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
at time
t is updated over time increment Δ
t according to
where
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
where
w is a nonnegative asymmetric matrix whose
wij element determines the interaction strength that particle
i exerts on particle
j.
wij >
wji 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 [−η
0/2, η
0/2] 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
.
Now, consider modified dynamics that modulate the dependence on θj(t) associated with follower-leader interactions that determine θi(t + 1): The leader influences the follower, but the follower does not influence the leader; i.e., wLF > 0, while wFL = 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
wLF ∈ [1,10]. We set
wLL = 1 and
wFF = 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
(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
wLL (
wFF) 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.
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(
Yt + 1,
Xt,
Zt) 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
NF is varied. L refers to the leader, and F refers to a particular follower.
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
NF 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
NF, 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).
Figure 4 (A to D) shows T
L→F as a function of η
0 for model A, T
F→L for model A, T
L→F for model B, and T
F→L for model B, respectively. At η
0 = 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 η
0 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 η
0 ≃ 0.7π, but the notable difference is that the bumps clearly decrease as a function of
NF 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).
The existence of bumps in T at η
0 ≃ 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 η
0 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 η
0 ≃ 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 η
0 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 η
0, the simultaneous knowledge of L and F has no predictive power as the dynamics are dominated by thermal noise. As
NF 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
NF 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
NF.
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.
0 Response to "Modes of information flow in collective cohesion - Science"
Post a Comment