., one that is based on a co-moving frame [38] . In what follows, we thus stick to this definition. In Fig. 5 , we report again the orientations, θ i (t) and their rates of changeθ i (t), as functions of the (initial) distance d i to the leader. The resulting curves are similar to those for the static network, as shown in Fig. 4 . However, the discontinuity in θ i (t) at the interaction range, d i = R, is now smeared out and the rateθ i (t) exhibits concomitant kinks. Both effects result from the anisotropy of the signal propagation caused by the motion of the leader. For v 0 > 0, both the leader and its "interaction zone" move. This affects differently particles in the direction of the leader than those in the opposite direction, which we do not distinguish when evaluating the shown results. The downward-kink inθ i (t) corresponds to the neighbors behind the leader, which leave its interaction zone, and the upward kink corresponds to the particles in front, entering its interaction zone. Averaging these two effects then induces the smearing of the discontinuity of the orientation θ i (t) as a function of distance, at the edge of the interaction zone.

A central quantity in the study of signal propagation is the signal speed, or, equivalently, the dispersion relation. However, to define these quantities for the VM is neither trivial nor unique. Consider the typical time evolution of the signal strength, measured in terms ofθ i (t). The signal reaches an agent i at time t, plotted in Fig. 6a . But its intensity is initially very low. It then grows over time until it reaches a maximum and starts to decay. For a slowly moving leader and flock, for which the information spreading is dominated by conduction, the time when the signal first reaches a particle at distance d i from its source can roughly be estimated as d i /R. This is the time required for the signal to arrive from the leader's initial position, if traveling along the shortest path. The signal gets damped by a factor 1/N int after each time step, since it is distributed among the N int ≈ N R 2 /L 2 F nearest neighbors, on average. This explains why it is very weak at its spreading front. As the number of possible paths connecting the initial position of the leader with the nearby particles in the flock increases over time, also their perceived signal intensity, as well as the rate of changeθ i (t) of their orientation, grows. Once the neighbors thereby become increasingly aligned with the leader, θ i (t) decays again. In what follows, we identify the time whenθ i (t) reaches its maximum as the signal propagation time T i . The corresponding signal speed is d i /T i . For more details concerning this choice, see Ref. [38] .

To inspect the signal propagation more closely, consider the time evolution of the signal strength at two given particles, as depicted in Fig. 6a -c. Each of the three panels corresponds to a different value of the delay. The most striking feature of the raw data (dotted blue and red curves) is the presence of sharp spikes, which become more frequent for long time delays. We reckon that these resonances are due to random inhomogeneities in the initial conditions. At a first glance, identifying the time T i with these spikes might be feared to lead to erroneous estimates for the signal speed. Therefore, we alternatively considered the smoothed profiles depicted by the solid yellow and dashed purple lines. They were obtained by first performing sliding averages over five raw data points and then substituting the remaining outliers (beyond five times the average θ i (t) t over the shown time interval) with an average value of their neighboring points. But the results for the dispersion relations remained qualitatively the same.

Besides the spurious spikes, the curves for time delays τ > 0 again exhibit oscillations, which can even reach negative values as the particle orientations tend to align to their previous values. The period of these oscillations is τ +1 and their magnitude increases with growing delay. For sufficiently long delays, the signal strength exhibits sharp positive and negative peaks as the particle is ei- ther aligned with its own initial orientation or with the initial direction of the leader. These sharp peaks are even better visible in Fig. 6d , where we show the average of θ i (t) N,Nruns taken over all N particles and N runs simulation runs, for the same values of time delay as in Fig. 6a -c. This quantity measures the total rotation rate in the flock at the given time. For short delays, the signal is initially stronger, but the envelopes of the peaks induced by long delays have the longer relaxation time.

In Fig. 7a -c, we show the main results of this section, which are the dispersion relations for different speeds v 0 and time delays τ . They were obtained as follows. We evaluated the times T i for the maximum signal strengtḣ θ i (t) for all N N runs particles in all N runs simulation runs. Then, we calculated the distances d i (T i ) from the leader at time t = 0 to particle i. Finally, we averaged the values obtained for the individual particles over N runs nearest neighbours (the • nn average introduced in Sec. V). The relation of both averages is shown in the plot.

For distances d i ≈ L F /2 = 10 near the boundary of the flock, the dispersion relations are strongly affected by boundary effects, leading to a "flattening" of the curves. For all considered speeds, a retardation of the interactions slows the signal propagation through the system. In accord with Ref. For longer delay times, the dispersion relations exhibit strong oscillations with period τ + 1 and an overall linear increase of the envelope. In Fig. 7d , we show that the dispersion relations roughly oscillate between the values attained for specific directions of the leader. To demonstrate this, we display dispersion relations calculated from the part of the flock in the upper (positive) and lower (negative) half-planes. For the chosen turning angle ϕ = 0.1, the leader moves toward the former and away from the latter. In accord with the findings of Ref. [38] , the dispersion relation for the positive direction is therefore linear, for large enough speed, while the dispersion relation for the negative direction remains diffusive, regardless of v 0 .

To put these results into a broader context, we note that, from the point of view of natural systems, the oscillations may be considered an artifact of choosing precisely the same delay time for all interactions. In practical applications, the time delay is more likely to be randomly distributed among the individuals, so that the oscillations are expected to average out, at least to some extent. With this in mind, the robust conclusion is that the retarded interactions embodied in the delay VM induce linear dispersion relations, similarly as to what inertia does in a generalized VM with underdamped orientations [25] .

So far, we have analyzed the response of a Vicsek flock with retarded interactions in terms of signal propagation, as appropriate in the most general context of its applications. However, in a more physics-focused context, it may be more natural to interpret the results within the framework of linear response theory and the fluctuationdissipation relations known for equilibrium systems. In the latter case, non-invasive measurements of correlation functions suffice to know the response functions for arbitrary weak perturbations [77] . However, for systems far from equilibrium, such relations are generally not available [1, 46] .

In accord with our previous discussion, we will investigate the response of the time delay VM to perturbations induced by a (or various) leader(s). To provide results independent of initial conditions, we consider perturbations that bring the system from one steady state to another perturbed one. The VM eventually converges to a steady state, which is independent of the initial conditions, for a non-zero rotational noise and if the models is solved, e.g., with periodic boundaries. The collective maneuver scenario of Sec. III requires that the size of the periodic box is inversely proportional to the curvature of the leaders' trajectory (so that it fits in the box), which is not a suitable situation for computer analysis. On the other hand, the perturbation scenario of Sec. IV induces a relaxation of the system to its (slightly rotated) initial steady state, and one can measure the characteristics of this relaxation, which have to some extent been discussed in Sec. IV. We now turn to the scenario of Ref. [59] , where one applies a permanent torque to the leader(s), which induces a steady state with a nonzero circular flux of the particles.

Specifically, we consider a system of N particles inside a square box with sides of length L and periodic boundary conditions. We again assume that the individual particles except for the leader obey Eqs. (1)-(2). However, different from the previous section, we consider a small but nonzero value of the noise η, such that the unperturbed system eventually reaches a highly ordered non-equilibrium steady state. We initialize the system at time t = 0 in a polarized state, let it evolve until time t I when it reaches a steady state, and switch on a perturbation affecting N L randomly chosen leaders. Specifically, we assume that velocities of the leaders obey the form

where the operator T ϕ applies a deterministic rotation by an angle ϕ to its argument. Following Ref. [59] , we study how this perturbation affects the discrete curvature in the overall velocity, defined as

where Φ(t) = 1/(N v 0 ) i v i (t) denotes the mean orientation of the flock at time t. The resulting dependence of the average curvature κ on the (relative) strength σ ≡ ϕN L /N of the perturbation is depicted in Fig. 8 . In this section, averages κ ≡ lim t I →∞ t −1 t I +t t I dt κ(t ) are calculated over time after the system reached the steady state, with t = 2 × 10 4 and t I = 10 4 . The shown error bars correspond to standard deviations of these averages computed from 10 independent realizations of the numerical experiment. Fig. 8a) illustrates that, for σ = 0, the flock turns randomly and κ(t) is a Gaussian random variable with zero mean, κ = 0. Imposing a small directional bias σ η to the leaders induces an overall non-zero mean curvature,

which increases linearly with σ [ Fig. 8b) ]. On the other hand, the variance, (∆κ) 2 ≡ κ 2 − κ 2 , in Fig. 8c ) remains constant. For our case of highly aligned flocks (small enough η and large enough density N/L 2 ), the slope α is approximately given by (1 + τ ) −1 . Notice how the individual curves in Fig. 8b depart from the initially common straight line. The linear-response regime is diminished for both long time delays and large directional biases, and the expression for α holds the better the smaller the product στ . The dependence of the variance on the time delay saturates. Its functional form is well parametrized by ∆κ 2 ≈ aτ /(τ + b) [ Fig. 8d) ]. Furthermore, for dense flocks, the variance increases with the noise as (∆κ) 2 ∝ η 2 (for the corresponding illustrations, see Ref. [59] ).

In App. A we show how this behavior can be understood analytically, based on a simplified model. Here, we provide the emerging intuitive explanation. For vanishing delay and in two dimensions, the curvature (5) can be rewritten as κ(t) = sin ∆θ N with ∆θ N = N −1 i [θ i (t)−θ i (t−1)] denoting the mean change of orientation during one time step. For small perturbations ϕ, ∆θ N 1 and thus κ ≈ ∆θ N . For weak noise, high average density, and vanishing delay, the local density of the VM is approximately homogeneous [24] . As a result, the total orientation (or information) N −1 i θ i (t) is for ϕ = 0 approximately conserved [38] . For ϕ > 0, it is then reasonable to assume that the change in the total orientation is directly given by the directional bias applied to the leaders, i.e., ∆θ N = 1/N N L 1 ϕ = σ. For a nonzero delay, the orientational change is spread over τ + 1 time steps, reducing ∆θ N by the factor τ + 1. Figure 8b can thus be regarded as a verification of the (approximate) information conservation in a highly ordered Vicsek flock. Large biases ϕ introduce growing density fluctuations which cause the breakdown of orientation conservation [38] and, as a consequence, of linear response. Similarly, as we have demonstrated above, long delays tend to decouple the leaders from their neighborhoods. One may thus conclude that the breakdown of linear response is associated with a loss in the orientational bias.

In this vein, the variance of the curvature can be approximated as

where κ i = ∆θ i . To expand the square, we assumed that the orientational fluctuations ∆θ i of individual agents are statistically independent, which is sensible for small perturbations ϕ of a highly aligned state, where ∆θ i can be assumed to be caused solely by the noise, i.e., ∆θ 2 i ≈ η −ηη 2 dη/(2η) = η 2 /3. Altogether, we thus find the variance ∆(κ) 2 ∝ η 2 /N , which explains the findings in Ref. [59] . According to our argument, this scaling should be a direct consequence of the law of large numbers. Introducing a finite delay enhances effects of the noise on the dynamics and triggers oscillations [37] . As a result, the curvature variance strongly increases with the delay. Unfortunately we were not able to rationalize the observed quantitative dependence of the variance on the time delay, shown in Fig. 8d , from the equations of motion.

In summary, we have studied the response to orientation perturbations in the 2-dimensional delay Vicsek model. We have found that a time delay diminishes the ability of strongly aligned systems to follow a leader moving along a predetermined path (different from inertia [23] ) and increases the stability of these systems to sudden local fluctuations of orientation. Both effects are caused by the decreased information (orientation) propagation through the system and the expanded timecorrelations in the system, due to the delay. Their combination explains the findings in Refs. [45, 61] that short delays foster and long delays hinder flocking in the delay VM.

In the scenario of a sudden turning pulse applied to a leader, the source of information or perturbation is well localized in space and time. Inspired by [25] , we used the induced orientational acceleration as a measure for the signal strength to study the propagation of information (orientation) through the system. As typical for delay systems [37] , a time delay causes oscillations. In the present context, they are manifestly seen in the orientational accelerations. Their amplitude, period, and decay time (and thus system memory) all increase with growing delay [45] . To measure the speed of information propagation through the system, we used the maximum of the acceleration pulse transmitted through the flock. We found that the dispersion relation for the signal spreading is diffusive (the distance traveled by the signal ∝ square root of the traveled time) for small speeds and short time delays, but becomes increasingly ballistic (linear) for long delays and high speeds, consistent with an increasing dominance of the convective over the conductive character of the flock dynamics.

We have further investigated the linear response of highly ordered systems to a weak orientational bias, applied to a subgroup of agents (leaders). We found that the overall change of the flock's orientation is a linear function of the perturbation for short enough time delays and weak enough orientational biases. The linear response regime coincides with the regime of low density fluctuations, where the total information (or average orientation) of the flock is (approximately) conserved. Furthermore, we provided intuitive and analytical arguments for the observed functional dependencies of the mean total orientational change and the corresponding variance.

While our work provides a first understanding of retardation effects in active many-body dynamics, it also raises some further questions. The choice of a universal value of the discrete time delay for all members of the flock certainly oversimplifies the conditions prevailing in natural flocks and swarms, and it would be desirable to get a better understanding on the concomitant internal averaging and its role in suppressing the oscillatory instabilities observed in our idealized model. Also, the reaction capabilities of natural agents will usually be described by more complicated memory func-tions [37, 49] . Further, it could be interesting to investigate more thoroughly the effects of time delay on the collective memory of the flock, which is an important ingredient in the understanding of so-called swarm intelligence [13, 14] . Our present results confirm the natural expectation that the system's relaxation time is sensitive to the delay time. Another direction could be to study information spreading in more complex interaction networks, e.g., with distributed individual weights for the flock members. Such leadership hierarchies are often observed in natural groups [26, 56] , and it has been argued that they increase the group fitness [7, 56, 67] . For instance, it is known that a few (well-connected) agents, known as superspreaders, lead to a rapid increase in the disease (or fad) spreading in epidemiology [41, 51, 58] .

We acknowledge funding through a DFG-GACR cooperation by the Deutsche Forschungsgemeinschaft (DFG Project No 432421051) and by the Czech Science Foundation (GACR Project No 20-02955J). VH was supported by the Humboldt Foundation. DG acknowledges funding by International Max Planck Research Schools (IMPRS) as well as by Deutscher Akademischer Austauschdienst (DAAD).

The dependence of the first two moments of the mean curvature (5) on the delay time τ , discussed in Sec. VI, can be obtained using a simple mean-field model. Consider a system of N mutually interacting particles with a single leader (index L) that interacts with the average flock orientation θ 0 (t) of the other N − 1 flock members. The orientations obey the dynamical equations θ L (t + 1) = 1 N [θ L (t) + (N − 1)θ 0 (t − τ )] + ϕ + ξ L (t),

The symbols ξ L (t) and ξ 0 (t) denote independent δ- 

where O(0) is the initial condition, which we set to 0 (all particles aligned with the x-axis). Our goal is to (A6) At late times, κ(t) reaches a steady state, in which all elements of the vector κ(t) have the same mean and variance, given by κ and ∆κ 2 . Specifically, we find

where

The right hand side of Eq. (A8) results from multiplying the sum in Eq. (A6) by itself and using the Markov condition ξ (t)ξ(t ) = ξ2 δ tt for the stationary noise correlation function.

It is straightforward to verify, e.g., by calculating left and right eigenvectors corresponding to the eigenvalue 1, that the stationary matrix M ∞ has all elements in its first column equal to c(N −1)/N , with c = [(N −1)/N +τ ] −1 . Its remaining elements are equal to c. Hence, we find that all elements of κ are given by σc, and thus, in agreement with the discussion in Sec. VI,

In Fig. 9a) , we show agreement of this prediction with a numerical solution to Eqs. (A1) and (A2). Equation (A8) is a scalar product of two identical vectors of length (τ +1) containing in all entries the variance (∆κ) 2 . Thus we find (∆κ) 2 = N 2 − 2N + 2 N 2 η 2 3(τ + 1) R 11 . (A10)

While we were not able to extract the τ -and Ndependence of the matrix element R 11 analytically, Fig. 9b) shows that the τ -dependence of the calculated (κ) 2 is well described by the empirical formula aτ /(τ +b), already employed in Sec. VI.

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