Self-reorganization and information transfer in large-scale models of fish schools

Domain: biology
Published: 2026-03-23
Editorial outcome: minor_revision

Abstract
The remarkable cohesion and coordination of moving animal groups and their collective responsiveness to threats are often attributed to scale-free correlations, where behavioral changes in one animal influence others in the group, regardless of the distance between them. But are these features independent of group size? Here, we investigate group cohesiveness and collective responsiveness in computational models of massive schools of fish of up to 50,000 individuals. We show that as the number of swimmers increases, flow interactions destabilize the school, creating clusters that constantly fragment, disperse, and regroup, much like in natural animal groups. Importantly, while spatial correlations in cohesive and polarized clusters are indeed scale free, fragmentation events are preceded by a decrease in correlation length, weakening the group’s collective responsiveness and leaving it more vulnerable to predation. We further show that information about directional changes propagates linearly in time among group members, thanks to the non-reciprocal nature of visual interactions between individuals. Merging events speed up this information transfer, while fragmentation slows it down. Our findings suggest that flow interactions may have played an important role in group size regulation, behavioral adaptations, and dispersion in living animal groups.

Introduction
Nature is in a perpetual state of reorganization. In animal groups on the move, such as in bird flocks[1–3], fish schools[4–6], and insect swarms[7–9], local social interactions result in cohesive global patterns [4, 10–14]. However, while these cohesive patterns are regularly documented in systems of small or moderate size [4, 6, 15–19], it is unclear how they scale with increasing group size[20]: do large groups remain cohesive or do they undergo dynamic reorganization? We address this question in massive simulations of schooling fish, where individual swimmers interact through self-generated flows and follow behavioral rules inferred directly from experimental data in shallow water environments [6, 16, 17, 21]. By optimizing our computational algorithms, we simulate over long times the motion of groups of up to 50,000 fish. We show that “more is different” [22, 23]. Where smaller groups maintain cohesive and polarized formations, larger groups spontaneously reorganize, constantly fragmenting, scattering and reassembling, similarly to empirical observations of large flocks of birds [1, 2] and schools of fish [20, 24–26]. We analyze how this self-reorganization influences the collective responsiveness and speed of information propagation between members of the group [1, 4, 27].

Collective responsiveness in self-organized animal groups manifests in long-ranged spatial correlations [1, 28, 29]. Correlation measures how the change in the behavior of one individual influences the behavior of others in the group. An animal group exhibits maximal responsiveness to a perturbation, say, caused by an attacking predator [20, 30], when correlations are scale-free, that is, when the range of spatial correlations scales with the linear group size [1, 30]. Analysis of empirical data of large bird flocks confirms that spatial correlations scale linearly with group size L [1]. But do these results translate to groups of swimmers?

In physical models of flow-coupled swimmers, microscopic [31–33] and inertial [18, 34–36], fluctuations get amplified as they propagate via the fluid medium, hindering group cohesion. These models do not enable individual swimmers to sense and actively respond to flows. Biological swimmers, on the other hand, are flow sensitive [37–39] and seem to correlate their tailbeat frequencies and phase [40–42], but the extent of flow-mediated correlations is limited in space [18]. Recent evidence suggests that vision is both necessary and sufficient for polarized schooling [43]. Even in robotic agents, active response to visual cues is sufficient to produce scale-free correlations in polarized collectives [44]. But are these scale-free correlations universal to groups of individuals with long-ranged visual and hydrodynamic interactions? If so, how does dynamic reorganization within the group, including splitting and merging event, affect the extent of spatial correlations? Statistical analysis in the context of our schooling model reveals that spatial correlations in cohesive and polarized groups are indeed scale free, much like in natural animal groups, but the correlation length decreases prior to fragmentation. This loss of scale-free correlations serves as a predictive marker of fragmentation.

Spatial correlations reflect the extent of information spread within a group but fail to capture the dynamic nature of information propagation. How fast does information travel within a polarized group? Motivated by analyses of information propagation in bird flocks [27], we approach this question by considering the behavior of our interacting swimmers during spontaneous collective turns. We find that the information about the change in direction propagates linearly in time across the group, at speeds much faster than the individual swimming speed.

This is in sharp contrast to the diffusive information propagation in symmetric consensus-based models [10], and in the absence of behavioral inertia [27]. We show that symmetry is broken due to the non-reciprocal nature of the interactions between individual swimmers [45, 46]. We further find that merging of separate clusters speeds up the transfer of information within each cluster by several folds, while fragmentation slows it down. Interestingly, flow interactions enhance the information travel speed beyond what would be expected from non-reciprocal visual interactions alone.

Results
Mathematical Model. Consider a system of N fish, where each fish is represented as a self-propelled particle moving at a constant speed U (m s−1) relative to the flow velocity. Each fish follows behavioral rules, derived empirically from shallow-water experiments [6, 15], modulated by an asymmetrical visual field representing frontal-biased perception [6]. Additionally, each fish creates a flow disturbance represented by its far-field potential dipole (Fig. 1A) [34, 47] and responds to the combined flow generated by all other swimmers [16, 17].

Let fish i be located at xi = (xi,yi) in an inertial (x,y)-frame and have a heading direction pi = (cosθi,sinθi) expressed in terms of the orientation angle θi measured from the x-axis. The velocity vi = x˙i, where ()˙ represents derivative with respect to time t, and the change of heading angle dθi/dt are governed by the equations of motion, written directly in non-dimensional form, with swimming speed U = 1,

x˙i = vi = Upi + Ui,

dθi/dt = ∑j∈Vi (sinϕij)(1+cosθij) / ∑j∈Vi (1+cosθij) + Ωi + In dWt.

The first term on the right-hand side of the orientation equation represents the fish change in heading in response to visual feedback: fish i only “sees” its Voronoi neighbors Vi, with attraction intensity normalized to one and non-dimensional alignment intensity Ia, both averaged with weight 1 + cosθij modeling continuously a rear blind angle [15] (Fig. 1B). The intermediate variables rij = ||xi − xj||, θij = ∠(xj − xi) − θi, and ϕij = θj − θi represent, respectively, the relative distance, viewing angle, and difference in heading angle between fish i and j. The non-dimensional noise intensity In scales a standard Wiener process W(t) modeling the fish “free will” [48].

The vector Ui represents the flow velocity generated by all other swimmers at the location of swimmer i and Ωi denotes the angular velocity

Ui = ∑j=1,j≠iN (If pi · dxij |xi| · p⊥i) / πrij2, Ωi = ∑j=1,j≠iN (If pi · dxij |xi| · p⊥i) / πrij2,

where p⊥i is a unit vector orthogonal to pi and If is the strength of fish-induced dipolar flow field, which, by definition, introduces a dimensionless length scale a = If/U reflecting the swimmer’s body length [47]. Eqs.(1)–(2) form a closed set of 3N differential equations governing the 3N unknowns (xi,yi,θi), where i = 1,...,N. These equations depend solely on three non-dimensional parameters, In, Ia, and If, representing the noise, alignment, and hydrodynamic intensities.

Fixing the hydrodynamic intensity and varying the noise and alignment intensities in groups of N = 100 fish produces distinct collective phases, from swarming to polarized schooling [16]. In polarized schools, the dipolar hydrodynamic interactions represented by Ui destabilize side-by-side configurations, favoring in-line arrangements (Fig. S1) [31, 34, 47]. This hydrodynamic drafting increases the school’s average velocity, given by ⟨v⟩ = 1/N ∑j=1N x˙j = 1/N ∑j=1N (Upj + Uj), causing the group to swim, on average, faster than the individual swimming speed U [16].

Dynamic reorganization, fragmentation, dispersal, and reassembly in large fish schools. Using (1)-(2), we numerically simulated the motion of a school of 50,000 fish coupled via visual feedback rules and flow interactions in an unbounded planar domain (Fig. 1, Suppl. Movie 1). Here, we used parameter values (In, Ia, If) that, in smaller groups of 100 fish, led to stable polarized schooling [16, 17] (Fig. 2A). We optimized our computational algorithms in order to scale our simulations to groups of the order of 104 swimmers (Methods).

In the group of 50,000 fish, starting from random initial conditions, the fish self-organized into coherent polarized structures that dynamically fragmented and reassembled, exhibiting large density fluctuations (Fig. 1, Suppl. Movie 1), comparable to empirical observations of large bird flocks [1] and fish schools [20, 24–26].

We systematically varied the number of swimmers N. In Fig. 2A-C, we report cohesive and highly polarized schools of 100 and 1000 swimmers and loss of global cohesion in a school of 10,000 swimmers, where distinct polarized clusters moved in different directions. Statistical results from sample simulations at N = 100, 1000, 10,000, and 50,000 are reported in Fig. 2D-G. The polarization order parameter P = ||1/N ∑j=1N pj|| is consistently close to 1 for N = 100 and 1000, indicating high polarization at all time. For N= 10,000 and 50,000, P fluctuates violently, reflecting the reorganization and constant splitting and merging in larger schools: a sharp decrease in P indicates a splitting event, while a sharp increase indicates a merging event.

Considering the average velocity ⟨v⟩ = 1/N ∑j=1N vj of the entire school, we found that, on average, schools swam faster than the individual self-propelled speed U for N = 100 and 1000 because of flow interaction, consistent with [16], but slower for N = 10,000 and 50,000 because these larger schools broke up into subgroups that themselves swam faster but in random directions. This increase in the speed of the polarized clusters is due to hydrodynamic interactions (1-2). For example, in the snapshots in Fig. 2A-C, the school moved at an average speed of 1.20 at N = 100, 1.08 at N = 1000, and 0.54 at N = 10,000. The highly-polarized clusters that formed within the larger schools could reach equally high speeds as their free counterparts; in Fig. 2C, the four clusters moved at speeds 1.14, 0.83, 0.86, 1.08, but in different directions leading to an overall speed of 0.54 (Suppl. Movie 2). The time evolution of cos(∠⟨v⟩), where ∠⟨v⟩ represents the school’s overall orientation, shows more frequent changes in orientation at smaller N, whereas in the larger schools, frequent splitting and merging events create subgroups that move in random directions, hindering the entire school from turning together cohesively. Fig. 2F and G show the number of subgroups per school identified by a density-based clustering algorithm (Methods, [49–51]) and the average number of fish per cluster. The larger schools at N= 10,000 and 50,000 exhibited wider distributions, reminiscent of empirical observations [25], suggesting the existence of a capacity of number of swimmers per polarized cluster that follows a distribution skewed towards moderate values, with a heavy tail beyond which the cluster breaks up and reorganizes. Because of the behavioral and statistical similarities between N= 10,000 and N = 50,000, and to save computational effort, hereafter we investigate the mechanisms responsible for this behavior in groups of up to 10,000 fish.

Transition from cohesion to fragmentation in larger schools. In Fig. 3A, we report the time-averaged values of the school polarization P as a function of N. As we varied N from 100 to 10,000, up to N ≈ 1000, the swimmers exhibited stable schooling, behaving mostly as an indivisible entity, with consistently high polarization values P greater than 0.95. Beyond N = 1000, the school began to fragment, forming locally polarized subgroups that dynamically rejoined and separated again. This indicates the existence of a bifurcation depending on school size, past which the dynamic reorganization within the school caused a decrease in the global polarization order parameter and an increase in its variance (Fig. 3A). In the highly polarized and cohesive regime, the school turned frequently and rarely fragmented, but as N increased, the frequency of global turning events decreased while the frequency of splitting and merging increased (Fig. 3B).

Flow interactions trigger spontaneous reorganization within the school. We next asked what mechanisms lead to the school fragmentation and self-reorganization at larger N. Given that our model accounts for vision-based rules of alignment and attraction, flow interactions, and individual noise, we set out to test the role of each in triggering the transition from the cohesive state to the state of self-reorganization with increasing school size. We first suppressed all hydrodynamic interactions, and considered a school of 10,000 swimmers interacting only via vision-based rules. We observed no fragmentation, reassembly, and reorganization, independent of noise levels (Supplementary Fig. S2D). At exceedingly large noise, the school transitioned to a swarming phase where all polarization was lost, consistent with classic models [4, 5, 13, 15, 16]. We thus concluded that the vision-based rules of attraction and aligning to Voronoi neighbors lead to no fragmentation of the group, independent of group size, and that noise alone is not sufficient for self-reorganization. Hydrodynamic interactions are important. With hydrodynamic interactions and no noise, the phenomena of dynamic reorganization, fragmentation, dispersal, and reassembly remain largely unchanged (Fig. S3). At a fixed dipole intensity, varying the noise intensity has little effect on the collective pattern (Fig. 3C). These results imply that the transition from cohesive schooling to self-reorganization is driven by flow interactions. To explore the threshold of this flow-induced instability, we maintained the same noise level and varied the intensity of the hydrodynamic interactions by increasing the dipolar field If across several orders of magnitudes from 10−4 to 5: since If ∝ a2U is proportional to the swimmer’s speed U and the square of the body length a, a weaker dipolar intensity represents smaller and slower fish and a larger dipolar intensity represents larger and faster fish [16]. In Fig. 3D, we report results across this wide range of If for N = 100, 1000, and 10,000 swimmers. Smaller schools maintain school cohesion at larger values of If. In larger schools, cohesion is lost at smaller values of If, indicating that the capacity for cohesive schools depends on the hydrodynamic intensity of individual swimmers, which in turn depends on their size and speed. That is, smaller fish can school cohesively in larger numbers (Fig. 3E).

Scale-free correlation breaks down during school self-reorganization. The range of spatial correlations in polarized flocks of birds was shown to scale with the maximal length of the flock [1]. This linear scaling of correlation length with group size implies that the effective perception range of each individual encompasses the entire group and enables transfer of information between members regardless of distance, ensuring collective response to perturbations [1, 2, 26]. We asked whether these conclusions are generic to emergent polarization in groups of self-propelled individuals, including our simulations of schooling fish, and how self-reorganization within the school, in the form of continuous fragmentation, dispersal, and reassembly, affects the range of spatial correlation and the ability to transfer information among school members.

To address these questions, we first considered cohesive and highly polarized groups of swimmers ranging in size from N = 100 to 1000, where consistent with [1], we analyzed snapshots with high degree of polarization (P > 0.9). For swimmer i, we defined the fluctuation δvi around the group’s mean velocity as δvi = vi − ⟨v⟩ (Fig. 4A,B). By construction, ∑i=1N δvi = 0, indicating no net fluctuations in the net motion of the center of mass of the school. We calculated the spatial correlation function C(r) of velocity fluctuations (Methods), where the span of r does not exceed the length L of the group defined as L = max ||xi − xj||. A positive value of C(r) close to 1 implies that the fluctuations are nearly parallel and strongly correlated. Conversely, a negative value of C(r) close to -1 implies that the fluctuations are antiparallel and anticorrelated. A value of C(r) ≈ 0 implies a random distribution of velocity fluctuations with no correlation. In Fig. 4C, we report C(r) versus r for the snapshot presented in Fig. 4A. At short distances, the correlation is close to 1 and decays with increasing r, becoming negative at large interindividual distances, indicating strong correlation at short distances and strong anticorrelation at large distances, and in no range of r are the velocity fluctuations uncorrelated.

To explain the behavioral implications of this form of C(r), we defined the correlation length ξ as the relative distance r at which C(ξ) = 0. By definition, the value of ξ is the maximal size of the positively correlated domain. In Fig. 4D, the resulting correlation length ξ is plotted versus school length L using simulations at various sets of parameters (If, Ia, In) and school size N ≤ 1000, provided P > 0.9 (Table S1). We found that ξ increases linearly with L, much like in the case of starling flocks [1]. We found no scale-free behavior in speed correlations, because speed fluctuations in our model are due to passive hydrodynamic interactions and do not arise from active interactions between the swimmers (Fig. S4). These results confirm that scale-free correlations in velocity fluctuations are generic. They reflect the rotational interactions encoded at the level of individual swimmers (Eq. 1), and can be attributed to the existence of a Goldstone mode associated with the breaking of rotational symmetry leading to group polarization [52, 53]. Interestingly, in our simulations, the slope of the best fit of ξ vs L is nearly one-third, similar to the slope reported in [1] for natural bird flocks.

But does this scale-free correlation generalize to larger groups that continuously reorganize? To answer this question, we revisited the simulation of N = 10,000 fish reported in Fig. 2 and identified cohesive and highly polarized clusters within the school that are about to undergo self-reorganization. In Fig. 4E, we report a snapshot where the entire school moves cohesively, at high polarization, preceding a splitting event, where the school fragments into three different clusters (highlighted in different colors). We calculated the time evolution of the polarization parameter P of the entire school and of the subgroups that later constituted the three separate clusters (Fig. 4F). The school maintained a high level of polarization until the time at which it fragmented, beyond which P decreased, but each cluster recovered quickly, exhibiting high polarization per cluster. Interestingly, a gradual decrease in the correlation length ξ far preceded the sharp decrease in P, while the school size L remained unchanged (Fig. S5E), inducing an overall decrease in ξ/L over time and loss of scale-free correlation prior to fragmentation. This loss in scale-free correlation is predictive of an upcoming splitting event in all cohesive clusters.

To verify this, we considered the time evolution of the school of 10,000 swimmers and, at each snapshot, we identified all clusters of cohesive swimmers, selected highly-polarized clusters for which P > 0.9, calculated the corresponding ξ and L, and plotted the joint probability density function of cluster size L and correlation length ξ as a heatmap over the (L,ξ) space (Fig. 4G). The (L,ξ) values are concentrated at and below the scale-free correlation line (dashed grey) obtained in stable schools in Fig. 4D. Highlighted on this plot are the (L,ξ) values corresponding to the fragmentation event reported in Fig. 4E,F: the correlation length starts at the scale-free line ξ/L ∼ 1/3 and decreases before the onset of splitting (grey arrow), emphasizing the loss of scale-free correlation during school reorganization.

Loss of scale-free correlation is a predictive marker of fragmentation. To support our finding that loss of scale-free correlation is a precursor to fragmentation, we revisited the case of N = 1000. At N = 1000, fragmentation events are isolated and thus simpler to analyze, as opposed to the school of 10,000 fish where multiple fragmentation events could occur simultaneously. We conducted 10 Monte Carlo simulations at N = 1000 for a total duration of 1000 time units each.

In Fig. 5A, we show snapshots of a splitting and merging event, and in Fig. 5B, we show the time evolution of the polar order parameter P, school size L, correlation length ξ and ratio of correlation length to school size ξ/L over a time window before and including a single fragmentation and merging event (highlighted in grey). Before the onset of fragmentation, the polarization P and school size L remain relatively stable. After fragmentation, the school size L increases and then decreases during merging. Importantly, ξ and ξ/L seem to decrease prior to the onset of fragmentation. To quantify this decrease in correlation length prior to fragmentation, we analyzed the time evolution of all 10 Monte Carlo simulations, discarding the initial 200 time units to ensure that the school has reached steady state. We then identified the time t* marking the onset of each fragmentation event and the time t† marking the end of merging following fragmentation, and took the time average of P, L, ξ, and ξ/L over a time window starting at the end of the preceding merging event and ending 100 time units prior to the onset of the next fragmentation event. We also averaged these values over a time window starting at t* − τ and ending at the onset of fragmentation t*. We tested four different values of τ = 10, 30, 50 and 100. Collecting the statistics from all simulations (Fig. 5C), we found that the polar order parameter P and school size L were largely insensitive to the temporal placement of the averaging window, far or right before a fragmentation event. However, the correlation length ξ and ratio ξ/L decreased significantly right before a fragmentation event. These results reinforce our conclusion that the correlation length decreases prior to fragmentation and that this decrease serves as a predictive marker of fragmentation.

Information propagates linearly in time in cohesive groups. Spatial correlations reflect the extent of information spread within a group but do not describe the dynamics of information propagation nor the efficiency of a collective response to environmental factors [1, 27, 52]. An efficient collective response depends on how fast localized perturbations succeed in modifying the behavior of the entire group. Take, for example, a group changing its overall heading direction (Fig. 6A and Suppl. Movie 3). The actual execution of such turn is not instantaneous, because a certain amount of time is needed to propagate the turn throughout the group. During this time, cohesion is strained by the mismatch between individuals who have already turned and those who have not yet done so, as reflected by a drop in polarization P (Fig. S5B). Therefore, the speed at which information is transferred from individual to individual plays a crucial role in maintaining group cohesion, which in return is key for scale-free correlation and collective responsiveness.

We set out to quantify information transfer in cohesive groups first, then to assess the effect of school reorganization – fragmentation and merging – on information transfer. To fix ideas, we analyzed, following [27], a collective turn in a cohesive group of N = 1000 swimmers. Given the trajectory of each swimmer i (Fig. 6A), we calculated the curvature κi as a function of time and identified the time ti of maximum curvature. For each pair of swimmers, i and j, we calculated their mutual turning delay, τij = ti − tj, defined as the amount of time by which swimmer j turns before (τij > 0) or after (τij < 0) swimmer i (Methods, Fig. 6B, inset, and S5C). From the delays τij, we ranked all swimmers in the group according to their turning order and labeled each swimmer i by its order o in terms of its absolute turning time ti with respect to the top-ranking swimmer. We found that the top-ranking swimmers – the first swimmers to turn – are physically close to each other (Fig. 6C, inset). That is, the collective turn has a spatially localized origin and propagates anisotropically from front to back through swimmer-to-swimmer transmission of information.

Given that the motion of the group is two-dimensional and that the turn has a localized origin, the information propagates a distance d = oi/ρ, where ρ is the school density, which remains nearly constant during the turn. Plotting di versus time (Fig. 6C), we found a clear linear regime at early and intermediate times, implying that the distance traveled by the information grows linearly with time d(t) = ct, where c is the speed of propagation of information; its value is about 20 times that of the self-propelled speed U of individual swimmers in our model. We repeated this analysis for various turning instances in schools ranging in size from 100 to 2000 swimmers (Fig. S6). The information transfer speed fluctuated with the number of swimmers but remained consistently an order of magnitude larger than that of the individual swimming speed (Fig.7B).

Information travels linearly thanks to non-reciprocal visual interactions. The linear and fast propagation of information within the school is a key factor in preserving school cohesion during turning. What is the mechanism responsible for this phenomenon? Theoretical models based on local alignment with neighboring individuals, such as the Vicsek model [10], lead to diffusive information propagation, with speeds that scale sublinearly with √t [27]. In [27], to account for the linear propagation speed of directional changes observed in empirical data of bird flocks, the authors introduced behavioral inertia into the alignment dynamics. This modification transformed the diffusive model into a wave equation, allowing information to propagate both forward and backward across the group. However, our results are based on a kinematic model (1)- (2) that ignores the inertia of the fluid and individual swimmers. In our model, the key ingredient enabling information to propagate linearly in time is the non-reciprocal nature of visual interactions between individuals.

To explain the role of non-reciprocal vision in information propagation within the group, we set out to derive a continuum partial differential equation governing the phase φ, where φi = θi − ⟨θ⟩, the perturbation from the school average heading direction ⟨θ⟩ = ∠⟨v⟩. Ignoring noise and flow interactions, and starting from the alignment dynamics in (1), we get

∂θi/∂t = Ia ∑j∈Vi (sinϕij)(1+cosθij) / ∑j∈Vi (1+cosθij). (3)

Considering, with no loss of generality, ⟨θ⟩ = 0 and assuming the swimmers are located on a two-dimensional square lattice of mesh size α, we coarse-grain (3) over a box containing a focal swimmer and four immediate neighbors and take the continuous limit to arrive at

∂φ/∂t = α2Ia/4 ∆φ + αIa/2 ∂φ/∂x + φ αIa/2 ∂φ/∂x. (4)

The equation governing φ has an anisotropic advection term, where φ is advected linearly in the longitudinal direction and non-linearly in the transverse direction, albeit at much smaller and thus ignorable speed (considering that φ ≪ 1). The diffusion term scales with α2Ia, while the advection term scales with αIa, implying that in dense schools, linear advection is dominant. That is, unlike the second-order wave equation in [27], information here is governed by a first-order transport equation. Ignoring diffusion and considering an initial perturbation φ0(x,y) = Asin(kx) in the longitudinal x-direction of amplitude A and wavelength k, the perturbation propagates from front to back at a speed c = αIa/2, following φ(t,x,y) = Asin(k(x+ct)).

To test the prediction that information propagates linearly from the front to the back of the school at speed c ∝ αIa in numerical simulations with noise and flow interactions, we systematically varied both the alignment and noise intensities and calculated the resulting polarization P in cohesive groups (Figs. 7A, S9). We found that P satisfies the relation P = 1 − In/Ia derived in [27] using the spin-wave approximation. We also calculated the information transfer speed c during turning as a function of Ia/In and found that, indeed, c scales linearly with Ia/In (Fig. 7C), demonstrating consistency between our simulations and the scaling law derived from the alignment model. Our results uncover a vision-based mechanism for linear information propagation, distinct from the behavioral inertia proposed in [27]. Indeed, our results differ from [27] in two ways: c scales linearly with alignment intensity Ia, in contrast to the sublinear scaling c ∝ √Ia in [27], and information propagates from the front to the back of the school, as opposed to the isotropic information propagation in [27]. Importantly, our theory shows an anisotropy in the information transfer speed, with information traveling faster in the longitudinal direction of the school. Experimental evidence in animal groups support our findings that information transfer is anisotropic, arising from the non-reciprocal nature of local vision-based interactions. In rummy-nose tetras, U-turns propagate front-to-back through the school at an approximately constant speed [54], whereas in pigeon flocks, the lateral placement of the eyes biases visual interactions toward side neighbors, leading to predominantly lateral information flow [55]. Assessment of the relative effects of inertia [27] and non-reciprocal visual interactions on the transfer of information in natural animal groups would require integrating models with empirical data [56].

Fragmentation slows down information propagation and merging speeds it up. We next examined splitting events during school self-reorganization. In Fig. 6D, we show trajectories of the splitting event pointed out in Fig. 4E, where the polarized school of 10,000 swimmers splits into three subgroups, each turning in a different direction. We analyzed each subgroup (Figs. 6D, S5D-F) and found that information travels linearly at nearly the same speed in all three subgroups, at about three fold faster than the self-propelled speed U, but at much slower compared to information travel speed during free turning (Fig. 6F, 7B). This is perhaps not surprising given the loss of spatial correlation prior to and during fragmentation (Fig. 4F,G).

We also examined information transfer during merging events. In Fig. 6G, we show trajectories of a merging event in the simulation of 10,000 swimmers, where two subgroups, starting from polarized states in different directions, turn and merge into a single subgroup. During merging, swimmers from different subgroups do not mix. Analyzing each subgroup (Figs. 6E,F, S5G-I), we found that the information transfer speed increases, reaching up to 40 fold that of the self-propelled speed, much faster than information propagation in free turning and during fragmentation. To further probe the robustness of these results, we analyzed multiple merging events in clusters of different sizes ranging from 1000 to 3000 (Fig. S7). We found that the information transfer speed is consistently higher during merging, indicating that continuous information input from neighboring clusters increases the speed of information propagation (Fig. 7).

Flow interactions enhance information travel speeds. Lastly, we explored the effect of flow interactions on information travel speed. In Fig. 7D, we fixed the alignment and noise intensities and systematically varied the hydrodynamic intensity If from 10−4 to 0.05. Results are shown in Fig. 7D inset on a semi-log scale. At small hydrodynamic intensity, the school stays cohesive and the information transfer speed follows closely that predicted by the vision-based alignment model c ∝ αIa, where we set α = ⟨VND⟩. However, as If increases, the information speed diverges from this model prediction.

To help explain the effect of hydrodynamic interactions on information propagation, we considered how a perturbation in phase φi propagates via the fluid medium only (Methods). In the continuum limit, we found that, as in the alignment model, hydrodynamic interactions alone cause information to travel from the front to the back of the school with speed c ∝ If/α2. However, this flow-based scaling does not correctly predict the information travel speed of the school, because of the non-trivial interplay between vision and hydrodynamic interactions. Indeed, hydrodynamics affects the average distance to Voronoi neighbors ⟨VND⟩ (Figs. S1 and S9D), which in turn affects the information travel speed due to alignment c ∝ αIa, where α = ⟨VND⟩. If hydrodynamic coupling between swimmers had no direct effect on c other than through its effect on ⟨VND⟩, we would expect c/⟨VND⟩ to be independent of If; rather, it increases linearly with If (Fig. 7D), indicating that hydrodynamic interactions, coupled to vision-based alignment, enhance information transfer speeds.

Methods
A. Computational method

To numerically solve the system of equations (1-2) for a large number of fish N, one needs a computationally efficient approach to handle the all-to-all hydrodynamic interactions and Voronoi tessellation at each time step. The computational complexity due to the hydrodynamic interactions in (2) scales with O(N2). To handle these interactions, we optimized and paralleled the code responsible for computing the direct sum in (2) using a just-in-time compiler called Numba [77]. Numba compiles, optimizes, and parallelizes the Python code to approach the computational performance of C or Fortran. Note that fast multipole methods (FMM) reduce the computational complexity of the hydrodynamic interactions from O(N2) to nearly O(N) [78, 79], but FMM algorithms do not have a significant advantage over direct sum in systems of the order of 104 agents [79], hence our choice to directly optimize the O(N2) sum in (2). For the Voronoi tessellation in two dimensions (2D), efficient algorithms exist for reducing this task to O(NlogN) [80]. We utilized the function Delaunay in Scipy [81]. We implemented these approaches in evaluating the right-hand sides of (1) at each time step dt, discretized the noise term using dW(t) = N(0,1)√dt, and used an explicit Euler–Maruyama method to integrate (1) forward in time at a timestep size dt = 10−2 [82]. We run our algorithm on an Exxact Valence Workstation with a 56-core Intel Xeon W9-3495X CPU. With this software and hardware setup, a timestep takes about 1 second for 10,000 agents, with hydrodynamic interactions and Voronoi tessellation taking about half of the computational time each. Integrating the motion of 10,000 agents over a time interval T = 1000 took about a day; integrating the motion of 50,000 swimmers for the same time interval took about three weeks.

B. Statistical and data-driven analysis

Polar order parameter. To quantify the degree of polarization within each group or subgroup of swimmers, we calculated the polar order parameter P = ||1/N ∑j=1N pj|| ∈ [0,1], where P =1 when all swimmers are heading in the same direction; it is nearly zero for randomly oriented swimmers.

Identifying splitting and merging events Fish remained cohesive in relatively small groups, but in large schools, we observed dynamic splitting and merging where the large school got divided into subgroups, each moving in a different direction that seemed to randomly rejoin and divide again for the entire simulation time. To identify these splitting and merging events, we examined the time evolution of the polar order parameter: P rapidly decreased or increased when a splitting or merging event occurred. To determine the time scale at which these events took place, we calculated the dominant frequency of dP/dt using Fast Fourier transformation (FFT). In the absence of splitting and merging events, such as at small number of fish, the FFT is characterized by high frequencies due to individual-level noise. We discarded these frequencies (equivalent to a low-pass filter) to identify the frequencies at which the splitting and merging events occurred in large schools. We discard all frequencies larger than 0.5. The inverse of this dominant frequency defines the time scale of splitting and merging.

Clustering algorithm. To identify the number of distinct subgroups in large groups of swimmers as a function of time, we used a numerical approach based on clustering methods [83]. Because in this active system, the individual clusters have versatile and time-varying shapes, we needed a computational approach that could handle arbitrarily shaped clusters. Classic clustering methods based on expectation–maximization algorithms [84], such as K-means [85] or Mixture Models [86], suffer in identifying intertwined clusters with time-varying shapes. Here, we used density-based methods that are designed to separate low- and high-density regimes in the domain and identify complex-shaped clusters [49–51, 87, 88]; particularly, we used the Hierarchical Density-Based Spatial Clustering of Applications with Noise (HDBSCAN) algorithm [49–51], implemented in the scikit-learn package [89], which has been successfully applied to identify clusters in simulations of the Vicsek model [83].

Spatial correlation in velocity fluctuations. The degree of polarization P provides little insights into the collective response in a school [1, 52]. To understand the collective response, we examined how fluctuations in each swimmer’s velocity correlate with those of others. For swimmer i, we defined the fluctuation δvi around the group’s mean velocity as δvi = vi − ⟨v⟩N, where ⟨v⟩N = 1/N ∑j=1N vj. By construction, ∑i=1N δvi = 0, which simply indicates no net motion in the center of mass reference frame of the school. We defined the spatial correlation function C(r) of fluctuations, which measures the average inner product of velocity fluctuations of swimmers at a distance r from each other,

C(r) = 1/Co ∑i≠j (δvi · δvj)δ(r − rij).

Here, the Dirac-delta function δ(r − rij), where rij = ||rij|| and rij = xi − xj, selects pairs of swimmers at mutual distance r, and Co is a normalization factor such that C(r=0)=1.

Time delays during turning and information propagation within the group. When a cohesive polarized group of swimmers performed a collective turn, to define the turn, we examined the time evolution of the curvature κi = ||vi × v˙i|| / ||vi||3 of the trajectory traced by swimmer i, where v˙i is the swimmer’s acceleration. In 2D, the curvature can be calculated directly in terms of the time derivatives of the coordinates (xi,yi), namely, κi(t) = (x˙iy¨i − y˙ix¨i) / (x˙i2 + y˙i2)3/2. The time-evolution of the curvature κi(t) of a swimmer i undergoing a turn exhibits a maximum at the time of the turn. Inspired by [27, 90], and given two swimmers i and j, we defined the mutual turning delay τij as the time required to shift the full curve of κj(t) to maximally overlap it with κi(t)

τij = argmaxτ ∫t1t2 κi(t)κj(t − τ)dt.

Here, τij < 0 means fish i turns ahead of fish j and vice versa. In the absence of noise, time ordering requires that τij = τik + τkj, for each triplet i, j, k. For example, if i turns 10 time units before k, and k turns 5 time units before j, then i turns 15 time units before j. Because we are dealing with a noisy system, this equality may not be strictly satisfied, but τij is equal to τik + τkj on average.

We next ranked the group of fish undergoing a turn based on their time of maximal curvature. For each fish i, we calculated how many other fish it has turned ahead of [27, 91]. The order of this number – the number of other fish a fish precedes in turning – defines a rank for the fish; the first-ranked fish is ahead of the largest number of fish and its turning time is used to set the time t1 of the onset of the turning event. In a perfect system, with no noise, the turning time ti of a lagging fish i can be calculated directly relative to the turning time of the first-ranked fish 1, ti = t1 + τi1. However, because the system is noisy, this method of calculating ti introduces small statistical errors. To minimize these errors, we define ti using the mutual delay τij with respect to all swimmers j ranked higher than i,

ti = 1/ranki ∑j<ranki (tj + τij), i>1.

C. Coarse-grained analysis of information propagation

Alignment model. Starting from the microscopic equation describing the time evolution of swimmer’s heading

θ˙i = Ia ∑j∈Ni (sinϕij)(1+γcosθij) / ∑j∈Ni (1+γcosθij),

where γ ∈ [0,1] is a parameter that controls the strength of vision-based bias, or non-reciprocity, toward neighbors in front: γ =1 is used in (1) while γ = 0 means no visual bias. We derive a continuum equation under the following conditions. Firstly, we consider a highly polarized school, which means that the orientation of each swimmer within the school can be decomposed into the average heading direction of the school ⟨θ⟩ and a small fluctuation φi of individual swimmers i about the average heading ⟨θ⟩, namely θi = ⟨θ⟩ + φi. Without loss of generality, we assume the ⟨θ⟩ = 0, which aligns the positive x-direction with the moving direction of the group. Based on this assumption, sinϕij = sin(θj − θi) = sin(φj − φi) ≈ φj − φi, and cosθij = cos(arctan(yj − yi)/(xj − xi) − θi) = cos(arctan(yj − yi)/(xj − xi) − φi). Substitute these relationships into (1), we get

∂φi/∂t = Ia ∑j∈Ni (φj − φi) / (1+γcos(arctan(yj − yi)/(xj − xi) − φi)).

Secondly, we assume that the swimmers are located on a 2d lattice of mesh size α, and mesh orientation aligned with the swimming direction. We aim to coarse-grain the discrete equations (1) over a coarse-graining box containing a focal swimmer and four immediate neighbors, such that a swimmer i responds to its direct front, left, back, and right neighbors, indexed by i1, i2, i3, i4. Their locations with respect to particle i can be written as xi1 − xi = (α,0), xi2 − xi = (0,α), xi3 − xi = (−α,0), and xi4 − xi = (0, −α). Plug it into (9) and reorganize to arrive at

∂φi/∂t = α2Ia/4 (φi1 + φi2 + φi3 + φi4 − 4φi) + γαIa/2 (cosφi(φi1 − φi3) + sinφi(φi2 − φi4)).

The finite difference can be approximated by first-order and second-order derivatives, such that at small φ where cosφ ≈ 1, sinφ ≈ φ, we arrive at

∂φ/∂t = α2Ia/4 ∆φ + γαIa/2 ∂φ/∂x.

The equation governing φ has an anisotropic advection term, where φ is advected linearly in the longitudinal direction x and non-linearly, albeit at much smaller speed (considering that φ ≪ 1) in the transverse direction. Ignoring the nonlinear term, we get

∂φ/∂t = α2Ia/4 ∆φ + γαIa/2 ∂φ/∂x.

The diffusion term scales with α2Ia, while the advection term scales with αIa, implying that in dense schools, linear advection is dominant. Ignoring diffusion and considering an initial perturbation in the longitudinal x-direction of form φ0(x,y) = A sin(kx x), the perturbation propagates from front to back at a speed c = γαIa/2, φ(t,x,y) = A sin(k(x+ct)).

Hydrodynamic interaction model. Considering the group is heading in the same direction and ignoring noise and all vision-based interactions in (1), a small perturbation in φi about the heading direction propagates via hydrodynamic interactions only following the simpler equation

∂φi/∂t = pi · dxij |xi| · p⊥i.

Here, to simply the analysis, we consider the swimmers to form an infinite one-dimensional lattice with equally-spaced potential dipoles of mesh size α, such that the flow field at location i is given by [33],

Ui = ∑j=−∞,j≠i∞ If/π (p⊥j sin2θji + pj cos2θji) / rij2.

Considering perturbations of wavenumber k and associated wavelength 2π/k = (K − 1)α, where K is the perturbed number of swimmers, we employ the analytical expression derived in [33], which transforms the infinite summation in (15) to a finite summation. Substituting back into (14), we get

∂φi/∂t = − 2If/πα3 ∑j=1,j≠iK cos[π(i − j)/K] / sin3[π(i − j)/K] (φj − 2φi).

Linearizing using sin(φj − 2φi) ≈ φj − 2φi, and approximating the finite difference by first-order derivatives, we arrive at

∂φ/∂t = 2If/πα ∂φ/∂x ∫απ/2 xcosx/sin3x dx.

Where the integral is a constant depends only on wavenumber k. This shows that perturbations propagate linearly while getting amplified by hydrodynamic interactions.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Please see attached PDF (\"review_Hang.pdf\")\n(Remarks on code availability)\nI do not see any code available in the ZIP file.", "reviewer_2": "This manuscript is a real puzzle: there are some interesting and nice results (especially one, see below), but unfortunately\nthese are diluted within a mix of irrelevant, misleading and (in some cases) plain wrong claims.\nThis make the final decision about the manuscript very hard. In principle, I am not totally opposed to publication In\nNat.Comm. but:\n- certainly not in the current form\n- the Authors should convince me that the supposedly interesting results are compelling enough\nHence, as I said, I recognize that the manuscript contains interesting contributions. However, the presentation is terrible,\nmaking it exceedingly difficult to extract the key messages without a lot of effort. A heavy revision, focused on summarizing\nthe content, highlighting the main insights and novelty, and streamlining the text, would significantly strengthen the\nmanuscript and *possibly* make it ok for Nat Comm.\nLet me start by listing here what are (in my opinion) the only three interesting results of the paper:\n1. flow interaction destabilises polarization as N increases, leading to fragmentation; this is quite interesting.\n2. a drop of the correlation length occurs *prior* to fragmentation; this is VERY interesting and it should be the punchline of\nthe paper: the Authors have found a precursor of fragmentation, which seems great.\n3. flow interaction increases information propagation speed, mildly interesting.\nThe Authors need to rewrite the manuscript using 1-2-3 above as a roadmap.\nOn the contrary, these are the problematic points of the manuscript:\nA. The introduction must be rewritten putting the emphasis on 1-2-3 above, and cutting all the vague and pseudo-holistic\nparts, which do not add any significance.\nB. It is unacceptable that NO details of the model are given in the main text. This is still a scientific paper, so the main\nmathematical details MUST be given in the main text; because much of the rest of the material can be cut (see below) there\nwill be ample space to do that. I cannot recommend publication if this is not done: this is an exclusively numerical paper, the\nmodel must be in the main. I am totally against equation-free science.\nC. I completely fail to get the relevance of non-reciprocity here; I realise that nowadays non-reciprocity is a hot topics, but\nagain - it is quite misleading in this context. Information propagates because of interaction, be it reciprocal or not (see\nbelow).\nD. Using \"More is different\" as a title of a Section is not helpful at all, and quite rhetorical. The Authors should use titles\nsummarizing the content of the section, not a famous paper by a Nobel prize.\nE. I am completely baffled by the statement that the school swims faster than the individual. Even after reading [16], I find no\nsense in this statement. We have been taught at school the inequality:\n|\\sum_i v_i| <= \\sum_i |v_i|\nbecause here |v_i| = U for each i, we get:\n(1/N) |\\sum_i v_i| <= (1/N) \\sum_i |v_i| = U\nmeaning that the mean speed of the group is smaller that the individual speed.\nso, what is going on here?\nthis seems completely wrong, or - at best - very poorly explained. If this is due to the interaction with the flow, then there is\nsomething wrong in the way the mean velocity is defined. Anyway, the point is completely irrelevant for the rest of the paper,\nhence I would cut it entirely (but I am still curious about how the above high-school-level inequality is violated)\nF. The 'more is different' section is too long; the Authors should keep only the material strictly needed to talk about the rest.\nG. In this disgraced Section there is another very suspicious - in my opinion wrong - claim, namely that the density increases\nwith size. According to the Supp Info no effort has been done here to cure and take care of the border bias. Therefore I am\n90% sure that what the Authors found here is simply an effect of the border bias; this is a very old bias indeed, that was\nalready recognised in the book Stoyan, D. & Stoyan, H. 1994. Fractals, Random Shapes and Point Fields. Chichester: J.\nWiley. To see this bias at work in a spectacular way see Fig.6 of Animal Behavior 76 (1), 237-248 (2008) and relative\ndiscussion on how to eliminate the bias. Therefore, either the Authors cure the border bias, or they cut any statements about\nhow density depends on size (which are irrelevant anyway).\nH. That scale free correlations break down when there is fragmentation is *completely* trivial, hence putting the emphasis on\nthis as the Authors do is very misleading. The Authors clearly know that scale free correlations of the orientations are due to\nGoldstone modes, because they write that; so, it is obvious that when there is fragmentation, one looses completely the\nspontaneous symmetry breaking of the Goldstone theorem, hence no Goldstone models, hence no scale free correlation. As\nI said, this is trivial. As I said above, what is FAR from trivial is that the correlation length decreases PRIOR to fragmentation.\nThis is the only reason I am not recommending straight out rejection of the paper: if true, this result is really compelling. So, I\nstrongly recommend to the Authors to concentrate the paper on this result, and on the handful of truly essential ancillary\nresults. However, because this result is so important, I urge the Authors to make an effort to strengthen the result with more\nand better numerical evidence: Fig.4F is definitely not enough to convince me.\nI. I think that the part (three sections) on information propagation is far far too long, and it should be cut entirely, or\nconsiderably trimmed, if the Authors insist on keeping it. Also it does not goes together very coherently with the result on the\nprecursor of fragmentation above. But, if the Authors decide to keep it and trim it, I recommend at least to cut - or make very\nvery short - the section about the fact that fragmentation slows down propagation: this too is completely obvious.\nJ. The statement that inertia is not necessary to have linear propagation is wrong. Even though here there is no behavioural\ninertia, there is of course inertia nevertheless due to the interaction with the flow. Hence, this claim should be cut, or at least\ncompletely reworded, as in the current form it is not true."}, "author_response": "SUMMARY of REVISIONS\nWe thank the referees fo r their ca reful consideration of our manuscript and helpful\nsuggestions.\nRESPONSE TO REFEREES\nReviewer #1 (Remarks to the Author):\nIn this manuscript, the authors study the collective behaviour of a large number of\npoint swimmers (on the order of ~O(1k) to ~O(10k) mostly) to better understand the\nbehaviour of fish schools. The authors use a potential flow model combined with\nbiased visual sensing and a set of empirical behavioural rules, to implement the\nmotion of each individual point swimmer. The primary focus is to examine scale-free\ncorrelations of the motion, group cohesion, and information transfer among the\nswimmers.\nResponse: We thank the referee for their careful and constructive review of our work and for\ntheir support of this manuscript.\nI have the following questions and comments for the authors:\n1. For the readers’ benefit, the authors should clarify in the main body of the text (not\nsupplementary) that the “school of 50,000 fish” is actually a collection of massless\npoint particles, and that the ‘flow interactions’ or ‘hydrodynamic interactions’ specific\nto this study are based on simplified two dimensional potential flow assumptions.\nResponse: We thank the referee for this suggestion. We revised the manuscript to include a\n“Mathematical Model” section that clearly spells out the details of the model.\n2. The mention of potential flow also brings another question to mind: how are the\nflow interactions damped in the simulations? Usually the influence decays with\ndistance, but viscous damping is not allowed in potential flow. This would allow the\nflow-mediated disturbances to accumulate indefinitely over time, leading to\nunrealistic behaviour.\nResponse: Thank you for bringing up this issue. In our model, flow interactions are\ninstantaneous – each particles creates a dipolar field along its heading direction (as shown in\nFig. 1A). There is no vorticity that is shed in the fluid medium that needs to be damped. The fluid\nhas no inertia; there is no accumulation over time or fluid inertia. This is a purely kinematic\nmodel where the far-field signature of a swimmer is captured by a dipolar flow singularity. This is\na hallmark of the potential flow model that we use: fish are self-propelled flow “singularities” that\nare fundamental solutions to the Laplace equation. We clarify this in the revised manuscript.\n3. The authors make an important point when they state “…in the context of our\nmodel…” at one point in their manuscript. This should be clarified very early on in the\nmain body, and maybe even in the abstract, to ensure that readers fully appreciate\nthat the conclusions being drawn are dependent on the model parameters.\nResponse: We thank the referee for this suggestion. We revised the manuscript to clarify in\nmultiple locations that all conclusions are drawn from our model and are rooted in quantitative\nresults, that are robustly produced by the model.\n4. The primary model for point swimmer motion depends on the three free\nparameters “(In, Ia, If)”. This brings about the question of having to manually tune\nthese coefficients until the desired/expected behaviour is achieved. For example, Fig\n3D shows that changing the value of I_f can change the behaviour substantially. So\nhow is the generalizable ‘correct’ value of these parameters determined, besides\ntuning to desired behaviour? The authors mention using parameters that maintain\npolarity for small groups, but then why would the same parameters expected to be\napplicable in the 50k case, where the behaviour is very different?\nResponse: We thank the reviewer for bringing up point.\nIn our previous work (Audrey Filella, François Nadal, Clément Sire, Eva Kanso, & Christophe\nEloy (2018) PRL), we conducted a parametric study over the parameter space of 𝐼 and 𝐼 ,\n! \"\nwhich we reproduce here for the referee’s convenience.\nFigure 1 Reproduction of phase diagram for the collective behavior model with hydrodynamic interaction for 100 fish\nfrom Filella et al. PRL (2018). Depending on the alignment intensity 𝐼 and noise intensity 𝐼 , the school self-\n! \"\norganizes into milling, schooling, turning and milling patterns. Parameter ranges are 𝐼 ∈[0,10] and 𝐼 ∈[0,1], 𝐼 =\n! \" #\n0.01, and N=100.\nBy changing (I_n, I_a), the fish school falls in disorganized swarming (low polar order parameter\nP, and low milling order parameter M), polarized schooling (high polar order parameter P, and\nlow milling order parameter M), rotationally ordered milling (low polar order parameter P, and\nhigh milling order parameter M), and polarized turning pattern (high polar order parameter P,\nand intermediate milling order parameter M), with the last one only existing when hydrodynamic\ninteraction is present.\nIn the current manuscript, we focus on the polarized schooling pattern and analyze how the\ngroup-level behavior changes with the increase in number of fish. We use parameter sets that\ngenerate polarized schooling pattern (𝐼 = 9, 𝐼 = 0.5, 𝐼 = 0.01) for N=100. With increasing N,\n! \" #\nthe emergent behavior of the fish school is different. The polarized school fragments and\nreorganizes themselves with the increase in number of swimmers. This behavior is robust as\nlong as the parameters 𝐼 , 𝐼 , 𝐼 produce polarized schooling in smaller fish groups.\n! \" #\nIn Fig.3, we performed parametric study of collective motion as a function of 𝐼 and 𝐼 . We found\n\" #\nthat with larger 𝐼 , the school fragments at smaller number N, and changes in 𝐼 does not\n# \"\nsignificantly influence the collective behavior. In Fig. 4D, we showed that at several discrete\nparameter sets (𝐼 and 𝐼 ) and same hydrodynamic intensity 𝐼 , the polarized school have the\n! \" #\nsame scale-free correlation. In Fig. 6, we showed that information transfer speed scales linearly\nwith 𝐼 / 𝐼 .\n! \"\nAs to why we don’t expect these parameter values with increasing number of fish, this is\nbecause we are probing exactly the physics of what happens when the individuals are the\nsame, but we increase their number.\nFixing the hydrodynamic intensity 𝐼 = 𝜋 (𝑎/2)$ 𝑈 , where 𝑎 is body length and 𝑈 is swimming\n#\nspeed, means simply that we are fixing the size and speed of the swimmers.\nFixing the alignment intensity 𝐼 means fixing the response of a swimmer to its immediate\n!\nneighbors, which is limited by the individual’s sensory and response abilities [1,2]. This\ninteraction model should work equally well for 100 fish and for 50,000 fish as a prior, unless\nthere exists some experimental finding that rejects this hypothesis (we are not aware of any). In\nfact, [3,4] show based on experimental evidence in fish schools and bird flocks that the coupling\nstrength does not change over the number of individuals.\nThe noise intensity 𝐼 may change over the number of fish, say due to crowding effects [5], but\n\"\nas we show in the manuscript, the change in noise intensity does not have significant effect in\nthe results (Fig. 3C) – noise is not the mechanism that drives fragmentation and reorganization.\n[1] Dickinson, Michael H., et al. \"How animals move: an integrative view.\" science 288.5463\n(2000): 100-106.\n[2] Burden, Samuel A., et al. \"Why animals can outrun robots.\" Science Robotics 9.89 (2024):\neadi9754.\n[3] Gautrais, Jacques, et al. \"Deciphering interactions in moving animal groups.\" (2012):\ne1002678.\n[4] Bialek, William, et al. \"Statistical mechanics for natural flocks of birds.\" Proceedings of the\nNational Academy of Sciences 109.13 (2012): 4786-4791.\n[5] Sosna, Matthew MG, et al. \"Individual and collective encoding of risk in animal\ngroups.\" Proceedings of the National Academy of Sciences 116.41 (2019): 20556-20561.\n5. Line 50: “In physical models of flow-coupled …” – I am having some trouble\nunderstanding the meaning of ‘physical models’. I assume the authors are referring to\nfish-shaped models used for laboratory experiments? Also, I am a bit unclear on\n‘…perturbations get amplified as they propagate…’ – perturbations die out due to\nviscous effects, them getting amplified would likely violate conservation of\nmomentum.\nResponse: Yes, the referee is correct. By physical models, we mean fish-shaped models used\nfor laboratory experiments, as well as fish-shaped mathematical models with flow interactions.\nSeveral recent studies (including work from our group) [1,2,3] have recently shown that\nswimmers interacting with each other through the fluid as medium can self-organize and\nposition themselves at regular spacing relative to each other. However, these equilibria are not\nstable. The fluctuations in the relative positions of these swimmers get amplified. This instability\ngrows downstream with more swimmers, yet there is not violation of conservation of\nmomentum. Conservation of momentum and balance laws do not prohibit growth of instabilities.\n[1] Tsang, A. C. H., Shelley, M. J., & Kanso, E. (2018). Activity-induced instability of phonons in\n1D microfluidic crystals. Soft Matter, 14(6), 945-950.\n[2] Newbolt, J. W., Lewis, N., Bleu, M., Wu, J., Mavroyiakoumou, C., Ramananarivo, S., &\nRistroph, L. (2024). Flow interactions lead to self-organized flight formations disrupted by self-\namplifying waves. Nature communications, 15(1), 3462.\n[3] Heydari, S., Hang, H., & Kanso, E. (2024). Mapping spatial patterns to energetic benefits in\ngroups of flow-coupled swimmers. Elife, 13, RP96129.\n6. Line 71: The authors liken message transmission to the game of telephone, and\nindicate that the message becomes distorted as it is passed among a large number\nof individuals. How does this happen in the authors’ model? Is there a stochastic\ncomponent involved that leads to message distortion? Does the cumulative ‘noise’\ngrow with number of times the message is propagated?\nResponse: We thank the reviewer for asking this question. The collective turning emerges from\nthe local noise and local interaction. The “message” is the turn itself. The message does not get\ndistorted (see Fig.6 inset): the curved trajectories of all swimmers are similar despite the time\ndelay in when a given swimmer in the group executes the turn.\nTo further convince the referee of this, we calculate here the correlation between curvatures as\n∫\n($\n𝑘 (𝑡)𝑘 (𝑡−𝜏) 𝑑𝑡\n𝑄 = 𝑚𝑎𝑥\n()(%*' % &\n%&\n' ∫\n($\n𝑘 (𝑡) 𝑑𝑡∫\n'\n𝑘 (𝑡) 𝑑𝑡\n()(%*' % ()' &\nIf this value is close to 1, it means that the curvatures are perfectly matched.\nIn the figure below, we show 𝑄 for the group of 1000 fish turning. We calculate 𝑄 between the\nfirst ranked fish and all other fish as a function of their turning order o. There is very little\ndistortion of the turning message (𝑄 ≈ 0.9).\nFigure 2 Correlation between the curvature of each fish relative to the first-ordered fish shows nearly no distortion in\nthe message (the curvature of the turn itself).\n7. Line 105: I am having some trouble understanding why the summation for P\ninvolves the complex exponential function (I assume ‘i’ represents sqrt(-1)). Why is\nthe imaginary part useful? Also, “θj is the orientation of swimmer j” with respect to\nwhat precisely, i.e., the group axis, the lab frame, or something else?\nResponse: The polar order parameter P measures how strong the alignment of the school is.\nIn vector form, it is\n+\n1\n𝑃 = ; =(𝑐𝑜𝑠 𝜃 ,𝑠𝑖𝑛𝜃 );\n𝑁 & &\n&),\nAlternatively, and for conciseness, we wrote it in complex form in the manuscript\n+\n1\n𝑃 = ; =𝑒%-&;\n𝑁\n&),\nBy definition, 𝜃 is measured in the lab frame (line 84-85 in main text of revised manuscript and\n&\nline 604 in original SI document). However, the choice of frame would not influence the value of\npolar order parameter P after taking the norm, namely, 𝑃 = F , ∑+ 𝑒%(-& * 0-)F=\n+ &),\nF , ∑+ 𝑒%-&𝑒%0-F = F , 𝑒%0-∑+ 𝑒%-&F =F , ∑+ 𝑒%-&F, where 𝛥𝜃 is an arbitrary angle. We modified\n+ &), + &), + &),\nthe text using the vector notation and clarify the definition of orientation.\n8. Lines 110 and 114: “we found that, on average, schools swam faster than the\nindividual swimming speed U” – this is a bit puzzling to me, how can the average\nspeed be faster than that of the individuals? Being slower would be understandable\nsince not all individual velocity vectors may be parallel to each other, but I don’t fully\nunderstand the school being faster.\nResponse: We thank the reviewers for pointing this out. We corrected this sentence by saying\n“we found that, on average, schools swam faster than the self-propelled speed U”.\nIn our model, the self-propelled speed U is fixed for each swimmer. When hydrodynamic\ninteraction is involved, each swimmer 𝑗 is also advected by the flow field created by other\nswimmers with velocity 𝑈 , thus, the speed of each swimmer is J𝑈(𝑐𝑜𝑠 𝜃 ,𝑠𝑖𝑛 𝜃 )+𝑈 J, which\n& & & &\ncan be larger than 𝑈. For the same reason, the magnitude of group velocity can also be larger\nthan 1. In both prior work and current work, we found that, in polarized groups, hydrodynamic\ninteraction increases the group speed.\n9. Line 138: “the school turned frequently” – what exactly induces a turning event in\nthe authors’ model? Is this driven by an external catalyst, or is it based on an internal\nmechanism? Also, a brief description of why the lead swimmers turn instead of those\nin the middle of the group would be helpful – I assume it is due to the biased vision\nand the imposed rules to maintain alignment with neighbours?\nResponse: There is no external catalyst. All reported behaviors arise spontaneously in the\nmodel. Turning arises naturally due to individual level noise and flow interactions. When noise\nis not present (Fig. S3), collective turning, fragmentation and rejoining arise from hydrodynamic\ninteractions, which are known to induce similar effects as noise (Filella et al. (2018) PRL).\nIn our model, and according to our analysis, the anisotropic information transfer in the\nlongitudinal direction is always from the front of the group to the back. Thus, turning is always\ninitiated at the front of the group because it is a vision-based effect. Perturbations initiated at the\nback of the school don’t propagate forward.\nThe fact that turning arises from the boundary of the school has been commonly observed in\nbiological and physical systems [1,2].\n[1] Cavagna, Andrea, Irene Giardina, and Francesco Ginelli. \"Boundary information inflow\nenhances correlation in flocking.\" Physical review letters 110.16 (2013): 168107.\n[2] Fahimipour, Ashkaan K., et al. \"Wild animals suppress the spread of socially transmitted\nmisinformation.\" Proceedings of the National Academy of Sciences 120.14 (2023):\ne2215428120.\n10. Line 187: The calculation of C(r) is limited to group sizes of 1000 or smaller, and\nonly to instantaneous snapshots of the data when the group is highly polarized.\nDoesn’t limiting the analysis to P>0.9 introduce a bias? It would be helpful if the\nauthors clarify the rationale for doing so. Also, I think small group sizes are\nconsidered since they remain coherent and polarized, but then again, what is the\nbenefit of selecting these? Wouldn’t it be more generalizable to consider 10k\nindividuals, since the authors state that group behaviour changes\nminimally beyond this number?\nResponse: We thank the reviewer for asking this! This is a deep point. It is perhaps not very well\nappreciated outside the statistical physics community: scale-free correlations in velocity\nfluctuations are guaranteed in cohesive and highly polarized groups (for which P>0.9). Scale-\nfree correlations in velocity fluctuations are related to Goldstone modes: it is imperative to start\nfrom a highly polarized state, for which there is a continuous symmetry breaking (rotational\nsymmetry is broken), and thus slow spatially-varying rotations cost little energy and produce\nGoldstone modes. For more details, we refer the review to read the original research paper of\nCavagna et al, PNAS (2010) and excellent review paper Cavagna et al. Physics Reports (2018)\n– full citation listed below.\nWith this as our starting point, we first verified that indeed scale-free correlations arise in our\nmodel in cohesive and highly polarized groups (whether they are of 1000 fish or 10000 fish –\nsee figure below)\nThe important contribution of our work is that when fragmentation occurs, there is a decrease in\nscale-free correlation prior to the fragmentation event.\nFigure 3 Correlation function over distance for a cohesive and highly polarized school of 10,000 fish without\nhydrodynamic interaction.\n11. At short distances, the correlation coefficient being close to 1 makes perfect\nsense since the point swimmers are instructed to maintain alignment using the\nmotion model. But a correlation coefficient of -1 at extremely large distances is very\nsurprising. What is the cause of this?\nResponse: We thank the reviewer for asking this question! This is actually the key point of\nscale-free correlation.\nBy construction, we are building our correlation function based on fluctuation of individual\nvelocity, and thus, the average or mean of velocity fluctuation is exactly zero. According to\nGoldstone modes, the correlation function should cross zero, and the integration of correlation\n2\nfunction over space should be zero ∫ 𝐶(𝑟)𝑑𝑟 = 0.\ndifferent ends of the school are anti-parallel, as illustrated in Fig.4B. This indicates that the\nschool acts as an entity, where any perturbation can propagate through the school.\nFor more details, we refer the review to read the original research paper and excellent review\npaper:\n1. Andrea Cavagna, Alessio Cimarelli, Irene Giardina, Giorgio Parisi, Raffaele Santagati, Fabio\nStefanini, Massimiliano Viale “Scale-free correlations in starling flocks” PNAS (2010).\n2. Andrea Cavagna, Irene Giardina, Tomas Grigera “The physics of flocking: Correlation as a\ncompass from experiments to theory” Physics Reports (2018).\n12. Line 211: “…the slope of the best fit of ξ vs L is nearly one-third, similar to the\nslope…” – What would happen if the authors consider the individual datasets shown\nin Fig 4D and get their corresponding slopes, i.e., separate slopes for dataset X, O,\nsquare, etc.? Do they show the same value of 1/3? Looking at the images, it seems to\nme the slopes would be very different, but a numerical value would be helpful to\nconfirm. If they are not the same, then again the issue arises of what the ‘correct’\nmodel parameter values are, since they can be selected arbitrarily.\nResponse: Yes, there is indeed difference in the fitting of different datasets:\nFor all the data, the slope is 0.365;\nFor 𝐼 = 9, 𝐼 = 0.5, the slope is 0.336;\n! \"\nFor 𝐼 = 7, 𝐼 = 0.5, the slope is 0.406;\n! \"\nFor 𝐼 = 5, 𝐼 = 0.5, the slope is 0.284;\n! \"\nFor 𝐼 = 9, 𝐼 = 0.3, the slope is 0.346;\n! \"\nFor 𝐼 = 9, 𝐼 = 0.7, the slope is 0.345;\n! \"\nClearly, the difference is not big. We included this data in SI Table. 2 in the revised manuscript.\n13. Line 268: The analysis (this time for turning) is again limited to groups with a\nsmall number of individuals (<2000). As the authors state, the behaviour changes\nconsiderably for 10k swimmers, including fragmentation and merging. It is difficult for\nme to understand why the smaller groups were selected, especially since the\nconclusions are presented as if they should be generalizable to all schools.\nResponse: We first analyzed these turning maneuvers on smaller schools of less than 2,000\nfish that stay cohesive during the turn for two reasons:\n1. To establish a baseline from which to compare what happens when the school splits and\nmerges.\n2. Because in larger school, the clusters that form do undergo collective turns, sometimes\nwithout splitting and rejoining, hence the analysis of a cluster of fish that turns is also\nrelevant in the context of the large schools that fragment, reorganize and reassemble.\nThe tools that we present for analyzing the turning only, splitting and turning, or merging and\nturning, are generally applicable to any group size.\n14. Also, in that vein, the statement in Line 348 (and similar elsewhere) “…massive\nschools of fish, consisting of up to 50,000 individuals…” - while technically true due to\nthe use of ‘up to’, could be tempered a bit, given that the majority of the analysis\nregarding correlation and information transfer is done using groups that are more\nthan an order of magnitude smaller.\nResponse: Yes, the reviewer is correct about this: the simulation of 50,000 fish is very costly.\nThe point of doing this simulation is to show our computational prowess but importantly to show\nthat beyond a certain number of swimmers, the phenomena of fragmentation and reassembly\nare always similar. Therefore, the analysis tools and the results that we present are applicable\nto any school size. Indeed, the analysis tools we employ in the current manuscript do not\ndepend explicitly on the total number of swimmers and are equally applicable to the observed\nphenomena, whether in a school of 10,000 or a school of 50,000.\n15. Line 296: The authors claim that “our theory predicts an anisotropy in the\ninformation transfer” – in my opinion it is not a prediction of theory, but rather a result\nof the sensing model imposed on the swimmers using biased vision.\nResponse: We thank the reviewer for bringing this up. We moved the theoretical derivation of\nthe continuum equation describing information propagation from the Supplemental to the Main\nmanuscript. Starting from ”the sensing model imposed on the swimmers using biased vision”\nand deriving a continuous model, the latter reveals an anisotropic transfer of information. We\nreworded the sentence to be more accurate.\n16. Line 374: “Here, we went beyond reporting the fragmentation-rejoining process to\nproposing a flow-based mechanism that drives…” – Maybe the authors could provide\nsome more discussion of the flow-based mechanism in the main text? I may have\nmissed it, and I don’t remember the precise flow mechanism being discussed.\nResponse: We thank the reviewer for asking. In Fig.3, Fig.S2, Fig.S3, we showed that the flow\ninteraction is crucial for the fragmentation and reorganization of the school. Without\nhydrodynamic interaction, the school will always swim cohesively together more swimmers.\nIn Fig.S2, we showed that a school composed of 10k swimmers remains cohesive without\nhydrodynamic interactions. In Fig.3, with hydrodynamic interactions, we showed that with larger\nhydrodynamic interactions, the school splits at smaller number of swimmers. In Fig.3 and Fig.\nS3, we showed that when hydrodynamic interaction is present and fixed in intensity, noise\nintensity does not have a large influence on collective behavior, and reorganization is present\neven if noise is completely removed.\nAll these results show that hydrodynamic interactions are the main mechanism responsible for\nfragmentation. We edited the manuscript to better explain this.\nReviewer #2 (Remarks to the Author):\nThis manuscript is a real puzzle: there are some interesting and nice results\n(especially one, see below), but unfortunately these are diluted within a mix of\nirrelevant, misleading and (in some cases) plain wrong claims.\nThis make the final decision about the manuscript very hard. In principle, I am not\ntotally opposed to publication In Nat.Comm. but:\n- certainly not in the current form\n- the Authors should convince me that the supposedly interesting results are\ncompelling enough\nHence, as I said, I recognize that the manuscript contains interesting contributions.\nHowever, the presentation is terrible, making it exceedingly difficult to extract the key\nmessages without a lot of effort. A heavy revision, focused on summarizing the\ncontent, highlighting the main insights and novelty, and streamlining the text, would\nsignificantly strengthen the manuscript and *possibly* make it ok for Nat Comm.\nResponse: We thank the referee for their careful and constructive review of our work and for\ntheir support of this manuscript. We thoroughly revised the manuscript to address all the\nconcerns raised by the referee. We hope that these revisions convince the referee that all the\nclaims we make in the manuscript are rooted in strong mathematical and numerical arguments.\nWe are thankful for the referee for pointing out that our presentation style might have given the\nimpression that some of the statements we made sounded “plain wrong.” We revised the\nmanuscript to clarify that all our statements and claims are rooted in statistical results or\nanalytical derivations based on our self-propelled particle model.\nLet me start by listing here what are (in my opinion) the only three interesting results\nof the paper:\n1. flow interaction destabilises polarization as N increases, leading to fragmentation;\nthis is quite interesting.\nThank you, we are also excited about this result. We revised the manuscript to clearly\nhighlight this finding.\n2. a drop of the correlation length occurs *prior* to fragmentation; this is VERY\ninteresting and it should be the punchline of the paper: the Authors have found a\nprecursor of fragmentation, which seems great.\nThank you for this positive and constructive feedback – we are thrilled that you find this\nvery interesting (with a capital VERY). We revised the manuscript and added Figure 5 to\nclearly explain and support this finding.\n3. flow interaction increases information propagation speed, mildly interesting.\nThank you! It is quite curious that flow interactions destabilize polarization, while at the\nsame time increase information propagation.\nThe Authors need to rewrite the manuscript using 1-2-3 above as a roadmap.\nResponse: Thank you very much for this careful and constructive feedback on the manuscript.\nWe are very grateful to the referee for their advice. We revised the manuscript accordingly and\nwe think the revised version is much stronger thanks to the referee’s comments.\nOn the contrary, these are the problematic points of the manuscript:\nA. The introduction must be rewritten putting the emphasis on 1-2-3 above, and\ncutting all the vague and pseudo-holistic parts, which do not add any significance.\nResponse: We revised the introduction per the referee’s suggestion.\nB. It is unacceptable that NO details of the model are given in the main text. This is\nstill a scientific paper, so the main mathematical details MUST be given in the main\ntext; because much of the rest of the material can be cut (see below) there will be\nample space to do that. I cannot recommend publication if this is not done: this is an\nexclusively numerical paper, the model must be in the main. I am totally against\nequation-free science.\nResponse: We included a mathematical description of the model in the main text.\nC. I completely fail to get the relevance of non-reciprocity here; I realise that\nnowadays non-reciprocity is a hot topics, but again - it is quite misleading in this\ncontext. Information propagates because of interaction, be it reciprocal or not (see\nbelow).\nResponse: Ah! This is an interesting result. We feel terrible that we didn’t explain it clearly\nenough in the first version of manuscript. This is not a pseudo-result. Non-reciprocity in the\nvision field is the mechanism that allows ballistic information transfer in the school, with\ninformation propagating from the front to the back of the school. We revised the manuscript to\nexplain how non-reciprocity affect information travel speed.\nD. Using \"More is different\" as a title of a Section is not helpful at all, and quite\nrhetorical. The Authors should use titles summarizing the content of the section, not a\nfamous paper by a Nobel prize.\nResponse: Indeed, we were playing on the title of the famous 1972 paper by physicist Philip W.\nAnderson. We are big fans of this paper – can you blame us? – and we thought that the concept\npresented in his paper captures the spirit of what happens in our schooling models as you\nincrease the school size. If the referee is strongly opposed to this, no problem. We changed the\ntitle of the section.\nE. I am completely baffled by the statement that the school swims faster than the\nindividual. Even after reading [16], I find no sense in this statement. We have been\ntaught at school the inequality:\n|\\sum_i v_i| <= \\sum_i |v_i|\nbecause here |v_i| = U for each i, we get:\n(1/N) |\\sum_i v_i| <= (1/N) \\sum_i |v_i| = U\nmeaning that the mean speed of the group is smaller that the individual speed.\nso, what is going on here?\nthis seems completely wrong, or - at best - very poorly explained. If this is due to the\ninteraction with the flow, then there is something wrong in the way the mean velocity\nis defined. Anyway, the point is completely irrelevant for the rest of the paper, hence I\nwould cut it entirely (but I am still curious about how the above high-school-level\ninequality is violated)\nResponse: Indeed, the school swims faster than the individual. The reason for that is that the\nindividual’s velocity is a combination of two effects: (1) its self-propelled velocity and (2) the\nadvection velocity due to the flow field at its location. On average, these flow interactions\nenhance the speed at which individuals move when polarized, thus enhancing the speed of the\nentire school (up to 20% for our choice of I_f). This flow-induced enhancement of overall speed\nis documented in Fillela et al PRL 2018 and mentioned in Huang et al PNAS 2024. We clarify\nthis in the revised manuscript.\nF. The 'more is different' section is too long; the Authors should keep only the material\nstrictly needed to talk about the rest.\nResponse: Done!\nG. In this disgraced Section there is another very suspicious - in my opinion wrong -\nclaim, namely that the density increases with size. According to the Supp Info no\neffort has been done here to cure and take care of the border bias. Therefore I am\n90% sure that what the Authors found here is simply an effect of the border bias; this\nis a very old bias indeed, that was already recognised in the book Stoyan, D. & Stoyan,\nH. 1994. Fractals, Random Shapes and Point Fields. Chichester: J. Wiley. To see this\nbias at work in a spectacular way see Fig.6 of Animal Behavior 76 (1), 237-248 (2008)\nand relative discussion on how to eliminate the bias. Therefore, either the Authors\ncure the border bias, or they cut any statements about how density depends on size\n(which are irrelevant anyway).\nResponse: We thank the reviewer for pointing out this effect of border bias -- we were ignorant\nof it. We carefully studied the paper suggested by the reviewer (Cavagna et al. (2008) Animal\nBehavior). To test whether this boundary bias plays a role in our results on the distance to\nnearest neighbor, we repeated our calculation, disregarding altogether the boundary fish in\nsumming over the nearest neighbor distance. The figure attached shows that, excluding the\nboundary fish, results only in a slight decrease in nearest neighbor distance, but does not\ninfluence the overall trend that we report in the manuscript, namely, that the nearest neighbor\ndistance is constant on average. We thus conclude that the effect mentioned by the referee is\nnot dominant here.\nAs a side note, we agree with the reviewer that scaling of density and nearest neighbor distance\nwith N is not a key point of our paper, but this data served as a basis for us to evaluate\ninformation transfer speed and connect to our analysis. We revised the manuscript to diminish\nthe prominence of these results and put them as a supplementary figure.\nFigure 4 Comparison between nearest neighbor distance with and without removing the boundary points.\nH. That scale free correlations break down when there is fragmentation is\n*completely* trivial, hence putting the emphasis on this as the Authors do is very\nmisleading. The Authors clearly know that scale free correlations of the orientations\nare due to Goldstone modes, because they write that; so, it is obvious that when there\nis fragmentation, one looses completely the spontaneous symmetry breaking of the\nGoldstone theorem, hence no Goldstone models, hence no scale free correlation. As\nI said, this is trivial. As I said above, what is FAR from trivial is that the correlation\nlength decreases PRIOR to fragmentation. This is the only reason I am not\nrecommending straight out rejection of the paper: if true, this result is really\ncompelling. So, I strongly recommend to the Authors to concentrate the paper on this\nresult, and on the handful of truly essential ancillary results. However, because this\nresult is so important, I urge the Authors to make an effort to strengthen the result\nwith more and better numerical evidence: Fig.4F is definitely not enough to convince\nme.\nResponse: We thank the referee for their constructive feedback. Yes, it is not surprising that\nscale free correlations break down when fragmentation occurs, but it is very interesting that the\ncorrelation length decreases prior to fragmentation.\nIn the revised manuscript, we provide additional data in support of the decrease in correlation\nlength prior to fragmentation. Specifically, we run additional Monte Carlo simulations with 1000\nfish, where the school occasionally fragments and re-assembles, as shown in the figure below.\nWe chose this school size because the fragmentation events are isolated as opposed to the\nschool of 10,000 fish, where multiple fragmentation events could occur simultaneously.\nIn the figure below, we show the time evolution of polar order parameter, school size, correlation\nlength and ratio of correlation length to school size after steady state was reached in a sample\nsimulation of 1000 swimmer. Intervals where the school fragments and reassembles are\nhighlighted in grey. During these intervals, the overall school size increases then decreases.\nAway from these events, the school undergoes cohesive polarized schooling and the school\nsize is relatively stable.\nTo quantify the drop in correlation length prior to fragmentation, we took the time average of P,\nL, 𝜉, and 𝜉/𝐿 over a time interval that long preceded a fragmentation event (namely, we\naveraged these quantities over a time window ending 100 time units prior to a fragmentation\noccurred). Then, we repeated this calculation over a time window that ended at the onset of a\nfragmentation event. For the latter, we tested four different time windows consisting of the 10\ntime units prior to fragmentation, 30, 50 and 100. We found that the polar order parameter P\nand school size L were largely insensitive to the temporal placement of the averaging window,\nfar or right before a fragmentation event. However, the correlation length 𝜉 and ratio 𝜉/𝐿\ndecreased significantly right before a fragmentation event. These results reinforce our\nconclusion that the correlation length decreases prior to fragmentation.\nFigure 5 Time evolution of A polar order parameter P, B school size L, C correlation length, and D ratio of correlation\nlength to school size for a school of 1000 fish.\nFigure 6 Statistics of A polar order parameter, B school size, C correlation length and D correlation length over\nschool size for 10 MC simulations of school of 1000 fish. The first column hind shows the results in time snapshots at\nleast 100 unit time from any splitting events; and the rest show the results within 100/50/30/10 time unit before a\nsplitting event at time t*. To evaluate the difference in statistical properties between close to splitting and far from\nsplitting, a two-sample t-test is used [1,2]. The null hypothesis states that there is no significant difference in success\nrates. A smaller p-value indicates stronger evidence against the null hypothesis, suggesting a more significant\ndifference in performance. * p < 0.05, ** p < 0.01, *** p < 0.001.\n[1] Student. (1908). The probable error of a mean. Biometrika, 1-25.\n[2] Fisher, R. A. (1922). On the interpretation of χ 2 from contingency tables, and the calculation\nof P. Journal of the royal statistical society, 85(1), 87-94.\nI. I think that the part (three sections) on information propagation is far far too long,\nand it should be cut entirely, or considerably trimmed, if the Authors insist on keeping\nit. Also it does not goes together very coherently with the result on the precursor of\nfragmentation above. But, if the Authors decide to keep it and trim it, I recommend at\nleast to cut - or make very very short - the section about the fact that fragmentation\nslows down propagation: this too is completely obvious.\nResponse: Thank you for this feedback. The motivation behind these last three sections is to\ndiscuss the dynamics of information propagation, as opposed to the static spatial correlations.\nThe manuscript is organized in two parts: the first part discusses spatial correlations and the\nsecond part discusses the dynamics of information propagation in the school.\nThe sections on the dynamics of information propagation are important because they provide\nthe following novel insights and contributions:\n1. Information propagates linearly from the front to the back of the school. The underlying\nmechanism is non-reciprocal vision. This mechanism is different from behavioral inertia\nwhere information propagates in both directions (Attanasi et al 2014).\n2. Information propagates faster when clusters merge together (not a priori obvious) and\nslower when a cluster fragments (predictable as noted by the referee).\n3. Flow interactions enhance information travel speed (not a priori obvious).\nWe agree with the referee that the manuscript covers a lot of ground, but we think that these\nfindings are important. Removing this discussion would take away fundamental understanding\nof the mechanisms driving information propagation in this schooling model.\nWe would not be opposed to splitting the manuscript into two papers:\n• Self-reorganization in Massive Schools of Fish: I. Scale-free Correlations\n• Self-reorganization in Massive Schools of Fish: II. Information Transfer\nBut we think that this should be an editorial decision. If the editor of Nature Communications is\nopen to this idea, we would pursue it.\nJ. The statement that inertia is not necessary to have linear propagation is wrong.\nEven though here there is no behavioural inertia, there is of course inertia\nnevertheless due to the interaction with the flow. Hence, this claim should be cut, or\nat least completely reworded, as in the current form it is not true.\nResponse: We appreciate the referee’s comments, but with all due respect, the referee is\nmistaken about this. Our statement that “inertia is not necessary to have linear propagation” is\na mathematical statement. This statement is supported by:\n• Numerical simulations of our self-propelled particle model, where there is no\ninertia: in our potential flow model, each swimmer generates an instantaneous potential\ndipole flow and is advected by the flow velocity field generated by all other swimmers at\nits location (Eq1, 2). There is no inertia in the fluid. Importantly, when in simulations,\nwe set flow coupling to zero, we still obtain linear propagation of information.\n• Continuum equation based on our kinematic self-propelled particle model: The\nanalytical model we derived in Eq 4 is purely based on the alignment interactions with\nfrontal bias, without hydrodynamics. Accounting for non-reciprocal vision introduces an\nadvection term responsible for information propagation from the front to the back of the\nschool.\nImportantly, the behavioral inertia introduced in Attanasi et al. (2014) Nature Physics leads to\ninformation propagation that is different from that due to non-reciprocal vision that we uncover in\nthis work: the propagation of information in Attanasi et al. (2014) is bi-directional and at a speed\nthat scales with R𝐼 while in the vision-based mechanism we uncover, information propagates\n!\nfrom front to back at a speed that scales with 𝐼 .\n!\nSUMMARY of REVISIONS\nWe thank the editor and referees for their effort in reviewing the manuscript. We further revised\nthe manuscript as follows:\n1. We clarify the reason for the enhancement in the average speed of polarized groups.\n2. We clarify the differences between the linear information propagation in our\nnon-reciprocal vision model (leading to a first-order transport equation) and the inertial\nmodel proposed in Attanasi et al. 2014 (leading to a second-order wave equation).\n3. We provide references to empirical evidence that support the front-to-back propagation\nof information we observe in our model.\n4. We made slight modifications to improve Figure 6 and its caption.\n5. Haotian Hang and I went very carefully over the Goldstone theorem and made sure we\nall understood the comments raised by referee 2 – the erroneous statements were made\nin the previous response letter, and not reflected in the manuscript – I apologize for this\noversight."}, {"round": 2, "reviewer_comments": {"reviewer_1": "The inclusion of a well-written ‘Mathematical Model’ section in the main body is an excellent\ndecision by the authors, and it clears up several of the questions that I had previously.", "reviewer_2": "The Authors have made several steps forward, although more is needed for me to recommend\npublication in Nature Communication. More specifically:\n* I am still completely baffled by the statement that the school swims faster than the individuals,\nand I see from Ref.1 report that I am not alone. In what reference frame this statement applies?\nAnyway, the current \"clarification\" in the amended version is not clear at all to me. I again\nsuggest that this (old, by the way) result be removed from the paper.", "reviewer_author_response": "All my points have been satisfactorily addressed, hence I now support publication in Nat Comm.\nI also congratulate the Authors for their great patience and cold blood in dealing with me, but I think that the final manuscript\nis significantly better than the original version."}, "author_response": ""}]

Decision letter