Modes of information flow in collective cohesion - Science

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With this background, we now turn to diagnose interactions between individuals in a collective system and between individuals and the collective.

Modified Vicsek model

To demonstrate the interpretability of informational modes I, σ, and S, we introduce a series of augmented Vicsek models. These extend the original (35) with asymmetric interactions and turn on-off the dependence on the present dynamics of interacting particles.

Consider N particles lying within a square box of length L with periodic boundary conditions. Particle i’s position ${\overrightarrow{\mathbf{r}}}_i^t$ at time t is updated over time increment Δt according to

$${\overrightarrow{\mathbf{r}}}_i^{t+1}={\overrightarrow{\mathbf{r}}}_i^t+{\overrightarrow{\mathbf{v}}}_i^t\mathrm{\Delta }t$$

(11)

where ${\overrightarrow{\mathbf{v}}}_i^t$ denotes particle i’s velocity at time t and i = 1,2, …, N. For simplicity, particles have uniform constant speed v₀, and only their orientations θ_i change.

Particle orientation is updated at each time by taking the weighted average of the velocity of neighboring particles within a given radius R

$${\mathrm{\theta }}_i(t+1)={\langle \mathbf{\theta }(t)\rangle }_{R,\mathbf{w},{\overrightarrow{{\mathbf{r}}_i}}^t}+\mathrm{\Delta }{\mathrm{\theta }}_i$$

(12)

where w is a nonnegative asymmetric matrix whose w_ij element determines the interaction strength that particle i exerts on particle j. w_ij > w_ji whenever particle i is a leader, and particle j is a follower in our setting. To model thermal noise, Δθ_i is a random number uniformly distributed in the range [−η₀/2, η₀/2] and is chosen uniquely for each particle i at each time step. In the original model (35), the right-hand side of Eq. 12 ensured that θ_i(t + 1) resulted from the configurations of all the particles j (including that of the same particle i) within the circle of radius R centered at ${\overrightarrow{\mathbf{r}}}_i^t$.

Now, consider modified dynamics that modulate the dependence on θ_j(t) associated with follower-leader interactions that determine θ_i(t + 1): The leader influences the follower, but the follower does not influence the leader; i.e., w_LF > 0, while w_FL = 0.

To graphically understand our models, we show Fig. 1 (Aa to Da), where each graph depicts one possible interaction protocol in a simple, two-particle system that determine θ_i(t + 1), where L and F denote leader and follower, respectively. There, if A or B are either L or F, then A → B signifies that A’s present state influences the B’s future state. We vary w_LF ∈ [1,10]. We set w_LL = 1 and w_FF = 1 for the models in which the present state of L (F) influences the future state of L (F). For models in which the present does not influence the future for the same particle (F or L)—see Fig. 1Aa (i.e., L’s and F’s dynamics), Ba (L’s), and Ca (F’s)—we replace θ_i(t), which appears in computing $<\mathbf{\theta }(t){>}_{R,\mathit{w},{\overrightarrow{{\mathbf{r}}_i}}^t}$ (see ∑′ term in Eq. 20) by a random number in the interval [0,2π] to erase any influence from θ_i(t)’s present. Note that the value of w_LL (w_FF) is inconsequential when the dynamics of leader (follower) do not depend on their present state, and θ_i(t) depends solely on a random number in the interval [0,2π]. In type A, neither the future dynamics of L nor F depend on their present (Fig. 1, Aa). In type B, only the future dynamics of F depends on its present (Fig. 1, Ba). In type C, only the future dynamics of L depends on its present (Fig. 1, Ca). In type D, both L’s and F’s future dynamics depend on their present (Fig. 1, Da). To further examine the effects of the history of L in types C and D, we also introduced interaction types C′ and D′, which are the same as interaction types C and D, respectively, except the future state of L depends on its present only when time step t is even, and L “forgets” its present in its future dynamics as in types A and B whenever t is odd. In types C′ and D′, the dependence of the future of F on its present are not changed, that is, the future of F does not depend on its present in type C′, and the future of F always depends on its present in type D′ regardless of the value of t.

Fig. 1. Graph representation of interaction types A, B, C, and D and the corresponding Venn diagrams representing information flow.

Areas of the circles are computed by integrating M and T over η₀ ranging from 0 to 2π and w_LF ranging from 1.0 to 10.0. The area of red circles and white striped circles in the Venn diagrams are equal to the (integrated) TDMI ∫M(η₀, w_LF) dη₀ dw_LF and TE $\int \mathrm{T}({\mathrm{\eta }}_0,w_{\text{LF}}){\mathit{d}\eta }_0{\mathit{dw}}_{\text{LF}}$, respectively. The centers of the two circles are determined as follows: First, each of the centers is connected with a horizontal line without being overlapped [the TE (TDMI) is located at the left (right)], and then a binary search algorithm was used to find the placement of those circles whose overlapping area is equal to the IMI ∫I(η₀, w_LF) dη₀ dw_LF by decreasing the distance between the centers. The part of the red (white striped) circle not overlapped with the white striped (red) circle has area equal to the synergistic information $\int \mathrm{S}({\mathrm{\eta }}_0,w_{\text{LF}})d{\mathrm{\eta }}_0{\mathit{dw}}_{\text{LF}}$ (the shared information ∫σ(η₀, w_LF) dη₀ dw_LF) (see Legend). (Aa) Type A. (Ba) Type B. (Ca) Type C. (Da) Type D. (Ab to Db) Venn diagrams from leader to follower for interaction types A to D. The information flows from follower to leader for types A and B are negligible and therefore are not shown. (Cc to Dc) Those from follower to leader for interaction types C and D. (Cd to Dd) Those from leader to follower for interaction types C′ and D′. (Ce to De) Those from follower to leader for interaction types C′ and D′.

Systems of more than two agents

The analysis up to now addressed only pairwise interactions. This is in accord with the theoretical basis of the information measures used; for example, T_{X → Y}(τ) in Eq. 2. The measures generalize straightforwardly to account for additional time series, say, of a third particle (or agent); see, for example, the causation entropy (26). Suppose the third variable Z, in addition to X and Y, are each symbolized by m discrete values. Then, for example, the dimension of the probability distribution p(Y_{t + 1}, X_t, Z_t) is m³ − 1 (−1 is because of probability normalization). This means that, the more the number of additional variables to be conditioned on increases, the more the dimension of the probability distribution required for computing the measures grows exponentially with respect to the number of additional variables. This requires increasingly large amounts of data to properly sample. Therefore, in multiagent systems, it is not usually feasible to condition on all or even a few other agents that interact with a given agent. In addition, even if additional variable(s) that indirectly affect(s) interactions between X and Y exist, it is nontrivial to look for this indirect “cause.” These hidden variables may be another agent entity, some past memory of the process of X and/or Y longer than being taken into account in the elucidation of TE, or something else.

Nonetheless, estimating two-agent information measures has been proven useful for monitoring influence in systems having more than two agents (5, 13, 14, 25, 27). We will now show how measuring I, σ, and S gives marked improvements even in these admittedly approximate settings.

Consider a collective in which L and F mutually interact with one another, but under model A, followers also directly interact with each other, and under model B, they do not. See, for example, Fig. 2 for the case of three agents. In the following discussion, there is one leader agent, and the number of follower agents N_F is varied. L refers to the leader, and F refers to a particular follower.

Fig. 2. Three-agent interaction diagrams.

(A) In model A, a leader influences both followers, both followers influence L, and followers influence each other. (B) Model B is similar to model A, but followers cannot influence each other. Weights are asymmetric: w_LF (leader to followers) is greater than w_FL and w_FF. We set w_LF = 4 and w_FL = w_FF = w_LL = 1.

Figure 3 (A and B) displays M_L→F for models A and B, respectively. The plots of σ (see fig. S9) are almost indistinguishable from those of M, indicating that a majority of M is actually coming from σ, which is due to shared history between L and F. As has been established for the Vicsek model (35), cohesive behavior increases as a function of density. Here, M_L→F and σ_L→F increase as a function of N_F in model A. Model B, however, is not the same as the original Vicsek model in that followers do not interact with each other, and therefore, M_L→F and σ_L→F decrease as a function of N_F, since the inclusion of additional agents that are not interacting decreases the overall cohesion between the present state of L and the future state of F. The plots of M_F→L for models A and B are not shown as they are not distinguishable by eye from those of M_L→F (see fig. S8).

Fig. 3. $\mathcal{M}$ as a function of noise level η₀ (in units of π radians) for models A and B with one leader.

Here, N_F = 1 (blue), 3 (red), 7 (yellow), and 15 (purple), where the number of leaders is always one. (A) M_L→F for model A. (B) M_L→F for model B.

Figure 4 (A to D) shows T_L→F as a function of η₀ for model A, T_F→L for model A, T_L→F for model B, and T_F→L for model B, respectively. At η₀ = 0, agent movements quickly reach a regular parallel flow independent of initial coordinates and velocities, and thus, any information about their present orientations are negligible (on average) in predicting the others’ orientational motions (see movie S1). In practice, all agents are subject to finite noise due to their environment (represented here by thermal fluctuation). Gradual decreases of T as η₀ increases simply arise from this natural stochasticity. In both models A and B, there are small bumps in T_L→F and T_F→L at η₀ ≃ 0.7π, but the notable difference is that the bumps clearly decrease as a function of N_F from L to F and F to L in model A (Fig. 4, A and B) and from F to L in model B (Fig. 4D) but not from L to F in model B (Fig. 4C).

Fig. 4. $\mathcal{T}$ as a function of noise level η₀ (in units of π radians) for models A and B with one leader.

Here, N_F = 1 (blue), 3 (red), 7 (yellow), and 15 (purple), where the number of leaders is always one. (A) ${\mathcal{T}}_{\mathrm{L}\to \mathrm{F}}$ for model A. (B) ${\mathcal{T}}_{\mathrm{F}\to \mathrm{L}}$ for model A. (C) ${\mathcal{T}}_{\mathrm{L}\to \mathrm{F}}$ for model B. (D) ${\mathcal{T}}_{\mathrm{F}\to \mathrm{L}}$ for model B.

The existence of bumps in T at η₀ ≃ 0.7π and the difference in their behavior between models A and B can be explained by decomposing T into I and S. Figure 5 (A to D) shows I_L→F as a function of η₀ for model A, I_F→L for model A, I_L→F for model B and I_F→L for model B, respectively. While the overall trend is very similar to that of T in Fig. 4, I does not contain bumps at η₀ ≃ 0.7π. Thus, the bumps in T are explained solely by S. This suggests that when such a bump exists in the values of T as a function of noise, this may be attributed to a difference in the location of the peak of I and that of S. Furthermore, when looking at how this structure changes as a function of the number of following agents, one can deduce whether follower agents mutually interact (i.e., model A) or not (i.e., model B) solely by observing the pairwise trajectories between the leader and one follower. Similar results are also obtained by more simplified binary models (see section S2 and fig. S14). Figure 6 (A to C) shows S_L→F as a function of η₀ for model A, S_F→L for model A, and S_L→F for model B, respectively (S_F→L for model B is indistinguishable by eye from S_F→L for model A and therefore not shown here) (see fig. S10). As an overall trend, S is negligibly small when η_{0 0}, simultaneous knowledge of L and F becomes relatively more important, and at high values of η₀, the simultaneous knowledge of L and F has no predictive power as the dynamics are dominated by thermal noise. As N_F increases in model A, S_L→F and S_F→L both decrease, as the future configuration of F (L) depends on more other agents and relies less on the simultaneous knowledge of L or F alone. Therefore, increasing N_F decreases the likelihood that simultaneously knowing the configuration of F and L has any additional predictive power on L or F. In model B, however, F is not affected by other followers, and therefore, S_L→F remains largely unchanged as a function of N_F.

Fig. 5. $\mathcal{I}$ as a function of noise level η₀ (in units of π radians) for models A and B with one leader.

Here, N_F = 1 (blue), 3 (red), 7 (yellow), and 15 (purple), where the number of leaders is always one. (A) I_L→F for model A. (B) I_F→L for model A. (C) I_L→F for model B. (D) I_F→L for model B.

Fig. 6. $\mathcal{S}$ as a function of noise level η₀ (in units of π radians) for models A and B with one leader.

Here, N_F = 1 (blue), 3 (red), 7 (yellow), and 15 (purple), where the number of leaders is always one. (A) ${\mathcal{S}}_{\mathrm{L}\to \mathrm{F}}$ for model A. (B) ${\mathcal{S}}_{\mathrm{F}\to \mathrm{L}}$ for model A. (C) ${\mathcal{S}}_{\mathrm{L}\to \mathrm{F}}$ for model B.

Now let us consider the case of multiple leaders in which the motility of follower agents are subject to more than one leader. Figure 7G exemplifies the case of four agents including one leader, while the three followers can interact with one another (model A). Figure 7H exemplifies the same interaction type, but with two leaders and two followers, where the leaders cannot interact with one another but the followers can. Graph representations of cases where the leaders and followers can all mutually interact, leaders can interact with one another but followers cannot, and neither leaders nor followers can interact with one another are shown in fig. S11. The values of S in these cases are shown in fig. S12, and the results are discussed in section SIF. Here, we study the effect of increasing the number of leaders in model A, where leaders cannot interact with one another but followers can (see Fig. 7, G and H). Figure 7 (A and B) shows S_L→F and S_F→L, respectively, for the case of four agents, including one leader and three followers (blue) and two leaders and two followers (red). As one may expect, S_L→F decreases as the number of leaders increases, since the dynamics of each follower results from the two leader agents, reducing synergistic effect between a leader-follower pair in the prediction of the follower motility. In the case of one leader and three followers, a follower is also subject to the interaction of an additional follower instead of the leader; however, since the weight of the follower is less, this does not reduce the synergistic effect as much as the case of two leaders and two followers. Counterintuitively, S_F→L increases as the number of leaders increases, as shown in Fig. 7 (B and D). Note that keeping the total number of agents fixed, there are fewer followers interacting with a given leader as we increase the number of leaders. It suggests that synergistic effects S_{X → Y} decrease as the weighted indegree of the agent Y increases, or, in other words, as more agents “participate” in determining the future of the target agent Y. In Fig. 7 (C and D), where there is a total of eight agents, the same trends are, respectively, seen as the number of leaders is increased; however, the overall values of S_L→F and S_F→L are lower than in Fig. 7 (A and B) due to the higher number of agents reducing the synergistic effects. In Fig. 7 (E and F), we keep the number of followers fixed to three and increase the number of leaders. Here, there is no change in S_F→L as the number of leaders increases, although the total number of agents increases, because the increase in agents does not increase the indegree of L.

Fig. 7. $\mathcal{S}$ as a function of noise level η₀ (in units of π radians) for model A with different numbers of leaders and followers.

(A and B) ${\mathcal{S}}_{\mathrm{L}\to \mathrm{F}}$ (A) and ${\mathcal{S}}_{\mathrm{L}\to \mathrm{F}}$ as a function of η₀ (in units of π radians) (B) for four agents with one leader and three followers (blue) and two leaders and two followers (red). (C and D) ${\mathcal{S}}_{\mathrm{L}\to \mathrm{F}}$ (C) and ${\mathcal{S}}_{\mathrm{F}\to \mathrm{L}}$. (D) for eight agents with one leader and seven followers (blue), two leaders and six followers (red), three leaders and five followers (yellow), and four leaders and four followers (purple). (E and F) ${\mathcal{S}}_{\mathrm{L}\to \mathrm{F}}$ (E) and ${\mathcal{S}}_{\mathrm{F}\to \mathrm{L}}$ (F) with three followers and one leader (blue), two leaders (red), and three leaders (yellow). (G) Graph representation of model A, where there is one leader and three followers. (H) Graph representation of model A, where there are two leaders and two followers.

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