Quasi-Attraction and Quasi-Alignment

Based on the aforementioned considerations, we define a new flocking model that naturally extends the previous models. In other words, because we have confirmed that two extremes (attraction and alignment) for the flocking model correspond to different time scales, our aim is to find smooth connections between them.

Let us define the “interaction sphere” for the connection (Fig 1B). First, we draw a Delaunay diagram from the positional relationships of the given agents [51]. The neighbors are connected to all the Delaunay triangles. Second, we compute the distance of each neighbor j from the agent of interest i, i.e., D_ij, and take its maximum. This value is the radius of . Mathematically, the interaction sphere of agent i, , where and is the set of Delaunay neighbors of i. Thus, , covers all neighbors directly linked with its Delaunay triangles.

Note that contains interaction domains from which the agent of interest determines its next direction. In our model, unlike most other models, we do not adopt the average values with external noise. Instead, we define a vaguer region on the interaction sphere; that is, a spherical cap . The possible directions must be selected from within this vague region (Fig 1B). This sphere enables the agents to tune their randomness according to their situations.

Next, we define the quasi-attraction, which consists of relatively small time-scale predictions such as genuine attraction (t = 0). First, the focal agent i determines from its Delaunay neighbors. The next step involves calculating the point at which the extended velocity vectors v_j intersect with ; i.e., t_j (∀j ∈ : Fig 1B: left). Here, we define the minimal spherical cap , which covers all these intersection points (). This cap can be computed using the cover function (Fig 1B); that is, . We also note that the cap can be defined as having two factors, the solid angle Θ and the cap center c(Θ), as shown in Fig 1B.

This cap is the interaction domain induced by the quasi-attraction (Fig 1C: left). We use the prefix “quasi-” because these intersections indicate longer time scales than that obtained for pure attraction. Here, we consider the time scales in the context of the prediction from the inside sphere. Although the travel time is relatively short, it is not necessarily t = 0. The travel time depends on the neighbor relative positions and the current direction of the focal agent only. For instance, in Fig 1C: left, agent 3 takes shorter time to achieve than agent 1. If the agent is on the , the reaching time becomes minimum, that is, t = 0.

In contrast, quasi-alignment pertains to long time-scale predictions. Recently, some studies have suggested the importance of future predictions of flocking movement [43, 46, 52–55]. In our model, each agent predicts the behavior of its neighbors based on their past movements; that is, their turning rates [33].

The turning rate can be represented using the rotation matrix R, which is defined as follows. According to the Rodrigues’ rotation formula [56], two factors determine the velocity rotation in three dimensions: the axis and the angle. The axis is in the direction that is unchanged by the vector rotation and the angle is the degree of rotation around that axis. If we have two vectors, we can construct R (Section 3 in S1 Appendix), and using R, we can then uniquely predict the future direction from the current direction. For example, if the current unit velocity vector is (Fig A in S1 Appendix), the future direction is expected to be .

For the quasi-alignment, we consider several prediction time scales Δt. The current direction can be described in several ways depending on Δt, with . The predicted future position , where is the current position. From this expression, if the agent behavior is predicted for long time scales (i.e., large Δt), is a great distance from . As the quasi-alignment yields predictions for long Δt, we compute all extending beyond ; that is, (, including self-prediction).

Let Δ_min represent the minimum time step that satisfies the above-mentioned condition. Then, we have a set (Fig 1C: right). By referring to , each prediction point on can be determined (see Section 3 in S1 Appendix). Finally, we also define the interaction domain () from its distribution on (Fig 1C: right).

The maximal sphere cap on , , is included in and gives the next direction of the focal agent (Fig 1C: lower): (1)

Let c(Θ_i) be a cap center and Θ_i be a solid angle of along c(Θ_i). We set the condition that a random direction directs the point, obeying a Gaussian distribution with variance Θ_i along the mean axis of c(Θ_i). On the sphere, the Gaussian randomness is given by the von Mises–Fisher distribution [57–59]. This function determines a random output based on a given mean c(Θ_i) and variance Θ_i (note that random values can also be outside the cap ). We define the random function Random_VM(Θ_i, c(Θ_i)), which obeys the von Mises–Fisher distribution (see Section 4 in S1 Appendix for the precise definition). In other words, the randomness intensity depends completely on the neighbor behaviors (the degree of Θ_i): if the neighbors have high alignment (i.e., small Θ_i), the noise naturally degrades, but if they have low alignment (large Θ_i), the noise increases. The noisy behavior of the focal agent is self-tuned according to its environment. Following this computation, each velocity is determined as follows: (2)

From this equation, it is clear that the deceleration depends on the degree of each turning direction (dθ). This condition is based on experimental data [24]. We only use the fact that the individual turn decelerates the agent speed from this study.

Finally, we define avoidance. In this study, avoidance rarely occurs because the agent speed exceeds that of the avoidance domains occurring in real-world situations. The definition of avoidance is opposite to that of quasi-attraction. In other words, the avoidance determines the direction in which the agent must travel to avoid the sphere cap region created by quasi-attraction, i.e., (, where n is the number of metric neighbors; Fig 1D: upper). The next direction given by the avoidance rule is the point of shortest distance to the edge of this sphere cap. Mathematically, where is the edge of and is the intersection of the extension of the vector v_i (i.e., focal agent i) with the repulsion area . The agent direction targets a point on the edge of this sphere cap (Fig 1D: lower).

Boundary condition

We set no boundary condition, wall, or periodic condition since the periodic (or wall) boundaries do not accurately capture the interaction factors inside the flock due to periodic effects. The agents can move without constraint.

Summary of algorithms

Fig 2 summarizes our algorithm. First, each agent checks if it has neighbors that must be repulsed. If so, an avoidance algorithm is applied. Following updating of all relevant agent positions according to the applied avoidance algorithm, a Delaunay triangle is constructed based on the current agent positions. Each agent computes the appropriate quasi-attraction and quasi-alignment. These two interactions determine the next direction of each agent. After all agent directions and positions are updated, one iteration is complete.

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Fig 2. Overview of algorithm.

One parameter (the maximum velocity v_max) exists when the repulsion radius R is fixed. Each agent checks if there are any other individuals in the repulsion zone, and if not, it executes the quasi-attraction/alignment rule. The update timings are synchronous. The algorithm framework is of the same type as those of previous Boids models and other methods.

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Group formations

Our model uses only three parameters: number of agents, maximum speed, and repulsion size. We reduced the number of parameters to two as the degrees of attraction and alignment were self-tuned. Because only Delaunay triangles determined the agent positional relations, the topological relation was more important than the metric relation. Therefore, adjustment of the velocity–repulsion ratio v_max/R was sufficient to alter the group behavior.

In the simulations reported in this section, v_max/R = 8 (this setting was based on the fish speed [24]) and the number of individuals was set to N = 1000. As already mentioned, the behavior of each agent varied dynamically according to the behaviors of its neighbors. This tendency indicates that the intrinsic noise of each agent could prevent perfect alignments and swarm-like noisy behavior.

Fig 3A confirms this characteristic. The group polarity P (blue) was unstable for a given condition; in other words, it increased to 0.9 and then decreased to 0.5. The group considered when using our model dynamically changed formation through ambiguous interactions. In addition to this dynamic group formation (red), the group size also varied (V_α is the volume of the group α-shape [60], the definition of which is given in Section 1 in S1 Appendix). The volume fluctuation did not yield a uniform expansion, but rather, a heterogeneous one. The flock distorted in various ways (e.g., twists denoted in Fig 3C). Fig 3B shows the negative correlation between group skewness and (correlation coefficient: −0.36). Group skewness can be defined as the ratio between the α-shape volume and its full convex volume. If the α-shape matches a full-convex formation, the skewness approaches zero. From the graph, the flock considered using our model tended to exhibit distorted shapes (such as twists or bursts, as shown in Fig 3C) when the value of increased.

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Fig 3. Group formation.

(A) Sample time series for group polarity and flock size . (B) The negative correlation between and its skewness 1 − V_α/V_conv, where V_conv is the full convex volume (Pearson’s correlation test: n = 8000, r = −0.36, p < 10⁻³⁰). Note that the inequality relation V_α < V_conv holds. (C) Examples of three types of group formation: alignment, twist, and burst.

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We identified three group formations often observed when using our model: (1) alignment: this formation is well-observed in all flocking models [37, 38]; (2) burst: this formation can be characterized as the one that occurs for low P but large and is often reported for fish schools [33, 61]; and (3) twist: this formation can be characterized as occurring for low P but high local P for bird flocks. From the outside, two groups in one collective interacted with each other. Sometimes, their directions were skewed; whereas the other times, they were directed opposite to one another, as if merging. We did not analytically examine these phenomena as it was sufficient to confirm that our model could successfully output various group formations from the given parameters.

Lévy-walk and 1/f fluctuation

First, we investigated the micro-level criticality, which does not pertain to the group, but rather, to the behavior of each individual.

We examined the internal trajectory of each agent for the center-of-mass reference frame. In our previous studies [24], we found that this type of a trajectory in a fish school indicates Lévy walks. We successfully replicated this result in the two-dimensional model [33]; however, it was unclear whether this finding extended to a higher dimension. Notably, a three-dimensional system has an additional degree of freedom than the two-dimensional case.

First, we briefly discuss Lévy walks observed in animal behavior. A Lévy walk is an optimal search strategy that balances exploration and exploitation [21–23]. It is a special type of a random walk and is in contrast to a Brownian walk, which exploits the surroundings of a specific location, as well as to a ballistic movement, which involves no exploitation and only exploration. The properties of the randomness of a Lévy walk lie between these two extremes. A Lévy walk is a characteristic property of animal behavior widely observed in nature [16–22]. Our previous study highlighted the possibility that a Lévy walk in a fish school facilitates efficient communication via super diffusion in the group [24].

Fig 4A shows an example of an agent trajectory for the center-of-mass reference frame. In this simulation, the trajectory spanned the entire group. In other words, each agent could have contact with neighbors in the flock. Fig 4B is an enlarged version of Fig 4A, where each dot on the line indicates the travel distance for one step. Here, the dot intervals were significant for the straight lines compared with the curved lines. This simple observation suggests that a Lévy-walk-like stop-and-go strategy may have been adopted.

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Fig 4. Lévy walk on the center-of-mass reference frame.

(A) Sample center-of-mass trajectory for 10000 steps. (B) Enlarged version of the blue-circle area (200 steps). The dots indicate individual steps. (C) Sample step-length l distribution for the mass-centered trajectories of 1000 agents. All l distributions obey the truncated power law distribution P(l) ∼ l^−μ. The average Lévy slope μ is 2.15. (D) The average μ with respect to group size for various velocity v_max parameters.

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To analyze this in greater detail, we applied the following method. We defined a “pause” (i.e., the zero-movement state) following the study conducted by Murakami and other researchers [24]. Thus, when dr > ||r_i(t) − r_i(t − 1)||, or when the travel distance in the center-of-mass reference frame for one step was smaller than a given threshold, dr, agent i was regarded to be pausing.

The rank distribution of the step length l was found to obey a power-law distribution. More precisely, because the group size confined the individual agent trajectories, the graph was expected to follow a truncated power law, satisfying the following relation: (4) where l is a step length, where μ is the power law exponent, l_min is the minimum step length of a series of the trial, and l_mix is the maximum step length of a series of the trial.

If the l distribution conformed to a Lévy walk, the exponential μ was approximately 2. Fig 4C shows a sample l distribution for the mass-centered trajectories of 1000 agents. We observe that all the 1000 distributions were well-fitted with the truncated power law (p > 0.10 for the Kolmogorov–Smirnov test and the Akaike information criterion (p) = 1 compared with the fitted exponential [62]). Interestingly, the average slope μ was 2.15 (see Fig 4D). Our result suggests that all the agents in the same group performed Lévy walks, even in the three-dimensional space.

Additional results support the aforementioned finding. The jittering behavior for the center-of-mass reference frame also has critical statistical properties. Fig 5A shows the time series of velocity variation, ||x_i(t) − x_i(t − 1)||, of agent i among 1000 agents, where x_i(t) = r_i(t) − r_CM(t) is the position of agent i in the center of the reference frame, and r_i(t) is the position of agent i at time t, r_CM(t) is the position of the flock center of mass at time t. Fig 5B shows the power spectrum of Fig 5A. The slope of this graph is approximately 1 for all the agents, which indicates 1/f fluctuation; this is also known as pink noise. When the slope is approximately 1, the time series fluctuation is said to be scale-free. In other words, a time series with strong self-similarity was obtained.

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Fig 5. Jittering behavior on center-of-mass reference frame.

(A) Sample velocity variation time series (i.e., ||x_i(t) − x_i(t − 1)||). (B) Power spectrum of (A): the slope of f^−γ is 0.98 (solid line). (C) Average slope 〈γ〉 according to group size for different velocity V parameters, with 〈γ〉 converging to 1.0. The jittering behavior of each agent is highly self-organized in time.

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Supper diffusion and scale free correlation

This subsection reports our examination of the macro-level criticality (i.e., the scale-free correlation). First, we examined the diffusive behavior of the flock. As some researchers have indicated, animal groups in nature exhibit high fluctuations (e.g., bird flocks [63], fish schools [12], and human crowds [25]). This instability in the intrinsic group behavior allows flocks to flexibly and dynamically respond to their dynamic environments [26, 27].

Super diffusion is an anomalous diffusion, which is high-speed compared with Brownian diffusion. To define this diffusion, the mean squared displacement must first be defined. Here, (4) where x_i(t) = r_i(t) − r_CM(t) is the position of agent i in the center of the reference frame, and r_i(t) is the position of agent i at time t, r_CM(t) is the position of the flock center of mass at time t,. We averaged over all N and overall time lags of duration t ∈ [0, T]. Here, (5) and the behavior of the mean square displacement δr²(t) fit the power law, where α is the diffusion exponent from 0 to 2 and D is the diffusion coefficient. If α was between 1 and 2, the system exhibited super diffusion [63].

Fig 6A shows that our model replicated super diffusion inside the flock (for mutual diffusion, see S1 Fig). In contrast with the empirical model, α was relatively high value around 1.78 for all parameter settings (Fig 6B). This high inner diffusion arose because no physical constraint was set on the model; that is, the condition for the xyz axis was homogeneous. However, actual flocks of starlings, for instance, have a high constraint on the z-axis because of gravity. In other words, movement in real flocks occurs in the medium region somewhere between two and three dimensions. The turning rate may have also impacted our result. In our model, the agents could change direction by up to π/2 radians (in the supplied Python code, this maximum turning rate can be tuned as a parameter S2 Appendix). Numerous flocking models adopt a maximal turning rate to replicate more realistic group behaviors. However, the aim of this study was not to mimic actual behaviors, but rather, to identify the origin of the criticality observed in nature while using minimal parameter settings. (Our model also behaved similarly to real flocks by exhibiting behaviors such as neighbor shuffling and boundary diffusion; see Section 5, 6, Fig Ca and Fig Cb in S1 Appendix).)

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Fig 6. Super diffusion.

(A) Mean square displacement in the center-of-mass reference frame. The diffusion coefficient D = 24.0, the diffusion exponent α = 1.79, the group size N = 1000, and the velocity parameter v_max = 8 (m/s). (B) The α results obtained for different v_max and N; hence, α depends on N but not on v_max.

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Next, we examined the scale-free correlation of the flock, which means that, even if the flock appears neatly aligned from the outside, the internal fluctuations (both orientation and speed) are not random but have a specific structure. Highly correlated fluctuation domains exist inside flocks and, in 2010, Cavagna et al. discovered that their domain sizes are scale-free [9]. Since Cavagna et al.’s discovery, numerous researchers have attempted to replicate their scale-free correlation results [14, 15, 40, 42]. Our interest is in confirming whether the high-fluctuation groups mentioned previously retain their scale-free properties.

To estimate the correlated domains inside a flock, we computed the correlation length ξ at which the correlation function of the fluctuation became zero, i.e., C(r = ξ) = 0, using the relation (6) Here, u_i is the fluctuation vector defined as u_i = v_i − (∑v_i)/N, δ(r − r_ij) is the delta function, and r_ij = |r_j − r_i| and c₀ are normalization parameters such that C(r = 0) = 1 holds. Further, C(r = 0) is 1 and gradually decays to negative values along with r. We computed the correlation function for the speed fluctuation in a similar manner (see Section 7 in S1 Appendix).

Fig 7A and 7B show that high scale-free properties were also observed for our highly fluctuated model, especially as regards their orientation. The orientation slope exceeded 0.40. In contrast, the speed slope was low (about 0.31) compared with available data for starlings [9]. This result may have been caused by our velocity algorithm. (A recent study has suggested another approach to coupling the velocity function with the restoring forces [64].) Nevertheless, our result indicates that both cases exhibited scale-free correlation.

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Fig 7. Relation between correlation length and flock size.

The flock size is defined as the maximum distance between two agents. Instead of trajectory smoothing to cancel the agent noise, we examined the velocity vector v = x^t+δt − x^t when δt was 2, 5, and 10 steps. (A) The correlation between the correlation length ξ and flock size L, where ξ is proportional to L: ξ = aL with a = 0.40 and δt = 2 (Pearson’s correlation test: n = 100, r = 0.93, p < 10⁻³¹), a = 0.40 and δt = 5 (Pearson’s correlation test: n = 100, r = 0.93, p < 10⁻³¹), and a = 0.40 and δt = 10 (Pearson’s correlation test: n = 100, r = 0.93, p < 10⁻³¹). (B) The correlation between the correlation length ξ_sp and L, where ξ_sp is proportional to L: ξ_sp = aL with a = 0.26 and δt = 2 (Pearson’s correlation test: n = 100, r = 0.92, p < 10⁻³¹), a = 0.30 and δt = 5 (Pearson’s correlation test: n = 100, r = 0.92, p < 10⁻³¹), and a = 0.31 and δt = 10 (Pearson’s correlation test: n = 100, r = 0.92, p < 10⁻³¹). For the other parameter settings, see S1 Table.

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The correlated domain in the group could be observed graphically. Fluctuations within the group could be divided into two groups (red and blue in S4 Fig). The next subsection reports our use of this property for analyzing information transfer between the two regions.

Information transfer between correlated subgroups

Constructing scale-free induced subgroups.

Thus far, we have investigated nested criticality in a group based on ambiguous interactions. This subsection reports our investigation of information flow in terms of critical properties.

To obtain a clearer information flow inside the flock, we divided the group into two subgroups using the scale-free correlation. Although we confirmed the existence of two highly correlated groups by their criticality, construction of a clear-cut criticality for the group using fluctuations alone was challenging. If scale-free correlations were used alone, some agents may have been mixed with opposite fluctuation groups. Without clear group divisions, we could not define the information flow among the subgroups. Therefore, additional information was required to construct distinct subgroups.

We applied the k-means method, which is a clustering method for given vector information, to construct two subgroups. We used the k-means function in Matlab (MathWorks Inc., Natick, USA), taking the position and unit fluctuation vectors as inputs and setting the division number (parameter k) to 2. As k-means clustering does not provide a unique division, we performed the same computation 20 times and selected the largest belonging member number in each case (see count(i) in Fig 8). Finally, we reindexed each group according to the alignment order in the mean direction: the top group was indexed as the “leader” and those behind were indexed as “followers.” S4 Fig shows that grouping based on the scale-free correlation yielded distinct divisions. We note that the number of subgroups was stable (the mean subgroup size was approximately 500 and the survival probability of the same subgroup decayed at a considerably slower degree than an exponential decay). More than 70% of the subgroup remained unchanged after 50 steps (S5 Fig). This stability confirms that our method successfully reflected the fluctuation vector property.

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Fig 8. Algorithm of the scale-free induced subgroup.

The input information is and (i.e., the fluctuation positions and directional information). The output index is the assignment group number.

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Mutual information for two input types.

We measured the information flow between two subgroups induced by scale-free correlation, considering the two types of average vectors. In other words, we prepared two patterns of interaction between a pair of average vectors for each reindexed group (i.e., two average velocity vectors {〈V〉_leader, 〈V〉_follower} and two average fluctuation vectors {〈F〉_leader, 〈F〉_follower}). This coarse-graining view elucidated the interaction between the two groups as an interaction between two virtual representative agents. The effect of these vectors could be evaluated as the future behavior of the group. We measured the behavior of this group as the degree of curvature K of the entire trajectory. We could compute the curvature vectors k at each point from the three position vectors of the center-of-mass trajectory of the flock (Fig 9A). We obtained a certain interval for the average to cancel its local zig-zag trajectories; in other words, . Therefore, the effects of the two vectors could be computed as mutual information: I({V_leader(t), V_follower(t)}; K(t)) or I({F_leader(t), F_follower(t)}; K(t)).

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Fig 9. Curvature vectors and mutual information obtained for different inputs.

(A) The flock center-of-mass trajectory and its curvature vectors at each point. (B) The probability density P(K) for all series of data where δt = 5 steps. We defined the region with curvature K < 0.05 as the ballistic region, and the region for which K > 0.10 as the group turning region. (C) Mutual information I_V ≡ I({V_leader(t), V_follower(t)}; K(t)) and I_F ≡ I({F_leader(t), F_follower(t)}; K(t)) for each region. The mutual information (i.e., I_V, I_F) in the group turning region is significantly larger than in the ballistic region. The detailed statistical test results are listed in S1 Table.

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Fig 9B shows the frequency distribution of K(t) extracted from a series of flocks (1000 agents). In the low-K(t) region, the flock exhibited ballistic movement. In contrast, in the high-K(t) region of Fig 9A, significant directional changes occurred. To understand the effect of information flow on flock turning, we identified two regions for the analysis: ballistic regions and turning regions, i.e., those with K(t) < 0.05 and K(t) > 0.10, respectively.

Fig 9C shows the mutual information from 100 trials of the velocity (blue) and fluctuation (red) vectors. Three patterns were identified for the analysis: whole, ballistic, and turning. The turning condition featured more mutual information than the other two conditions. In particular, relatively little mutual information was required for ballistic movement. This observation indicates that the velocity and fluctuation information transfer between the subgroups is significantly related to changes in the whole-flock turning behavior. Next, we focused on high-K(t) (> 0.1) regions only, as their mutual information was sufficiently large for the application of PID.

Application of PID to rapid group turning.

To understand the turning condition in more detail, we applied PID to the given mutual information. PID ensures that any mutual information can be decomposed into four elements: redundancy, two unidirectional information flows, and synergy [34, 35, 65, 66]. The use of transfer entropy to measure the information flow between two systems has been criticized [67–69] as over- or under-estimation may occur because the redundancy and synergy are omitted [67, 70]. To elucidate the information flow between two groups, the transfer entropy may be insufficient as the pure information transfer from one side to another must be estimated. Thus, the redundancy and synergy effects in the system must be considered. In our analysis, we used the PID Matlab code given in [66] (https://github.com/robince/partial-info-decomp).

The PID equation for the two inputs is given as follows: (7) where R(t) is the redundancy, U₁(t) is the unique information flow from X₁(t) to K(t), U₂(t) is the unique information flow from X₂(t) to K(t), and S(t) is the synergy.

For an easy understanding of the PID, a simple binary logic circuit is suitable (we present a figurative interpretation in S6 Fig; alternatively, see [69]). However, this interpretation is valid for discrete inputs only. Thus, we briefly describe PID here for readers unfamiliar with this approach.

Note that redundancy increases when the two given inputs (i.e., X, Y) are correlated. This explains the use of the term “redundancy” as when two inputs are highly correlated, the output (i.e., Z) contains unnecessary information and the system is highly redundant. The lack of one input does not affect the system output in terms of redundancy. If we subtract the redundancy R from I(X; Z), we obtain a unique information flow U₁. Compared to transfer entropy, this quantity can more accurately measure the flow of information. The synergy is defined as the remainder obtained when subtracting R, U₁, and U₂ from I(X, Y; Z). In contrast with redundancy, this quantity measures the non-linear effect generated by the combination of two inputs. One input cannot generate the same output as the other; i.e., redundancy and synergy play different input roles. The input relation of redundancy is symmetric as each input has the same function as regards the output, whereas the input relation of the synergy is asymmetric as two inputs act as a set for the output.

We applied the PID to the rapid group turning region (i.e., K(t) > 0.10). According to our settings, if the information flow was unidirectional from the leader to the follower, U₁({V_lead(t)}; K(t)) was the largest and the other three were smaller; however, Fig 10A shows a different result. For the velocity inputs, the largest was the synergy information S({V_lead(t), V_follow(t)}; K(t)) and the second-largest was the redundancy R({V_lead(t), V_follow(t)}; K(t)) (the same degree for the fluctuation inputs). The two unidirectional flows for both cases were relatively low. For the velocity inputs, U₁({V_lead(t)}; K(t)) was larger than U₂({V_follow(t)}; K(t)); however, both the values were small. Our result suggests that the group information transfer did not reflect the leader–follower relation. In other words, it was confirmed that positional information is not a crucial factor in group turning.

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Fig 10. PID for group turning phase.

(A) The PID decomposition for I_V (blue) and I_F (red) of the turning result in Fig 9C. The velocity synergy is higher than that of the redundancy (paired t-test: t(99) = 2.75, p < 0.01); however, the fluctuation synergy does not exceed the fluctuation redundancy (paired t-test: t(99) = −0.934, p > 0.1). The other statistical test results are listed in S1 Table. (B) The relation between the velocity synergy S({V_lead(t), V_follow(t)}; K(t)) and fluctuation redundancy R({F_lead(t), F_follow(t)}; K(t)). The correlation efficient is 0.93 (Pearson’s correlation test: n = 100, p < 10⁻³¹). The other parameter settings are reported in S1 Table.

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By now, it is clear how the velocity and fluctuation inputs affected group turning, and we also examined the relationship between the two inputs. Fig 10B indicates a high correlation between the synergy of the velocity input S({V_lead(t), V_follow(t)}; K(t)) and the redundancy of the fluctuation input R({F_lead(t), F_follow(t)}; K(t)). We discuss the significance of this relationship in more detail in the Discussion section. Here, however, we note only the correlation. That is, the group turning couples the asymmetry of the velocity inputs to the symmetry of the fluctuation inputs. This dual aspect effectively reflects the critical nature of the system, which we identified throughout this study. The system maintains its high susceptibility to external stimulus from the fluctuation perspective; from the velocity perspective, our findings suggest that the system responds to some non-linear effect. Interestingly, the inverse relationship does not hold (S1 Table). The different roles of the velocity and fluctuation inputs may be an essential property of the system criticality.