Origin of alignment and attraction

First, we provide a brief overview of alignment and attraction in flocking models. Although alignment and attraction have been widely used in many flocking models, there have been few discussions of what alignment and attraction mean. Alignment is the coordinating of the direction of motion with an agent’s neighbors and attraction is the tendency of the agent to move toward a neighbor’s current position. Some studies suggest that alignment has a weak effect in fish schools [31–33]. This may be because, in the captive environments in which fish schools are usually observed, the concept of infinite-future positions may not be appropriate. However, for starling flocks, alignment fits well with the empirical results because birds have effectively infinite space in the sky [15–17].

Although attraction and alignment seem quite different because of the difference between the properties on which they act, we can describe alignment in a way that is similar to the description of attraction. If an agent adjusts its direction of motion to that of its neighbor, the agent points toward the infinite-future position of its neighbor (see Fig 1A; note that the dashed line connecting the agent of interest at O to the infinite-future position of its neighbor is only curved because of the constraint of representing a point at infinity on a finite page). Alignment is, therefore, the result of a kind of attraction: “attraction” is an attraction to the neighbor’s current position, Q₀, while “alignment” is an attraction to the infinite-future position of the neighbor, Q_∞. Thus, the difference between alignment and attraction is not in the interaction properties but in the time scales.

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Fig 1. Schematics of attraction, alignment, and their generalization.

(A) Schematic of attraction (left) and alignment (right). The agent (red dot) is attracted toward its neighbor’s (black dot) current position (Q₀: left) and infinite future position (Q_∞: right) (B) Schematic of attraction and alignment with a neighborhood. The circle indicates the boundary of the agent’s neighborhood. In the attraction case (left), the point P₀ is precisely the same as Q₀. In the alignment case (right), the point P_∞ reflects the information of Q_∞ as the point on the boundary the agent would reach if it moved in a direction parallel with its neighbor’s motion. (C) Schematic of the generalization from P₀ and P_∞ to P_τ and P_T. P_τ is the crossing point of the agent’s extended direction vector at the boundary of the neighborhood. P_T is the point where the vector from the agent to a certain prediction point Q_T crosses the boundary of the neighborhood.

https://doi.org/10.1371/journal.pone.0195988.g001

Short-term and long-term prediction on the edge of agent’s neighborhood

These two time scales hint at other possible flocking models. Taking this view of attraction, allows us to consider attraction at arbitrary time scales t between the two extremes Q₀ and Q_∞, namely, attraction to Q_t. To implement these intermediate points of attraction in our model, we convert each point Q_t to a point P_t on the edge of an agent’s neighborhood. In Fig 1B a circular neighborhood is superimposed upon the diagrams in Fig 1A. Attraction point P₀ is the position of a neighbor on the edge of an agent’s neighborhood, and P_∞ is the point that the agent’s neighborhood intersects with the dashed line from O to Q_∞ (O is the position of the agent of interest). The two extremes Q₀ and Q_∞ are thus converted to points on the edge of the neighborhood as P₀ and P_∞, respectively. Note that attraction point P₀ is derived from information Q₀ from inside the neighborhood and alignment point P_∞ is derived from information Q_∞ from outside the neighborhood.

Fig 1C shows a generalization of our discussion. There is one neighbor within the agent’s neighborhood. We define the quasi-attraction point P_τ as the point where the extended neighbor’s direction vector crosses the agent’s neighborhood. When the neighbor is near the edge of the neighborhood, the quasi-attraction point is almost the same as the attraction point. Next, the quasi-alignment point P_Τ is defined as the intersection point where the vector to a prediction point (white circle in Fig 1C) from the center crosses the boundary of the agent’s neighborhood. The type of prediction determines the flocking behavior. In this paper, we use the alignment prediction and anticipation methods. Note that τ and T represent the two most extreme prediction times. T is always larger than τ because τ is related to a short-term prediction while T is related to a long-term prediction: a neighbor would only take a short time (minimum time of zero) to reach the edge of the agent’s neighborhood from the inside while it would take longer to reach a point outside of the neighborhood (e.g., the white circle in Fig 1C). Not also that, since Q_T and P_τ are only points on a neighbor’s predicted trajectory, the neighbor need not actually travel to either of these points. However, one implication of these predictions should be noted. Through the generalization process shown in Fig 1C, the relation of the two extremes P₀ and P_∞, which we have observed, is weakened while preserving its original properties.

Prediction methods

The time scales τ and Τ can take many values according to the neighbor’s position relative to the boundary of the agent’s neighborhood and the prediction method. In this section, we introduce the alignment prediction and anticipation methods. There may be other prediction methods; however, these are the only examples that we have found. In this paper, we only use the anticipation method for our analysis. The alignment method is only used to show that this method can create self-propelled particle-like behavior.

1. Alignment prediction: This prediction method is similar to the alignment interaction that is widely used. This method produces group behaviors like the self-propelled particle model (S1 Movie). Fig 2A shows a schematic of the alignment prediction method. The end point of an extremely elongated neighbor’s direction vector v_i is used as the prediction point Q_T (white circle in Fig 2A). Thus, the prediction point is Q_T = x_i(t) + Tv_i(t), where x_i(t) is neighbor i’s position and v_i(t) is its velocity vector at time t. In this paper, we fixed T = 300 because this method is used only to provide an example of the generation of self-propelled-particle-model-like behavior.
2. Anticipation: This prediction method is similar to that of Morin et al. [35). Past direction turning rate dφ_i(t) affects the future direction turning rate dφ_i(t + 1), assuming dφ_i(t+1) = dφ_i (t) for the prediction (Fig 2B). For precision we write dφ_i(t) as dφ_i^s(t) because the degree of dφ_i depends on how long the agent of interest refers to its neighbor’s past movements (s steps), that is, dφ_i^s(t) = arg(v_i^s(t)) − arg(v_i^s-1(t)) where v_i^s(t) = x_i(t) − x_i(t−s) and v_i^s-1(t) = x_i(t−s) − x_i(t−2s). The prediction point for anticipation is Q_T = x_i(t) + T || v_i(t) || u_i(t), where u_i(t) is a unit vector with argument arg(v_i¹(t)) + dφ_i(t). The symbol ||-|| indicates the vector norm. The value of T is given by s*r₁(t)/ || v_i(t) || where r₁(t) is the radius of the neighborhood at time t. This definition means that enlarging s, that is how long the agent refers to its neighbor’s past, enlarges T. The value of s is uniquely determined as the minimum value such that all neighbors’ Q_T’s lie outside the neighborhood. In S1 Fig, we give the graph of the distribution of s, which shows the frequency of s values for our simulations. We use this method for all the results (S2 Movie).

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Fig 2. Schematics of the two prediction methods.

(A) Alignment prediction: the prediction point Q_T is the end point of the elongated neighbor’s direction vector v_i(t). (B) Anticipation: the prediction point Q_T is the predicted landing point if the turning rate of the agent’s neighbor is the same as its previous rate, that is, dφ_i^s(t) = dφ_i^s(t + 1). The value of s indicates how long the agent refers to its neighbor’s past movement in terms of a number of time steps. The quasi-alignment point P_T is the point where the segment from the agent to the prediction point Q_T crosses the boundary of the neighborhood.

https://doi.org/10.1371/journal.pone.0195988.g002

Method descriptions

In this section, we describe our flocking model algorithm. The parameter settings and other more complicated mathematical expressions are provided in the Method section and S1 Text.

First, we define the functions and symbols that we use in our model. We define four functions: (i) The distance function d(x, y); its output is the Euclidean distance between position vectors x and y in Rⁿ (in this paper, n = 2). (ii) The mean angle function mean(X); its output is the mean angular direction in set X. (iii) The covering function Cov(X); its output is the set {θ| min({θ_i − mean(X)| θ_i ∈ X }) ≤ θ − mean(X) ≤ max({θ_i − mean(X)| θ_i ∈ X })}. The relation X ⊆ Cov(X) always holds. (vi) The random function Ran(S): its output is an element x randomly selected from set S.

Next, we define four sets for our model. N = {1, 2, …, N} are the tags for the agents and N = |N|, where |-| indicates the cardinality of a set. E = {(i, j)| i, j ∈ N, i and j are Voronoi neighbors} is a network constructed from all directly connected Delaunay triangles. The set n_i = {j| j ∈ N, (i, j) ∈ E or i = j} represents all of agent i’s neighbors, including itself. The set o_i = {j| j ∈ N, d(x_i, x_j) < R or i = j} represents all of agent i’s neighbors inside its repulsion zone, including itself. R is a parameter giving the size of the repulsion zone, which is fixed in our simulations. Our model uses the Voronoi neighborhood to determine neighbors because recent studies, especially in fish schools, suggest that the Voronoi neighborhood provides a more valid description of collective behavior than other neighborhoods [36, 37].

We need to define a circular neighborhood including all Voronoi-connected-neighbors n_i(t) for agent i at time t to obtain quasi-attraction points (i.e., P_τ’s) and quasi-alignment points (i.e., P_T’s) on the boundary of this neighborhood. This circle is C_i(t) = x_i(t) + {(r_i(t)cos(θ), r_i(t) sin(θ))| 0 ≦ θ<2π} where r_i(t) = max({d(x_i(t), x_j(t)) | j ∈ n_i(t)}) + c (c is a constant parameter). All short-term (i.e., P_τ) and long-term (i.e., P_T) predictions for an agent i lie on this circle C_i(t).

Here, we describe the two interaction methods (Fig 3A and 3B).

1. Direction through common information: A quasi-attraction point collection of P_τ’s for i is a collection of directions, θ_i(t) = {θ₁(t), θ₂(t), …, θ_|ni(t)| (t)}, because only directional information matters for C_i(t) in our model (Fig 3A). Similarly, a quasi-alignment point collection of P_Ts for i is also a collection of directions, Θ_i(t) = {Θ₁(t), Θ₂(t), …, Θ_|ni(t)| (t)} (Fig 3B). Each P_T is given from Q_T using the prediction method, that is, alignment or anticipation (see the section Prediction Methods). We take the intersection between Cov(θ_i(t)) and Cov(Θ_i(t)), I_i(t) = Cov(θ_i(t)) ∩ Cov(Θ_i(t)). These common short-term (i.e., P_τ) and long-term (i.e., P_T) predictions determine agent i’s next direction of motion, φ_i(t + 1) = mean(φ_i(t), Ran(I_i(t))). In particular, in the case I_i(t) = ∅, φ_i(t + 1) = mean(φ_i(t), Ran(C_u—J_i(t))), where J_i (t) = Cov(θ_i(t)) ∪ Cov(Θ_i(t)) and C_u is a unit circle.
2. Repulsion: Repulsion in our model is a deflection of an agent’s movement direction from the interval. Because the neighbors in its repulsion zone are o_i, a quasi-attraction point collection of P_τ on gives . Then, agent i’s next direction is , where C_u is a unit circle.

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Fig 3. Schematics of the showing short-term and long-term prediction on the circle.

(A) The circle shows quasi-attractions as internal information for agent 1. The numbers under the dots are the tags for the agents. The bold blue arc is a covering set for θ₁. (B) The circle shows quasi-alignments as external information for agent 1. Each prediction point determines Θ_i. The bold red arc is a covering set for Θ₁. (C) The complete chart of our algorithm. In the first step, find an agent which has its neighbors within its repulsion range () colored green) and apply the repulsion algorithm (the updated agent is shown in red). Repeat this until there is no agent which has its neighbors within its repulsion range. Then, construct a Delaunay triangulation from all agents’ positional information. Apply quasi-attraction (Cov(θ_i(t)) colored blue) and quasi-alignment (Cov(Θ_i(t)) colored red) and take their intersection (I_i(t) colored yellow). The agent of interest (colored blue) selects its direction from this yellow interval.

https://doi.org/10.1371/journal.pone.0195988.g003

The intersection I_i(t) in (I) is constructed from both short-term predictions (i.e., quasi-attractions) and long-term predictions (i.e., quasi-alignments) on the circle C_i. Each agent can choose any direction from the interval I_i(t). The interaction in our model only determines a bundle of directions in an obtained interval, but not a certain direction. This fact suggests that the direction that the agent chooses always has some uncertainty, even if it can precisely compute and predict all its neighbors’ behaviors, because the agent can only obtain interval information I_i(t) and select one element from this interval.

Noise definition.

In this paper, we regard noise as arising from an agent’s ability to resolve space divisions (i.e., a cognitive resolution of space). The bundle of directions in interval I gives the next direction of motion for an agent. Treating noise as cognitive resolution blurs this interval (see S2 Fig). This assumption comes from the finite nature of the accuracy of animal cognitive systems (e.g., animal visual perception) [38–41].

Consider the p-division of unit circle C_u for agent i, where p is a natural number. The p-division acts on circle C_u centered on the agent’s position at time t. Each fragment of the circle is , where the agent direction of motion is φ_i(t) and 0 ≦ a<p. Note that if a≠b and . The size of each fragment, , indicates the ability of the agent to divide space (i.e., the degree of directional resolution). Thus, intervals I can be partitioned into , where . Applying this method, we can rewrite Eqs. (I) and (II) as follows.

Large p preserves the original structure of I. Interval I and partitioned interval are generally equal. However, when p is small, the partitioned interval always includes the original interval I. This mismatch generates more choices from Cov(I) than from the original interval I because the selection area from which each agent chooses a direction expands (S2 Fig). In this paper, a noise parameterζdetermines the value of , where [–] is the floor function. Note that noise strength is not the same as used in traditional models. Noise in traditional models acts on an agent’s pre-determined direction of motion. On the other hand, the role of noise in our model is only to expand the interval Cov(I) and increase the set of directions which the agent can take.

Algorithm

1. Distribute all agents randomly in a two-dimensional space, which is 100(m)×100(m) in our model. Each agent also has a random velocity.
2. Each agent checks whether it has its neighbors in its repulsion zone or not. If it does, go to 2.1. Otherwise, go to 3.
   1. 2.1. Compute the agent’s quasi-attraction points on for each agent in o_i(t) = {j|d(x_i(t), x_j(t)) < R} to obtain . Make a covering set, , and determine the agent’s next direction, .
   2. 2.2. Determine the agent’s velocity: v_i(t + 1) = Vcos(φ_i(t + 1)–φ_i(t)) where V is a maximum velocity.
   3. 2.3. Update the agent’s position: x_i(t + 1) = x_i(t) + v_i(t + 1).
3. Draw Delaunay triangles (i.e., find the Voronoi neighbors) E(t) = {(i, j)| i, j ∈ N, i and j are Voronoi neighbors} from agent’s distribution to find its directly connected neighbors, n_i(t) = {j| j ∈ N, (i, j) ∈ E(t) or i = j}. Obtain the edge of neighborhood C_i(t) = x_i(t) + {(r_i(t)cos(θ), r_i(t) sin(θ))| 0 ≦ θ<2π} where r_i(t) = max({d(x_i(t), x_j(t)) | j ∈ n_i(t)}) + c (c is a constant parameter). Compute the agent’s quasi-attraction and quasi-alignment points for n_i(t) and get θ_i(t) and Θ_i(t) by using the method (I). Make two covering sets (Cov(θ_i(t)) and Cov(Θ_i(t))) and take the intersection, I_i(t) = Cov(θ_i(t)) ∩ Cov(Θ_i(t)).
   1. 3.1. If I_i(t) ≠ ∅, the agent’s next direction is .
   2. 3.2. If I_i(t) = ∅, the agent’s next direction is
4. Update the rest of the agent states v_i(t + 1) = Vcos(φ_i(t + 1)–φ_i(t)) and x_i(t + 1) = x_i(t) + v_i(t + 1) synchronously. Then update t → t + 1 and return to 2.

In Fig 3C, we describe the chart of our algorithm. The parameters are listed in the Method section. This model has no boundary conditions. Almost all agents are simultaneously updated by using the three steps; however, some agents apply step 2 before proceeding. This procedure is introduced to avoid the situation of agents’ positions coinciding. Thus, our model uses a semi-synchronous update rule.

Schooling, milling, and swarming

First, we examine the effects of noise as cognitive resolution (the ability to divide space; in this case, circle division) and the maximum velocity V on the collective behavior of our model. To summarize the agents’ behavior, we use two measures, the degree of polarity O_P and torus O_T, which have been widely used for collective behavior. High polarity O_P means that the groups are in high alignment (ordered state) and high O_T means that groups are in a milling state, which is sometimes observed in fish schools. In contrast, low O_P and O_T values indicate swarming (i.e., the flock is in a completely disordered state). Note that 0 ≦ O_P, O_T ≦1. O_P and O_T are expressed mathematically as (1) (2) where N is the number of agents (|N| = N), u_i is the unit velocity vector of agent i, and q_i is a unit vector pointing from the group’s center of mass toward agent i. Most flocking models have suggested that these qualitatively different behaviors (schooling, milling, and swarming) are parameter-dependent, which means a certain set of parameters corresponds to each group’s behavior. The parametric region for producing multiple states at the same time is a restricted one [37].

However, Fig 4A suggests that agents in our model (N = 100) can switch between schooling (high alignment), milling, and swarming behaviors. Agents can have a high O_P, a high O_T, or a low O_P and O_T (S2 Movie). Almost all parameter settings (velocity and noise intensity) have the same tendency in terms of the coexistence of three different behaviors (Fig 4B and 4C). Even if the agents are in a highly ordered state on average (O_p>0.9), their minimum O_P is lower than 0.3 in most cases (S3 Fig). This fact suggests that high alignment flocks are not always stable. Although milling states are observed more often in low-velocity regions (S3 Fig), high O_T values above 0.5 remain in both the high-velocity and noise regions. To achieve various collective behaviors in flock models, many researchers have suggested that the balance between attraction and alignment is crucial. In contrast, the agents in our model can tune the balance between attraction and alignment as the time scales (τ and T) by themselves, as discussed in the previous section.

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Fig 4. Averaged polarity O_P and torus O_T values and their time series.

(A) Time series of O_P and O_T values (N = 100). (B) Polarity for the conditions V = 1.0, 2.0, 3.5, 5.0, 10.0, and 20.0 with respect to noise of 0.01, 0.05, 0.10, 0.50, and 1.00. V is the maximum velocity that each agent can have. The graph shows low polarity in the low-velocity regions because of repulsion effects as we discussed in the main text. (C) Torus structure for the conditions V = 1.0, 2.0, 3.5, 5.0, 10.0, and 20.0 with respect to noise of 0.01, 0.05, 0.10, 0.50, and 1.00 (628, 124, 62, 12, and 6 divisions of the unit circle). The 17,000 steps were run 30 times for all cases.

https://doi.org/10.1371/journal.pone.0195988.g004

Noise as cognitive resolution works differently compared with past flocking models. Usually, increasing the noise intensity results in disordered groups. Although this trend is also observed for high noise intensity (light green in Fig 4B), it is not always the case. In low-velocity regions, noise increases the O_P on average.

In the low-velocity region, the agents collide with each other because agents tend to stick together. Collisions happen less frequently in the high-velocity case. It is true that each agent tends to move closer in this model, but high velocity decreases the possibility of becoming trapped in another agent’s repulsion region because high-velocity agents pass through other agents’ repulsion areas (in our simulation, the average collision rate in a group of 100 agents per step is 4.54, 1.49 and 0.77 for V/R = 4/4, 7/4 and 10/4, respectively). These collisions destabilize the flock alignment.

The effect of this misalignment by collision would be greater for relatively low noise intensity than for relatively high noise intensity. Indeed, low noise agents show high alignment, but the unexpected large direction changes (repulsion in this case) are easily spread all over the flock because of the high space division ability. This perturbation spreading tendency prevents low noise agents from creating high alignment in the low-velocity region.

Diffusion in flocks

Recent findings about real flocks and schools have shown that individual behavior inside the group is disordered rather than highly ordered. Seamless shuffling with neighbors and bustling movements have been little reported in flocking models [11]. In particular, the maximum entropy approach is known to be able to describe these collective properties well, but it does not provide individual perspectives to describe what is going on inside these collective behaviors. Murakami’s recent finding on the Lévy walks inside fish schools shows the statistical regularity in these movements [19]. Exploring and exploiting the behavior of each single fish in the school suggest individual intelligent behavior within the group. It is difficult to understand their finding without considering each individual’s decision under imperfect and fragmented information.

Our model explains these collective behaviors. Fig 5A shows super-diffusion in a flock, which is described as (3) where x_i(t) is the position of agent i at time t, x_CM(t) is the position of the center of the mass of the flock at time t, and r_i(t) = x_i(t) − x_CM(t) is the position of agent i in the center of the reference frame. We averaged over all N and over all time lags of duration t in the interval [0, T], where T is the total time interval (T = 2000 steps). The behavior of the mean square displacement δr²(t) is well-described by the following power law (4) Where α is the diffusion exponent (0–2) and D is the diffusion coefficient. The value of α is important. If α is 1, the process is normal diffusion, and if α is >1, the process is super-diffusion (α = 2 is purely ballistic diffusion). If the agents undergo super-diffusion collectively, information transfer is much faster than for normal diffusion.

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Fig 5. Shuffling behavior inside the flock.

(A) Mean square displacement in the center of mass reference frame up to 30 steps. D = 26.0 and α = 1.59. (B) Neighbors overlap. Q_M against time t for groups of 100 individuals. Full lines represent Eq (6) for 0.66 (fitted value) with α_m = 1.55 and . (C) Neighbors overlap: Q_M against number of neighbors M for 100 individuals. Values are not fitted for high M because M is too large for the group size (M = 80 and N = 100).

https://doi.org/10.1371/journal.pone.0195988.g005

Fig 5A shows that the diffusion exponent of our model is >1 (i.e., super-diffusion). The diffusion exponent, α, is 1.55 on average and its value matches the experimental results well (Plecoglossus altivelis: 1.34 < α <1.73 and starlings: α = 1.73; other parameter results for Eq (4) and mutual diffusion are listed in S4 Fig and S1 Table).

This super-diffusive behavior in flocks results in neighbor shuffling. Neighbor shuffling means that each agent’s neighbors change with time. Cavagna and other researchers [17] described this using the time-dependent overlapping neighbors rate Q_M (5) where M_i(t) is the number of neighbors of agent i at time t + t₀ that share the same neighborhood. members as agent i’s member M at time t₀. Q_M decreases over time. According to their calculation, the distribution of Q_M along time t is well fitted by (6) where is an effective dimension, α_m is a mutual diffusion exponent, and c is a constant parameter to fit Q_M (the method of obtaining and α_m is shown in S5 and S6 Figs). This function is also satisfied in our results. Agents in our model change position dynamically inside the flock.

Lévy walk in the flock

The Lévy walk is an optimal searching strategy found widely in nature, from the movement of bacteria to human movements [43–46], especially in foraging strategies. This behavior can be described by (7) where l is step length and μ is a power law exponent. Step length is the cumulative traveling distance between two pausing points. This procedure is based on the intermittent behavior of the Lévy walk. The definition of the pause (i.e., the state of not moving) is the same as the definition used by Murakami and other researchers [19, 43, 44]). Namely, when dr > ||r_i(t) − r_i(t—1)||, or when the traveling distance in the center of mass reference frame for one step is smaller than a given threshold, dr, agent i is regarded as pausing.

Fig 6A is an example of one agent’s trajectory in the center of the mass reference frame for a group size of 100. The graph shows that the trajectory has ballistic movements (non- entangled) and entangled lines. The trajectory in Fig 6B resembles the trajectories of Plecoglossus altivelis in Ref. [19]. This observation is also confirmed through our statistical analysis. Fig 6B shows the step length distributions for 100 agents (gray lines), which are similar. The red line in Fig 6B is the fitting curve with the truncated power law (slope: μ = 2.11), which is often used to estimate the Lévy walk power law distribution, of averaged data for 100 distributions. S3 Table lists an example of all the statistical tests for 100 agents and averaged results over 20 simulations for 100 agents for other threshold dr settings (see detailed discussion in the Method section or Refs. [47, 48]).

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Fig 6. Lévy walk behavior inside flocks.

(A) Example of an agent’s trajectory in the center of a mass reference frame for 4000 steps. The vertical bar is the mean flock size during the simulation. (B) Power law distribution of step lengths. The vertical line is the step length rank for each individual. The gray lines correspond to each individual’s step length for its trajectory. The bold red line is the fitting curve with averaged parameters for the truncated power law function. Details of the statistical tests for the power law are given in S2 and S3 Tables and the Method section.

https://doi.org/10.1371/journal.pone.0195988.g006

Our result of an average value of μ = 2.11 (see S2 Table for various parameter settings) matches Murakami’s experimental data well (1.86 < μ < 2.33). Furthermore, an exponent value of μ ≈ 2 is the value for an optimal foraging strategy in a certain environment [20, 21]. The whole trajectory covers the mean flock size in Fig 6A. As Murakami and other researchers suggested [19], the bustling behavior inside the flock is not restricted to a local region, but occurs throughout the group. Thus, the agents in our model can communicate with each other in the most efficient way.

Local equilibrium of the flock

We have shown that our model can produce the noisy behavior of individuals. However, the highly ordered macroscale behavior must be connected to the noisy microscale behavior. To bridge this gap, we use Mora and other researcher’s findings [42]. According to their study, parallel processing of different time scale events for micro- and macroscopic behavior explains the apparent paradox of an ordered state being based on a disordered state. They defined two time scales: the alignment time scale, which corresponds to an individual’s behavior over short time scales; and the neighbor exchange time scale, which corresponds to the network (connected topological neighborhood] rearrangement time as a whole flock. We use the name neighbor exchange time instead of using network rearrangement time, following [42].

The alignment time is the relaxation time that eliminates the effect of divergent modes because of the Goldstone theorem (see Ref. [42] for a detailed discussion). The alignment time is approximately τ_relax = (Jn_c)⁻¹, where J is the overall interaction strength and n_c is the number of nearest neighbors of each agent. This formula is based on an approximation assuming that the group is in a state of high polarity, O_p ≈ 1. Therefore, this method must be applied carefully to our model. We discuss the validity of using this method in the Method section.

The neighbor exchange time is defined as the characteristic decay time of the autocorrelation function C_network(t) = ∑_ijn_ij(t₀)n_ij(t + t₀), where . Here k_ij(t) is the topological distance at t, given as the time-dependent rank of agent j among agent i’s neighbors ranked by distance. This function decays exponentially, C_network(t) ≈ C₀ exp(−t/τ_network) (see S7 Fig).

The relationship between τ_relax and τ_network is crucial. When τ_relax ≈ τ_network, the flock shows only highly polarized behaviors without any rapid shuffling of neighbors in the flock. When τ_relax ≤ τ_network, the flock shows swarming because the network arrangement is faster than the local alignment speed. The most interesting case is when τ_relax ≪ τ_network, which is an adiabatic system or local equilibrium. The alignment speed is much faster than the network arrangement speed. This gap between the two time scales enables the flock to have consistent bridging between highly ordered macroscopic behavior and disordered microscopic behavior.

Fig 7A shows our results. About 200 ordered differences of the two time scales (average of τ_relax ≈ 180τ_network) are observed in our model. The size of this gap in our model is similar to Mora’s experimental results (τ_relax ≈ 100τ_network). In addition, we found that there is no correlation between τ_relax and τ_network, as Mora and other researchers found (Fig 7B; Pearson correlation coefficient: 0.02).

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Fig 7. Local equilibrium in flocks.

(A) Histogram of alignment time τ_relax (red bars) and neighbor exchange time τ_network (blue bars). The vertical line is the frequency for 50 simulations. Alignment time 1/Jn_c is almost equal to 1 step in our simulation where n_c = 5. (B) The scatter plot of τ_relax versus τ_network shows no correlation (Pearson correlation coefficient: 0.02).

https://doi.org/10.1371/journal.pone.0195988.g007

Unit length

We define the unit length as the norm of a unit velocity vector and its indication as m.

Parameter setting

The parameters are as in Table 1 unless indicated otherwise. The ratio between the repulsion radius and the maximum velocity is the most definitive factor for our model because, as we have suggested in the manuscript, collisions (or repulsion) in flock largely depend on an agent’s velocity. In our setting, the ratio between them is fixed at 7/4. Considering the repulsion radius as an agent’s body size, the speed would not be very high compared with its body size. In the Supporting Information, we discuss three ratio values do not affect our result and suggest the possibility that a low ratio would not be good for our analysis.

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Table 1. Parameters for Figs 4 to 7.

https://doi.org/10.1371/journal.pone.0195988.t001

Threshold value dr for Lévy walk

We set the threshold parameter dr for each trial to obtain enough step length data: dr = 0.8D, where D is and T = 9000. The parameter dr determines the flock’s average travelling distance during the whole process in the center of mass of the reference frame. We examined three cases (dr = 0.9D, 0.8D and 0.7D) to demonstrate the threshold parameter-independence of our results.

Validity of the approximation for τ_relax ≈ 1/Jn_c

Mora’s approximation for the relaxation time can apply at high polarities, that is, O_p ≈ 1. To apply this method to our flocking model, the method requires modifications because the noisy behaviors in our model prevent stable high-polarity behaviors from being obtained. To confirm the validity of the method for our results, we used two treatments: (i) creating a highly ordered group with O_p > 0.9, and (ii) omitting the out values that come from highly noisy behavior. These are discussed in more detail below. Compared with τ_relax, obtaining τ_network does not need such careful treatment. All the graph curves apparently satisfy exponential approximations (S7 Fig).

1. Setting high-velocity agents (V = 20 and R = 2): Under this condition, the average O_p is 0.93. The approximation of the overall interaction strength, J, is valid under this condition. This condition rarely happens in real flocks, especially in two dimensions, because the agent’s speed (V) is 10 times faster than its body length (R). This condition is only used to examine the validity of the approximation.
2. Out values: The out value is far from the average of a series of trials (2000 steps). In S8 Fig and S4 Table, the average J values with no out values are around 0.1. However, out values with very low (<<0.1) or even negative J values occur because the average O_p is far from 1.0 (e.g. O_p ≈ 0.5, when V = R = 4). Fortunately, these values are rare in our model, occurring at most about once in 2000 steps (V = 4 and R = 4). Furthermore, we have no out values at all when V = 20 and R = 2. This proves that Mora’s approximation works when V = 20 and R = 2. Therefore, we expect that the approximation is valid for most cases.