How does the interaction radius affect the performance of intervention on collective behavior?

Caiyun Wang; Jing Han

doi:10.1371/journal.pone.0192738

Abstract

The interaction radius r plays an important role in the collective behavior of many multi-agent systems because it defines the interaction network among agents. For the topic of intervention on collective behavior of multi-agent systems, does r also affect the intervention performance? In this paper we study whether it is easier to change the convergent heading of the group by adding some special agents (called shills) into the Vicsek model when r is larger (or smaller). Two kinds of shills are considered: fixed-heading shills (like leaders that never change their headings) and evolvable-heading shills (like normal agents but with carefully designed initial headings). We know that with the increase of r, two contradictory effects exist simultaneously: the influential area of a single shill is enlarged, but its influence strength is weakened. Which factor dominates? Through simulations and theoretical analysis we surprisingly find that r affects the intervention performance differently in different cases: when fixed-heading shills are placed together at the center of the group, larger r gives a better intervention performance; when evolvable-heading shills are placed together at the center, smaller r is better; when shills (either fixed-heading or evolvable-heading) are distributed evenly inside the group, the effect of r on the intervention performance is not significant. We believe these results will inspire the design of intervention strategies for many other multi-agent systems.

Citation: Wang C, Han J (2018) How does the interaction radius affect the performance of intervention on collective behavior? PLoS ONE 13(2): e0192738. https://doi.org/10.1371/journal.pone.0192738

Editor: Andrew Baggaley, Newcastle University, UNITED KINGDOM

Received: September 30, 2017; Accepted: January 30, 2018; Published: February 15, 2018

Copyright: © 2018 Wang, Han. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper and its Supporting Information files.

Funding: Our work is supported by National Natural Science Foundation of China (No. 61374168)(JH) and the Frontier Science Key Programs of the Chinese Academy of Sciences (No. QYZDJ-SSW-JSC003) (JH). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

In recent years, collective behavior has drawn a lot of attention from scientists in many areas. It is a significant feature of self-organized multi-agent systems (MASs) where agents usually interact with each other based on local rules, i.e., each agent interacts with its neighbors. At the macroscopic level, new phenomenon will emerge in MASs which can not be found in a single agent, such as flocking of birds [1, 2], schooling of fishes [1], crowd panic [3], swarm intelligence [4], pattern formation [5, 6], synchronization [7–9], etc.

A classic example is the Vicsek model [10], a self-driven multi-agent model proposed by Vicsek et al. This model consists of autonomous agents with discrete time update rule: at each time step, velocity of all agents are updated as the vector average velocity of its neighbors with some random perturbation added and at the same time. At the same time, each agent updates its position following its velocity. The neighborhood is defined based on the interaction radius r, which means the distance between any two neighboring agents should be less or equal to r. In other words, agents only interact with their neighbors within a distance of r. They found an interesting phenomenon: if the density of the group is high and the noise is small, headings of all agents will converge to a same value and reach consensus. That means, with a larger interaction radius r, i.e., the interaction network gets more links, the group will be more likely to reach consensus.

In fact, the interaction radius has a big impact on the collective behavior of many other self-organized MASs. For example, Couzin et al. [11] found that as the alignment radius increases, agents will present different collective behavior such as swarm, torus, and consensus in a self-organizing animal group. Zúñiga and Krishnamachari [12] studied the optimal transmission radius for flooding in large scale wireless sensor networks. They found that there exists an intermediate transmission radius which minimizes the settling time for large scale wireless networks. Bagchi et al. [13] investigated the transmission radius condition that can achieve connectivity in duty-cycled wireless sensor networks. Shen et al. [14] calculated the optimal radius for caching scheme which achieves significant performance improvement in power saving, network throughput and load balance.

Another important issue is the intervention of collective behavior of MASs: if the self-organized collective behavior is not what we expect, how can we intervene the system and change the collective behavior? One way is to redesign the MAS [15–17] or put some local control in the system (pinning control) [18, 19]. The other way is nondestructive, which is called ‘soft control’ [20] proposed by Han et al. It has been successfully applied in some MASs: lead the group to converge to an expected heading of the Vicsek’s model [20, 21]; promote cooperation of multi-person prisoner’s dilemma game models [22, 23]; change the convergent opinion value in the weighted Degroot model [24]. The soft control method does not change local rules of the already-existing agents in the system, but adds one or several special agents, called shills, into the group. Shills can be redesigned and controlled, but they are treated as normal agents by normal ones. Therefore normal agents do not need to pay special attention to shills, they are not aware of the intervention. Shills can only affect their neighboring agents with normal influential power. For those MASs whose neighborhood is defined based on the interaction radius r, r not only has significant effect on the collective behavior of self-organized system, but it might also affect the soft control performance.

In this paper, we will study the problem of “whether and how the interaction radius r affects the performance of soft control based on the Vicsek model”. As we know the interaction radius r is an important factor for consensus of the Vicsek model. Tang et al. [25] proposed that as long as the interaction radius satisfies some conditions, the linearized Vicsek model will reach consensus with large probability. Later, Liu et al. [26] gave the similar condition for the Vicsek model. Furthermore, Chen et al. [27] studied the smallest interaction radius for the synchronization behavior of the Vicsek model.

As a starting point, this paper is based on specific cases of the Vicsek model: n normal agents with initial heading θ₀ are randomly placed inside an M × M periodic boundary square area. It means the group has reached consensus (heading θ₀). Now the intervention purpose is to change the consensus heading. Using the soft control idea, one or several shills with initial heading θ_s are added into the group. Two scenarios are considered: (i) evolvable-heading-shill, which means the shill will be affected by its neighbors and its heading is updated following the same update rule as normal agents, so the group will converge to θ′ which will be on average inside the interval between θ₀ and θ_s (0 = θ₀ < avg. θ′ θ < θ_s) if the population is suitably large [26]. In this scenario, changing the interaction radius r will change the convergent heading value of the system. Therefore, we use the change of convergent heading of normal agents Δθ (Δθ = θ′ − θ₀) to measure the soft control performance. Larger Δθ means better soft control performance; (ii) fixed-heading-shill, which means the shill’s heading is fixed during evolution. They are persistent, like the leader in [28]. Therefore, the group will eventually converge to θ_s. In this scenario, changing the interaction radius r will not change the convergent heading value but it will change the number of convergent time steps (relaxation time) T. So we use T to measure the soft control performance. Smaller T means better soft control performance. In each of these two scenarios, two strategies are considered: (i) centered, which means shills are staying together and initially placed at the center of the square, so they are neighbors of each other; (ii) distributed, which means shills are regularly distributed inside the square, so they are separated. Therefore, we have 4 combinations of different scenarios and strategies. Then our problem of “how does the interaction radius affect the soft control performance” is split into two specific sub-problems:

(1) what is the optimal interaction radius that can maximize the change of convergent heading of normal agents Δθ = θ′ − θ₀, i.e., , in the ‘evolvable-heading-shill’ scenario for both strategies?

(2) what is the optimal interaction radius that shills can lead normal agents to converge to θ_s with the smallest number of time steps, i.e., , in the ‘fixed-heading-shill’ scenario for both strategies?

We find answers to the above two problems through simulations and some related theoretical analysis: for the centered fixed-heading-shill case, larger r leads to better soft control performance (smaller T); for the centered evolvable-heading-shill case, smaller r gives better soft control performance (larger Δθ); for the distributed strategy (both in the fixed-heading-shill and the evolvable-heading-shill scenarios), the effect of r on the control performance is not significant. On the other hand, we also find some interesting results by comparing the distributed strategy and the centered strategy: for the fixed-heading-shill scenario, the distributed strategy outperforms the centered strategy; while for the evolvable-heading-shill scenario, the centered strategy outperforms the distributed strategy because of the ‘reciprocal team’ effect—the neighboring evolvable-heading shills support and interact with each other.

There are some related studies about intervention on flocking models. Kyriakopoulos et al. [29] studied the response of flocks, described as hydrodynamic equation which contains the Vicsk model, to a small homogeneous external field of amplitude h. The function of external field is like the fixed-heading shill, if the external field is constructed by continuously adding the same number of fixed-heading shills to the neighborhood of every normal agent. On the other hand, the fixed-heading shill with velocity v > 0 in our approach can be regarded as a moving local external field. However, our fixed-heading shills methods is different to external field in general. Shills are more like an agent than external field because they are moving and their influential area is decided by the interaction neighborhood radius r as same as that of all normal agents. Even though in the distributed strategy, normal agents usually do not have the same number of neighboring fixed-heading shills, especially when v > 0. That makes the ‘external field’ constructed by fixed-heading shills very complicated. Pearce et al. [30] considered some leaders (informed individuals [31]), who update their headings by taking average of the heading of their neighbors and with a biased specific direction, based on the Vicsek model. They studied how the flock responses to the leadership. Their approach has some essential differences with our paper. First, these leaders who are biased to turn toward a specific angle are different from our shills: fixed-heading shills always head to a fixed desired direction; evolvable-heading shills follow the same updating rule as normal agents but with well-designed initial headings. Second, [30] considered the Vicsek model with a fixed interaction radius r = 1; while in this paper a wider range of r is investigated to study the impact of r on the control performance. Third, because the dynamic of leaders is different from the shills, the conclusions on linearity are different: [30] found that the group shows a linear response to leadership; while in this paper we will show that the response of the group to shills is not linear.

Methods

The basic model

Vicsek et al. proposed a self-propelled multi-agent model in 1995 [10]. Suppose there are n agents (x_i(t), θ_i(t)), labelled from 1 to n, are placed inside an M × M periodic square area, where x_i(t) denotes the position of agent i at time t and θ_i(t) denotes the heading of agent i at time t. The velocity of agent i at time t is defined by v_i(t) = (v cos(θ_i(t)), v sin(θ_i(t))) is constructed to have an constant absolute value v and a heading given by the angle θ_i(t) ∈ (−π, π]. The neighborhood of agent i is defined as (1) where r is a given interaction radius. This means two agents are neighbors if the distance between them is not larger than r.

For any agent i, its position and heading are updated by Eqs (2)–(3) below: (2) (3) where 〈θ(t)〉_r denotes the average direction of the velocities of neighbors of agent i. In other words, 〈θ(t)〉_r has the following form: (4) with some necessary regulations reflecting heading of (−π, π]. In Eq (3), δ_i(t) represents noise and it is a random number chosen with a uniform probability from the interval [−η/2, η/2].

Here is the evolution process of the Vicsek model: after the initial heading and the position of each agent are given, all agents update their headings using Eqs (1)–(4) at every discrete time step t = 1, 2, ⋯. The simulation results in Vicsek’s paper [10] show that if the density is large and noise is small, the group will reach consensus. In Liu’s paper [26], they prove that without noise, the Vicsek model will almost surely reach consensus for any r > 0 and v > 0.

Model with shills

Suppose the group has reach consensus (heading θ₀). Now we want to change the consensus heading. Then we add l (l ≥ 1) shills with heading θ_s into the group and reset the clock as t = 0. Therefore we now have θ_i(0) = θ₀ for i = 1, 2, ⋯, n. Because normal agents regard shills as normal ones while updating their headings, the neighborhood of all agents should be redefined as follows: (5) where is the total number of agents, agent 1, 2, ⋯, n are normal agents, agent n + 1, n + 2, ⋯, n + l are shills. The absolute value of the velocities of shills is also set to be v here. After adding shills into the system, normal agents will evolve following Eqs (2)–(5).

The effect of interaction radius r

Since the neighborhood is decided based on r, the neighborhood graph might be different if r is different. Therefore, r also has significant impact on the effect of soft control. For any r, two factors describe how much the influence of one shill bring to the system: the influential area S and the influence strength P. The influential area S is the number of normal agents which are affected by one shill, i.e., S = πr² ρ_n, where ρ_n is the density of normal agents. The influence strength P is the influence that one shill puts on its neighboring agents. The measurement for P is much more complicate than S, because from the second step, headings of normal agents might not be the same, and headings of evolvable-heading shills are no longer θ_s. So for simplicity, we consider P only for the first step influence in an ideal case: suppose the initial heading θ₀ of normal agents neighboring to the shill is zero, then we have . Lager P means more powerful influence strength that one shill puts on its neighboring normal agents. Apparently, P decreases with the increase of S. From this simplified analysis, there is a tradeoff between P and S.

Smaller (or larger) interaction radius r has both the negative and positive effects when we consider of how much influence of one shill brings to the group:

The positive effect for smaller r: the influence that one shill puts on its neighboring agents P is larger. With the increase of r, S becomes larger which makes P smaller.
The negative effect for smaller r: the influence area of one shill S is smaller. This means there are less normal agents directly influenced by one shill.

Therefore, if the positive effect of smaller r dominates, smaller r will give better soft control performance. Otherwise, larger r will give better performance. But which effect dominates? And in what circumstances?

To answer above questions, we consider our study in the circumstance of two different soft control strategies (centered and distributed) in two scenarios (fixed-heading-shill and evolvable-heading-shill):

Centered: l shills are initially centered inside an M × M periodic boundary square. This means all shills are initially located in the same position, the center point of the square (example is Fig 1a).
Distributed: l shills are initially regularly distributed inside an M × M periodic boundary square (example is Fig 1b). In this way, the M × M square is evenly divided into l small squares, where each of them initially locates one shill. These shills will not be neighbors of each other if r is not large enough.

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Fig 1. Examples of initial positions of adding 25 shills into a group of 225 normal agents inside a 15 × 15 square.

Blue dots represent normal agents. Red stars represent shills. Short lines connected to the blue dots or red stars represent their headings. Here initial headings of all normal agents are set as zero and initial headings of shills are π/2. (A) Centered shills; (B) Distributed shills.

https://doi.org/10.1371/journal.pone.0192738.g001

When l shills are initially placed at the center, they will have strong influence on their neighboring normal agents. These shills act like a ‘core’, the neighboring normal agents will align with the ‘core’ quickly and become ‘virtual shills’ in some sense. The ‘core’ and the ‘virtual shills’ form an ‘alliance’ in the center. When l shills are distributed in the area, their influence is evenly distributed among the group, the ‘core’ effect of each shill is much weaker.

Two scenarios are:

Fixed-heading-shill: l fixed-heading shills with heading θ_s are added into an M × M periodic boundary square. It means headings of shills are not affected by their neighbors. They are always equal to θ_s.
Evolvable-heading-shill: l evolvable-heading shills with initial heading θ_s are added into an M × M periodic boundary square. It means headings of shills are initialized as θ_s, and then they will be updated following Eqs (2)–(5) without noise, just like normal agents.

If l evolvable-heading shills are initially distributed inside the group, each of them will be easily affected by its neighboring normal agents and lose the shill effect within the first few steps. But if they are initially placed together at the center, they will interact with each other and become a reciprocal team, which forms a strong ‘alliance’ around them. We call this the ‘reciprocal team’ effect, which becomes significant with the increase of the number of neighboring evolvable-heading shills in the team. It cannot be found in the fixed-heading-shill scenario, because these shills do not interact with each other.

Results and discussions

To study effects of interaction radius on the soft control performance based on the Vicsek model, a number of simulations are performed for four combinations of two strategies (centered and distributed) in two scenarios (fixed-heading-shill and evolvable-heading-shill). And then we extend this study to the linearized Vicsek model and the non-periodic boundary model cases. Some related discussions and theoretical analysis are given.

Here are settings of simulations in this section: v = 0.03 which is the same as the setting of [10]; η = 0.02; θ₀ = 0. n normal agents with initial heading θ₀ are randomly placed inside the M × M square. Soft control performance is measured as the average of 100 runs on random position distributions of normal agents. For each simulation, we add l shills into the system. For the centered strategy, all l shills are initially placed at the same position (example is Fig 1a with M = 15 and l = 25); for the distributed strategy, each shill is initially placed at the center of every square (example is Fig 1b with M = 15 and l = 25). In [10], Vicsek et al. proposed an order parameter , where v_i(t) = (v cos θ_i(t), v sin θ_i(t)), to measure the consensus level. In our simulations, the system is regarded as consensus when 1 − ϕ(t) ≤ 10⁻⁴, then the simulation is terminated.

Fixed-heading-shill scenario

In this subsection, we add l fixed-heading shills with heading θ_s into an M × M square with periodic boundary. The number of convergent time steps T is used to measure the soft control performance. We find that r has big impact on the value of T in the following simulations.

Fig 2 shows avg. T for different θ_s, M and ρ_n with η = 0.02, and has the following results (results are also hold for η = 0.1 (see S1 Fig for details)):

For the centered strategy, larger r gives better soft control performance (smaller avg. T).
Although smaller r has positive effect, there is a big delay of shill influence for normal agents located far away from shills. Therefore it needs more time steps to spread the shill effect to all normal agents. In this case, the negative effect overcomes the positive effect of smaller r, so larger r gives better soft control performance.
For the distributed strategy, the effect of r on the soft control performance is not significant.
In this case, the positive effect and the negative effect of smaller r are balanced in some way, so T varies very little while r changes. The mean-field analysis is given in S1 Appendix.

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Fig 2. Soft control performance (avg. T) for different θ_s, ρ_n and M in the fixed-heading-shill scenario with l = 25.

https://doi.org/10.1371/journal.pone.0192738.g002

Then we investigate the finite system size effect. The system size is the number of normal agents n (n = ρ_n M²). Fig 3a and 3c show that avg. T and M approximately obey positive power function regardless of r. Fig 3b and 3d show that in general the avg. T increases with the increase of ρ_n. This is because more normal agents are needed to be influenced with the increase of ρ_n. But in Fig 3b, there is a special phenomenon when r and ρ_n are small: avg. T decreases with the increase of ρ_n. This is because the neighborhood graph is too sparse that it takes more time steps for the shill influence to spread to all normal agents.

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Fig 3. System size effect on soft control performance (avg. T) in the fixed-heading-shill scenario with l = 25.

The system size is n = ρ_n M². (A) and (C) show how soft control performance (avg. T) change when n increases (by increasing M with fixed ρ_n); (B) and (D) show how soft control performance (avg. T) change when n increases (by increasing ρ_n with fixed M).

https://doi.org/10.1371/journal.pone.0192738.g003

Furthermore, in Fig 4 we compare the performance between centered and distributed strategies, and investigate whether adding l shills can improve the soft control performance by the factor of l comparing to the case of adding one shill. We find the following results with θ_s = π/4 (results are also hold for other θ_s (see S2 Fig for details)):

Distributed strategy (blue line) outperforms centered strategy (green line).
25 centered shills (green line) do not enhance 25 times of the soft control performance of one single shill (red line).
We compare the number of convergent time steps of adding 25 centered shills with 1/25 of the number of convergent time steps of adding one shill. We find that the number of convergent time steps of adding one shill is smaller than 25 times of the number of convergent time steps of adding 25 centered shills. Therefore, the performance of l fixed-heading shills is less than the linear superposition.

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Fig 4. Comparisons of different soft control strategies in the fixed-heading-shill scenario.

https://doi.org/10.1371/journal.pone.0192738.g004

Evolvable-heading-shill scenario

In this subsection, we add l evolvable-heading shills with initial heading θ_s into an M × M square with periodic boundary. These shills will be affected by neighboring normal agents and update their headings during evolution like normal ones but without noise. Given θ₀ = 0 and θ₀ < θ_s in our simulations, heading of agents in the group will converge to value θ′ (0 < avg. θ′ < θ_s) if the population is suitably large [26]. Therefore, the change of convergent heading of normal agents Δθ (Δθ = θ′ − θ₀ = θ′) is used to measure the soft control performance. Larger Δθ means better performance of soft control. We find that r has big impact on the value of Δθ in the following simulations.

Fig 5 shows avg. Δθ for different θ_s, M and ρ_n with η = 0.02, and has the following results (results are also hold for η = 0.1 (see S1 Fig for details)):

For the centered strategy, smaller r gives better soft control performance (larger Δθ).
This is unlike the result in the fixed-heading-shill scenario. It is because the smaller r gives stronger ‘reciprocal team’ effect. While in the fixed-heading-shill scenario, there is no ‘reciprocal team’ effect.
For the distributed strategy, the effect of r on the soft control performance is not significant.
This is similar to what we have found in the fixed-heading-shill scenario. It is worth mentioning that for θ_s > π/2, increasing θ_s decreases Δθ. This is caused by the nature of the updating rule of agents—each agent updates its heading by taking average direction of the velocities of its neighbors. The mean-field analysis is given in S1 Appendix.

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Fig 5. Soft control performance (avg. Δθ) for different θ_s, M and ρ_n in the evolvable-heading-shill scenario with l = 25.

https://doi.org/10.1371/journal.pone.0192738.g005

Then we investigate the finite system size effect. The system size is n = ρ_n M². Fig 6a and 6c show that avg. Δθ and M approximately obey negative power function regardless of r. Fig 6b and 6d show that avg. T and ρ_n approximately obey negative power function regardless of r.

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Fig 6. System size effect on soft control performance (avg. T) in the evolvable-heading-shill scenario with l = 25.

(A) and (C) show how soft control performance (avg. Δθ) change when n increases (by increasing M with fixed ρ_n); (B) and (D) show how soft control performance (avg. Δθ) change when n increases (by increasing ρ_n with fixed M).

https://doi.org/10.1371/journal.pone.0192738.g006

Again, to compare the soft control performance between centered and distributed strategies, study the linear superposition property, and demonstrate the importance of ‘reciprocal team’ effect, we compare the performance (Δθ) for four situations in Fig 7 with θ_s = π/4 (results are also hold for other θ_s (see S2 Fig for details)):

Centered strategy (green line) outperforms distributed strategy (blue line).
It is because the significant ‘reciprocal team’ effect strongly strengthen the shill effect for the centered strategy.
Centered (interactive) shills (green line) outperform centered non-interactive shills (purple line).
This pattern shows the importance of the ‘reciprocal team’ effect. Interactive shills have the ‘reciprocal team’ effect, while non-interactive shills do not. Without the ‘reciprocal team’ effect, the performance (purple line) is much worse (than the green line) and the pattern is more like the single shill case (red line). Therefore, the interactions among evolvable-heading shills significantly enhance the soft control power.
25 centered shills (green line) v.s. one single shill (red line) for small r.
Unlike the result in the fixed-heading-shill scenario, we find that the performance of adding 25 centered shills is much better than 25 times of adding one centered shill when r is smaller than a critical point r_c₀. This can be regarded as a ‘1 + 1 > 2’ phenomenon, and shows how important it is for shills to stay together and support each other when their headings are evolvable (not persistent), especially when r is small. This means that the significant ‘reciprocal team’ effect brings in a nonlinear enhancement for the soft control power especially when r < r_c₀. When r > r_c₀, increasing numbers of neighboring normal agents of shills will strongly dilute the shill influence in first few steps. Notice that the critical point r_c₀ varies with different initial heading of shills (see S3 Fig for details).

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Fig 7. Comparisons of different soft control strategies in the evolvable-heading-shill scenario with ρ_n = 1 and θ_s = π/4.

https://doi.org/10.1371/journal.pone.0192738.g007

Extension to the linearized Vicsek model

Jadbabaie et al. [28] proposed the linearized Vicsek model by changing Eq (4). For any agent i, its heading is updated as the average heading of its neighboring agents: (6) where |N_i(t)| is the number of agent i’s neighbors.

The simulation results for the linearized Vicsek model is similar for these of the Vicsek model and we give theoretical analysis for these patterns (see S2 Appendix for details).

Extension to non-periodic boundary model

What we have studied above are for periodic boundary cases (based on both the Vicsek model and the linearized Vicsek model). For non-periodic boundary cases, agents located near the boundary have less neighboring agents. Our earlier conference paper [32] has shown some simulation results for the non-periodic boundary Vicsek model with zero absolute velocity for n = 200 and l = 25. To be consistent with the parameter settings of this paper, we re-do simulations for the non-periodic boundary Vicsek model with zero velocities and noise (Fig 8a and 8b) and the non-periodic boundary linearized Vicsek model with v = 0 (Fig 8c and 8d).

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Fig 8. Soft control performance of different strategies in different scenarios based on two different models (the Vicsek model and the linearized Vicsek model) with non-periodic boundary.

ρ_n = 1, M = 15, l = 25, v = 0 and η = 0.02. (A) Fixed-heading-shill scenario for the Vicsek model. (B) Evolvable-heading-shill scenario for the Vicsek model. (C) Fixed-heading-shill scenario for the linearized Vicsek model. (D) Evolvable-heading-shill scenario for the linearized Vicsek model.

https://doi.org/10.1371/journal.pone.0192738.g008

Patterns for non-periodic boundary cases are similar but not exactly the same as patterns that we have found in the periodic boundary cases because of the boundary effect. But they share similar patterns in many ways: (i) for the centered fixed-heading-shill case, larger r gives better soft control performance when r is smaller than a relatively large value (r < r_c₁); (ii) for the centered evolvable-heading-shill case, smaller r gives better soft control performance except a short interval of r in the middle (r_c₃ < r < r_c₄); (iii) for the distributed strategy (in both the fixed-heading-shill and the evolvable-heading-shill scenarios), the effect of r on the soft control performance is not significant; (iv) the centered strategy outperforms the distributed strategy in the evolvable-heading-shill scenario; the distributed strategy outperforms the centered strategy for small r (r < r_c₂) in the fixed-heading-shill scenario. The ‘reciprocal team’ effect are also found.

Conclusions

Based on a classic multi-agent system model, the Vicsek model, we study the effect of the interaction radius on the soft control performance. Then we extend the approach to the linearized Vicsek model and the non-periodic boundary model cases. We consider two different soft control strategies (centered and distributed) in two scenarios (fixed-heading-shill or evolvable-heading-shill). We obtain the following results (see Fig 9 for details) through simulations and analysis:

Answer to the question of “how does the interaction radius affect the performance of soft control based on the Vicsek model” (with periodic boundary) is shown as conclusion 1 in Fig 9.
To explain these results, we analyze and discuss the positive and negative effects of the smaller/larger interaction radius r on the soft control performance. These two effects are competitive: the positive effect of smaller r—there will be fewer connections among normal agents, so the strength of influence that a shill puts on a neighboring normal agent will be stronger since the normal agent has fewer neighbors; the negative effect of smaller r—the influential circle of a shill is smaller, so fewer normal agents are directly influenced by a single shill. These two factors (the shill’s influential area and the shill’s influence strength) are contradictory: as one falls, another rises. Therefore, there is a tradeoff between the negative and positive effects of smaller/larger r on the soft control performance. The optimal value of r is different for different strategies and scenarios. And we give mean-field analysis for the distributed strategy.
Another interesting question is: should we place shills together at the center or distribute them evenly among the group while considering different r? We find the following answers regardless of r (shown as conclusion 2 in Fig 9): (i) for the fixed-heading-shill scenario, the distributed strategy outperforms the centered strategy; (ii) for the evolvable-heading-shill scenario, the centered strategy outperforms the distributed strategy. Fixed-heading shills are persistent individuals. They affect their neighbors but never be affected. While evolvable-heading shills are not that ‘strong-minded’. They interact with their neighbors and will be affected by them. Therefore, it will make a big difference if evolvable-heading shills are staying together to support each other—this is called the ‘reciprocal team’ effect, which brings in a nonlinear enhancement for the soft control power. We can see that the soft control performance is much worse in the centered non-interactive evolvable-heading-shill case where interactions among evolvable-heading shills are removed. This interesting phenomenon can be concluded from anther aspect: the performance of l fixed-heading shills is worse than linear superposition; the performance of l evolvable-heading shills is better than linear superposition when r is small.
We extend our study to the linearized Vicsek model, and we find similar results. Besides simulations, we also give mathematical analysis to these patterns in the evolvable-heading-shill scenario with v = 0 and prove that these results are true for v > 0 under some conditions. We also extend our study to the non-periodic boundary model and find its patterns are similar but not exactly the same as patterns that we have found in the periodic boundary cases because of the boundary effect. The ‘reciprocal team’ effect are also found in non-periodic boundary cases.

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Fig 9. Main conclusions.

https://doi.org/10.1371/journal.pone.0192738.g009

This paper explores a new question which is about the relationship between the interaction radius and the intervention performance in different cases. It will help intervention on collective behavior of MASs in a more efficient and effective way. We believe this study can be extended to other MASs, especially MASs whose neighborhood relationships are also defined by interaction radius.

Supporting information

S1 Fig. Three-dimensional plot for soft control performance with respect to r for different θ_s based on the Vicsek model with ρ_n = 1, M = 15, v = 0.03 and η = 0.1.

The simulation terminates when 1 − ϕ(t) ≤ 10⁻³.

https://doi.org/10.1371/journal.pone.0192738.s001

(EPS)

S2 Fig. The soft control performance for different θ_s (θ_s = π/4, π/2 and 3π/4) with ρ_n = 1 and M = 15.

(A) is for the fixed-heading-shill scenario; (B) is for the evolvable-heading-shill scenario.

https://doi.org/10.1371/journal.pone.0192738.s002

(EPS)

S3 Fig. The critical point r_c₀ varies with different θ_s in the evolvable-heading-shill scenario.

(A) is the pattern for the comparison of the soft control performance between 25 centered shills and 25 times of one single shill in the evolvable-heading-shill scenario. (B) is the r − θ_s projection of (A). Soft control performance is measured as the average of 500 runs on random position distributions of normal agents.

https://doi.org/10.1371/journal.pone.0192738.s003

(EPS)

S1 Appendix. Mean-field analysis for the distributed strategy soft control performance in the Vicsek model.

https://doi.org/10.1371/journal.pone.0192738.s004

(PDF)

S2 Appendix. Simulations and analysis for the soft control performance based on the linearized Vicsek model.

https://doi.org/10.1371/journal.pone.0192738.s005

(PDF)

S1 File. This file contains the data used in figures.

https://doi.org/10.1371/journal.pone.0192738.s006

(7Z)

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[ref1] 1. Reynolds CW. Flocks, herds and schools: A distributed behavioral model. ACM SIGGRAPH computer graphics. 1987 Aug 1;21(4):25–34.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Morse DH. Ecological aspects of some mixed-species foraging flocks of birds. Ecological monographs. 1970 Feb 1;40(1):119–168.
View Article
Google Scholar

[5] View Article

[6] Google Scholar

[ref3] 3. Helbing D, Farkas I, Vicsek T. Simulating dynamical features of escape panic. Nature. 2000 Sep 28;407(6803):487–490. pmid:11028994
View Article
PubMed/NCBI
Google Scholar

[8] View Article

[9] PubMed/NCBI

[10] Google Scholar

[ref4] 4. Bonabeau E, Dorigo M, Theraulaz G. Swarm intelligence: from natural to artificial systems. Oxford university press; 1999.

[ref5] 5. Meinhardt H, Meinhardt H. Models of biological pattern formation. London: Academic Press; 1982.

[ref6] 6. Gierer A, Meinhardt H. A theory of biological pattern formation. Kybernetik. 1972 Dec 1;12(1):30–39. pmid:4663624
View Article
PubMed/NCBI
Google Scholar

[14] View Article

[15] PubMed/NCBI

[16] Google Scholar

[ref7] 7. Arenas A, Díaz-Guilera A, Kurths J, Moreno Y, Zhou C. Synchronization in complex networks. Physics reports. 2008 Dec 31;469(3):93–153.
View Article
Google Scholar

[18] View Article

[19] Google Scholar

[ref8] 8. Gómez-Gardeñes J, Zamora-López G, Moreno Y, Arenas A. From modular to centralized organization of synchronization in functional areas of the cat cerebral cortex. PLoS One. 2010 Aug 26;5(8):e12313. pmid:20865046
View Article
PubMed/NCBI
Google Scholar

[21] View Article

[22] PubMed/NCBI

[23] Google Scholar

[ref9] 9. Wang Q, Chen G, Perc M. Synchronous bursts on scale-free neuronal networks with attractive and repulsive coupling. PLoS one. 2011 Jan 6;6(1):e15851. pmid:21253015
View Article
PubMed/NCBI
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[25] View Article

[26] PubMed/NCBI

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[ref10] 10. Vicsek T, Czirók A, Ben-Jacob E, Cohen I, Shochet O. Novel type of phase transition in a system of self-driven particles. Physical review letters. 1995 Aug 7;75(6):1226. pmid:10060237
View Article
PubMed/NCBI
Google Scholar

[29] View Article

[30] PubMed/NCBI

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[ref11] 11. Couzin ID, Krause J, James R, Ruxton GD, Franks NR. Collective memory and spatial sorting in animal groups. Journal of theoretical biology. 2002 Sep 7;218(1):1–11. pmid:12297066
View Article
PubMed/NCBI
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[34] PubMed/NCBI

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[ref12] 12. Zúñiga M, Krishnamachari B. Optimal transmission radius for flooding in large scale sensor networks. Cluster Computing. 2005 Jul 1;8(2–3):167–178.
View Article
Google Scholar

[37] View Article

[38] Google Scholar

[ref13] 13. Bagch A, Galhotra S, Mangla T, Pinotti CM. Optimal radius for connectivity in duty-cycled wireless sensor networks. ACM Transactions on Sensor Networks (TOSN). 2015 Mar 2;11(2):36.
View Article
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[ref15] 15. Ceccarelli N, Di Marco M, Garulli A, Giannitrapani A. Collective circular motion of multi-vehicle systems. Automatica. 2008 Dec 31;44(12):3025–3035.
View Article
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[ref17] 17. Liu J, Jing H, Tang YY. Multi-agent oriented constraint satisfaction. Artificial Intelligence. 2002 Mar 31;136(1):101–144.
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[52] PubMed/NCBI

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View Article
Google Scholar

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[56] Google Scholar

[ref20] 20. Han J, Li M, Guo L. Soft control on collective behavior of a group of autonomous agents by a shill agent. Journal of Systems Science and Complexity. 2006 Mar 1;19(1):54–62.
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[59] Google Scholar

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[ref25] 25. Tang G, Guo L. Convergence of a class of multi-agent systems in probabilistic framework. Journal of Systems Science and Complexity. 2007 Jun 1;20(2):173–197.
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Figures

Abstract

Introduction

Methods

The basic model

Model with shills

The effect of interaction radius r

Results and discussions

Fixed-heading-shill scenario

Evolvable-heading-shill scenario

Extension to the linearized Vicsek model

Extension to non-periodic boundary model

Conclusions

Supporting information

S1 Fig. Three-dimensional plot for soft control performance with respect to r for different θs based on the Vicsek model with ρn = 1, M = 15, v = 0.03 and η = 0.1.

S2 Fig. The soft control performance for different θs (θs = π/4, π/2 and 3π/4) with ρn = 1 and M = 15.

S3 Fig. The critical point rc0 varies with different θs in the evolvable-heading-shill scenario.

S1 Appendix. Mean-field analysis for the distributed strategy soft control performance in the Vicsek model.

S2 Appendix. Simulations and analysis for the soft control performance based on the linearized Vicsek model.

S1 File. This file contains the data used in figures.

References

S1 Fig. Three-dimensional plot for soft control performance with respect to r for different θ_s based on the Vicsek model with ρ_n = 1, M = 15, v = 0.03 and η = 0.1.

S2 Fig. The soft control performance for different θ_s (θ_s = π/4, π/2 and 3π/4) with ρ_n = 1 and M = 15.

S3 Fig. The critical point r_c₀ varies with different θ_s in the evolvable-heading-shill scenario.