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

RESEARCH ARTICLE How does the interaction radius affect the performance of intervention on collective behavior? Caiyun Wang1,2, Jing Han1,2* 1 LSC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China, 2 School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS 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. * Abstract The interaction radius r plays an important role in the collective behavior of many multiagent 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. Data Availability Statement: 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. 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. PLOS ONE | https://doi.org/10.1371/journal.pone.0192738 February 15, 2018 1 / 19 Interaction radius and intervention performance Competing interests: The authors have declared that no competing interests exist. 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 selforganized 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 selforganizing 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 “ (...truncated)