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The authors have declared that no competing interests exist.

Conceived and designed the experiments: JH. Performed the experiments: JH. Analyzed the data: JH. Contributed reagents/materials/analysis tools: JH. Wrote the paper: JH LW. Mathematical proof: LW.

For a given multi-agent system where the local interaction rule of the existing agents can not be re-designed, one way to intervene the collective behavior of the system is to add one or a few special agents into the group which are still treated as normal agents by the existing ones. We study how to lead a Vicsek-like flocking model to reach synchronization by adding special agents. A popular method is to add some simple leaders (fixed-headings agents). However, we add one intelligent agent, called ‘shill’, which uses online feedback information of the group to decide the shill's moving direction at each step. A novel strategy for the shill to coordinate the group is proposed. It is strictly proved that a shill with this strategy and a limited speed can synchronize every agent in the group. The computer simulations show the effectiveness of this strategy in different scenarios, including different group sizes, shill speed, and with or without noise. Compared to the method of adding some fixed-heading leaders, our method can guarantee synchronization for any initial configuration in the deterministic scenario and improve the

Multi-agent methodology is a natural and popular way to model a system consisting of many locally interacting individuals (units). Collective behavior such as phase transition, flocking/schooling/herding

In a distributed multi-agent system, there is no central controller. Usually agents interact locally. The set of rules (or mechanisms) describing the interaction between two agents are called ‘local rules’. If the local rule is elaborately designed, the system will show expected and useful function, such as swarm intelligence. However, for some systems, the self-organized collective behavior is not what we expect. Then, how do we intervene in the system and change the collective behavior? One way is to re-design the multi-agent system. For example, re-design the local rule of the agents. That is actually about the problem of how to design a distributed system. Examples include formation control

The other way is to use nondestructive intervention methods. In many real world systems, such as birds and crowds, the interaction rules among individuals are part of the natural mechanism. They can not be re-designed to achieve the desired collective behavior. Therefore, the coordination should not change the interaction rules of the existing agents (normal agents). Meanwhile, in these decentralized systems there is no central controller who sends orders to agents, nor can global parameter be adjusted to change the collective behavior either. In this case, how do we softly intervene in the system and guide the collective behavior?

For some multi-agent systems, adding a few agents into the system is allowed. One way to intervene in the system is to add one (or more) special agent(s), called a ‘

Synchronization is one of the most basic yet important collective behavior, which has profound impacts on many systems

In situations where the initial configuration of the system does not satisfy the synchronization condition, intervention is needed to help synchronize the system – shills are added into the system. There are four related approaches: (1)For the Vicsek's model without noise, an early attempt to intervene and guide synchronization is described in

In the above approaches, shills of

In this paper, a comprehensive algorithm for an efficient and intelligent shill is introduced: it has a new and subtle strategy called ‘consistent moving’, which uses online information of normal agents' locations to determine and update its heading at each step. The shill periodically affects every normal agent and will eventually synchronize the whole group towards the desired heading. Both the mathematical analysis and the simulation results prove that the system can be synchronized by adding one intelligent shill with a limited speed. Merits of this new approach are as follow:

synchronization is guaranteed for any initial configuration in the deterministic scenario by adding only one intelligent shill;

the strategy is much more clever and the heading of the shill is consistent with its actual moving route, so this is the first comprehensive approach for intelligent shill with theoretical analysis;

by using feedback information, the intelligent shill is possible to handle noise in the Vicsek's model, which cannot be achieved with fixed-heading shills.

We will demonstrate these advantages in the computer simulations by comparing with the method of adding some fixed-heading shills (leaders). It shows that one intelligent shill can perform better when measured by the synchronization level, especially in low density groups. Besides, the intelligent shill has significant advantages in the case of noise. It implies that feedback information is essential for intervention in the model with noise.

For nondestructive intervention of collective behavior of multi-agent systems, adding shills is a feasible method. Although shills have only the same strength of influence on neighbors as normal agents do, they can have a bigger impact on the collective behavior of the group, depending on the number of shills that are added into the system and how intelligent the shills are. Without using feedback information, shills are not intelligent and more than one shill is needed to guide the group

Flocking of birds, schooling of fishes and herding of sheep are ubiquitous in nature. In 1987, Reynolds, a computer scientist, might be the first to propose a computer simulation which can show flocking phenomena

The Vicsek-like model we adopt in this paper is described bellow: there are

Agent

The computer simulation of the above model shows that given different initial configurations (initial locations and headings of

Velocities of agents are displayed for two cases: a self-organized group and a group with one shill. The number of agents is

considering the Viksek-like model defined above. Given any initial configurations that consist of

As we have pointed out, the difficulty is how the shill decides the moving direction for each step using online feedback information. Especially, how a shill moves from one target agent location to another without putting negative effects on others is challenging. This paper gives a solution by proposing a novel strategy for the shill called ‘consistent moving’ (demos see

The position, heading and speed at time

Therefore, the way normal agents treat a shill is the same as the way they treat normal ones. So shills have only the same strength of influence on neighbors as normal agents do.

There are differences between the shill and the normal agent: (a)The shill does not need to follow the normal agent's local rule of

To affect the heading of a normal agent

The big dash-line square indicates the current group area (note that to show ideas of the shill route, for convenience, the group area shown here is supposed to be static. In fact, the actual consistent moving strategy considers the cases that normal agents are moving when the shill moves, i.e., the group area keeps changing, which is much more complicated.). Two moving routes for the shill are shown:

(1) When to affect a normal agent? Does the shill need to affect all normal agents one by one repeatedly? Which agent should be selected as the next target to be affected? What are the criteria for selection? Jadbabaie et al.

In a period, the order for target agent selection does not affect whether or not the shill can complete the task, but it might affect the convergence time. The simplest schedule in a period is to affect agents in a fixed order:

(2) What is the moving route of the shill from one location to the next target agent location? This is the core of the strategy. Note that if the shill is not always moving with the desired direction (heading zero), it will have negative effects on its neighboring normal agents. However, if the shill moves with a fixed heading of zero, it could never get to some locations. One solution is to allow the shill to change heading but make sure not to put negative effect on any normal agent during its movement. This can be described as the second principle: ‘

The simplest idea for a shill to move from one target agent location to another is shown in

However, this simple route is not efficient. In fact, the shill can find a shorter route shown in

With a shill, the overall system evolves in this way: firstly,

The section of theoretical result below will give a rough upper-bound for the speed of the shill which can guarantee the evolution will stop in finite steps for any

For compact construction and easy understanding, we will present the theoretical result in this subsection, and give a detailed mathematical proof in

From the above section, we know the following two principles, denoted by

Note that in (i),

An essential problem we concerned about is that can the shill with a limited speed accomplish periodical intervention? We know that if all normal agents can be covered by a limited circle in every time step, the shill can finish its task with a limited speed. On the other hand, if the minimal circle which covers all normal agents becomes bigger and bigger during evolution, the shill will have to keep accelerating to make sure it can catch up and affect all agents in a fixed period. Thus, the question can be translated to whether all normal agents can be covered by a limited circle during the whole evolution process.

For the group, we define a dynamic reference point

Solid-line circle represents the location area of time

The mathematical result shows that with the help of a shill using ‘consistent moving’ strategy, the size of the group area is always limited during the evolution. This means the distance between any two normal agents is always limited. So with a limited speed, the shill can move from one agent's location to any other agent's location within limited steps.

It can be found that with some speed not larger than

Note that

The settings of parameters in simulations are as follow: neighborhood radius

How does the shill maximal speed

With

The impact of group size

It is worthy to point out that small

As mentioned in the

In simulations,

In the original Vicsek' model

Three scenarios are considered for each case: (1)self-organized; (2)one intelligent agent with

This result shows that for small noise, one intelligent shill using ‘consistent moving’ strategy can significantly promote synchronization and perform much better than the method of adding some fixed-heading shills, because the intelligent shill can use online information to adjust its heading and try to keep up with every agent. Therefore, online feedback information of agent location is important in this sense, especially in the procedure of next target agent selection. This shows the advantages of intelligence through feedback mechanism. We notice that tightly connected group(group with high density) has low tendency of dispersion caused by noise because each agent in the group has many neighbors that can cancel out most of the noise effect. It suggests future study of the intelligent shill to achieve better synchronization level in the noise model: for example, first the shill tries to drive all normal agents moving towards a center to form a tightly connected group, and then the shill use the ‘consistent moving’ strategy to guide synchronization.

In a word, the most important difference between methods of adding a number of fixed-heading shills and adding one intelligent shill is the feedback mechanism which brings big benefits but requires observation of information (in fact, global information is not necessary during the procedure of shill moving from one location to the next target agent location, see

We have studied nondestructive intervention by adding one intelligent shill and proposed a smart strategy for the intelligent shill to coordinate synchronization of a Vicsek-like model using online feedback information, which is the first complete and feasible algorithm for this purpose. The strategy obeys two principles: (1)the shill should directly or indirectly affect all normal agents periodically; (2)by using online feedback information of normal agents, the shill should avoid putting negative effects on normal agents when it is moving in a non-desired direction. According to these two principles, we have designed a finite-state machine for the dynamics (moving algorithm) of the shill which can produce refined shortcut route based on the simple U-turn route. This is called ‘consistent moving’ strategy. The mathematical analysis gives a bound

Based on our approach, there are a number of possible future studies: (1) The goal of this paper is to prove that one intelligent shill can guide the group to synchronization and there exists such a strategy for the shill. We believe that the efficiency of the algorithm for the shill can be improved by changing the heuristics and some parameter settings. (2) In this paper, the desired heading is zero. Actually, theorem 1 is true for any desired heading

In fact, the intention of this paper is not only to complete a special coordination task of a specific MAS, but also to suggest a nondestructive intervention method by adding intelligent shills for other MASs as well. For example, adding intelligent shills in a multi-player group to change the group decision, adding intelligent shills in crowd to avoid panic, adding intelligent shills on internet to intervene public opinion, etc. On the other hand, this idea also provides a possible solution for the design of man-made MASs. To design a self-organized MAS to achieve an expected collective behavior requires a lot of skills in the design of local interaction rules. Adding one or more intelligent shills to coordinate the collective behavior can release designers from the hard work of subtle design.

In one word, this approach demonstrates the importance of using feedback for intervention. Intelligent shills can significantly outperform simple fixed-heading agents. As more attention has been paid to control and intervention of collective behavior, we need to design intelligent strategies for effective and efficient intervention for complex multi-agent systems and varieties of intervention purposes, not limited to synchronization only, but other patterns of collective movements as well.

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The authors thank the anonymous reviewers for their helpful comments and suggestions. We also appreciate comments from Prof. John Holland, discussions on noise and fixed-heading shills from Prof.Zhixin Liu, revision ideas from Prof.Ming Li. Thanks to Prof.Yuhong Feng, Dr.Mingjia Guo and Dr. Xiaodong Yang for their careful proof reading.