# The Role of Sensory Cues in Collective Dynamics: A Study of Three-Dimensional Vicsek Models
## 3 Results
The largest cluster size is limited to the total number of agents \(N=1250\), which occurs when the entire group forms a single large cluster. For both auditory and visual modalities, when the sensing region is small (\(\phi\leq 3\pi/15\)), the cohesion values are generally low. In particular, in the auditory mode (Figure 4d), cohesion values remain consistently low across all noise levels and sensing angles. As the sensing region increases (\(3\pi/15\leq\phi\leq 10\pi/15\)), cohesion values in the visual mode (Figure 4e) are significantly higher and remain robust to some extent even under increased noise intensity. This observed discrepancy between the two modalities can be attributed to the formation of multiple clusters within this range of sensing angles in the visual mode. Within these clusters, particles are densely packed, leading to higher cohesion values. However, since these clusters are disjoint, the size of the largest cluster is relatively small, as seen for small sensing angles (\(3\pi/15\leq\phi\leq 5\pi/15\)) in Figure 4h. The formation of multiple clusters in the visual mode can lead to lower group alignment, as these individual clusters may move in different directions. However, the high polarization and moderate cohesion values shown in Figure 4b,e (for example at \(\eta=0.3\) and \(\phi=5\pi/15\)) indicate that these clusters are not incredibly distant from each other, an effect expected with periodic boundary conditions. In the auditory mode, the entire group typically forms a single large cluster (Figure 4g), specifically when \(\phi\geq 5\pi/15\). However, the particles are more evenly distributed in space, resulting in relatively lower cohesion values compared to the visual mode. In the composite sensing mode, shown in Figure 4f, we observe that group cohesion is lower, and particles begin to form a single large cluster at a relatively smaller sensing angle of \(\phi\geq 5\pi/15\) when compared with the visual mode, which has large cluster sizes when \(\phi\geq 6\pi/15\). A summary of the key findings for each modality is provided in Table 1.  
These findings suggest that the group behavior in the composite mode closely resembles that of the auditory mode, where particles coalesce into a single large cluster with common heading direction, resulting in a near-perfect polarization value. However, the particles within the cluster are uniformly dispersed throughout the entire arena, resulting in reduced cohesion. Noticeably, in the deterministic case (\(\eta=0\)) of the composite mode, cohesion values in the are slightly higher, resembling a perfect combination of the \begin{table}
\begin{tabular}{l l l} \hline \hline  & \multicolumn{2}{c}{**Summary of Results**} \\ \hline
**Mode** & **Polarization** & **Cohesion** \\ \hline Auditory & decreases with increasing \(\eta\) & very low overall \\  & smooth phase transition with increasing \(\phi\) & \\ \hline Visual & decreases with \(\eta\) & low overall \\  & sharp phase transition with increasing \(\phi\), then decreases & higher for region of \(\phi\), even with increasing noise \\  & high at specific range of \(\phi\) & \\ \hline Composite & decreases with \(\eta\). & very low overall \\  & smooth phase transition at smaller \(\phi\) & slight increases in the deterministic case \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of results. corresponding auditory and visual modes, but deteriorate when noise is introduced. In other words, in the composite sensing mode, auditory mechanics play a dominant role in determining group behavior, especially in the presence of noise.