# The Role of Sensory Cues in Collective Dynamics: A Study of Three-Dimensional Vicsek Models
## 3 Results
At even higher noise intensities with \(\eta=0.7\) and 0.8 (Figure 3d,e), the group alignment deteriorates substantially across all three sensing modalities. Polarization remains low even when the sensing region is increased from \(\phi=5\pi/15\) (Figure 3d) to \(\phi=6\pi/15\) (Figure 3e). This low polarization indicates random walk behavior, attributable to the low density of the system (\(\rho=2\)) used in this analysis. These results indicate an interplay between noise intensity and sensing angle that dictates the resulting group behavior. For a more comprehensive analysis, we next conduct a thorough investigation using a higher density (\(\rho=10\)) and a broader set of parameters, examining all three order parameters, including cohesion and cluster size. Figure 2: Snapshots of the 3D Vicsek model implementing (a) auditory, (b) visual, and (c) composite sensing modalities, with parameters \(\rho=10\), \(\eta=0.2\), and \(\phi=6\pi/15\), at time step \(t=1500\). To compare three distinct modes, we conduct simulations by first assigning particles' initial positions and initial heading directions sampled from a uniform distribution as before. These initial conditions are generated once and remain identical in all simulations. The simulations are run over the time interval \([0,\,10,000]\), with the initial transient period of 9000 time steps excluded from the analysis. To compare group-level behavior, we calculate three observables, namely, the mean polarization, mean cohesion, and mean size of the largest cluster, averaged over the final 1000 time-steps. The results are illustrated in Figure 4. The left, middle, and right columns correspond to the auditory, visual, and composite modes, respectively. The first, second, and third rows display the mean polarization, mean cohesion, and mean largest cluster size, respectively. In each plot, the two control parameters, noise intensity \(\eta\) and sensing angle \(\phi\), are varied. Specifically, \(\eta\) ranges from 0 to 1 in increments of 0.1, and \(\phi\) ranges from 0 to \(\pi\) in increments of \(\pi/15\). Figure 3: Monte-Carlo simulations of polarization over 20 iterations for five different pairs of parameters comparing three modalities at \(\rho=2\), \(L=5\), and \(R=0.5\). By examining the three plots in Figure 4a-c, we observe that the mean polarization values are zero when \(\phi=0\), regardless of the noise intensity \(\eta\). This occurs because \(\phi=0\) implies that the particles lack any sensing capability, resulting in no interactions and  
Figure 4: Results of the order parameter analysis comparing three modalities. Mean polarization (**top row**), mean cohesion (**middle row**), and mean largest cluster size (**bottom row**) are calculated for auditory (**left column**), visual (**middle column**), and composite sensing (**right column**) modes. therefore a random walk behavior. Furthermore, a sensing angle of \(\phi=\pi\) establishes a spherical sensing neighborhood for the particles in all three sensing schemes, similar to the original 3D Vicsek model. As expected, with a sensing angle of \(\phi=\pi\), we observe that polarization decreases with increasing noise intensity, which is consistent with the findings from the original Vicsek model [23]. This trend of decreasing polarization with increasing noise is also observed for other values of sensing angles for all three modalities. Interestingly, for small noise intensities, we observe that polarization values are also sensitive to the sensing angle \(\phi\). For example, in all three sensing modalities, when the sensing region is reduced from a sphere (\(\phi=\pi\)) to a cone, the polarization values decrease. The decrease in polarization with decreasing sensing regions can be attributed to the limited number of neighbors available for each particle to gather information, which disrupts group alignment. To compare the sensing modalities, we observe notable differences in polarization values when noise intensities are low and sensing angles are small.