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

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.


Fixed-heading-shill scenario
The updating rule (without the noise) of the heading of normal agents, i.e., can be written as the following form: where i is the imaginary unit.Suppose each normal agent has πr 2 ρ n normal agent neighbors and πr 2 ρ s shill neighbors, where ρ n is the density of normal agents and ρ s is the density of shills.Therefore, for normal agent k, we have: Furthermore, It follows from Eq (4) and the Euler's formula that According to the nature of the imaginary, we have: For convenience, let , through equation ( 9), the convergent time step T satisfies 1 : Therefore, Figure 1 shows the comparison between simulation results and mean-field results for different density ratios between shills and normal agents in the fixed-heading scenario.The 1 The system without noise is regarded as reaching consensus when max corresponding numerical values are given in Table 1.In general, the mean-field results conform to the simulation results especial for the lower bounds.And max T − min T is relatively small.It proves that r does not have significant influence on the soft control performance in the fixed-heading scenario.

Evolvable-heading-shill scenario
Similarly, for each normal agent, suppose it has πr 2 ρ n normal agent neighbors and πr 2 ρ s shill neighbors.Through the updating rule (without noise) of the heading of normal agents, i.e., Eq 1, we have Figure 2 shows the result of comparisons between simulation results and corresponding mean-field results for different density ratios between shills and normal agents in the evolvable-heading scenario.Mean-field results are roughly in accordance with simulation results especially.So it proves that r does not have significant influence on the soft control performance in the evolvable-heading scenario.
We can see from Fig. 5D of the main text that: for θ s > π/2, increasing θ s decreases ∆θ.This means a larger perturbation perturbs less the system.We will give mean-field proof for this counter-intuitive phenomena in the following.
By Eq (13), regard ∆θ as a function of θ s , then we have: Therefore, when θ s > θ 0 + arccos(−ρ s /ρ n ), ∆θ decreases with the increase of θ s .Especially, ρ s : ρ n is set to be 1 : 9 in the Fig. 5D of the main text, when θ s > θ 0 + arccos(−ρ s /ρ n ) ≈ 0.5354π, ∆θ decreases with the increase of θ s .So our mean-field proof for this counter-intuitive critical point is consistent with the simulation result.

Fig 1 .
Fig 1. Simulation results v.s.mean-field results for different density ratios between shills and normal agents in the fixed-heading scenario.ρ n = 1, l = 25, v = 0.03, η = 0, θ 0 = 0 and θ s = π/4.Dotted lines represent corresponding upper and lower bounds calculated by mean-field analysis.The simulation results are measured as the average of 100 runs on random position distributions of normal agents 1 .

Fig 2 .
Fig 2. Simulation results v.s.mean-field results for different density ratios between shills and normal agents in the evolvable-heading scenario.ρ n = 1, l = 25, v = 0.03, η = 0, θ 0 = 0 and θ s = π/4.The simulation results are measured as the average of 100 runs on random position distributions of normal agents.

Table 1 .
Simulation values v.s.mean-field value