Various Vicsek Models with Underlying Network Characteristics

TL;DR

The paper investigates how restricted perception and hierarchical heterogeneity affect synchronization in Vicsek-type swarms by mapping interactions to complex networks. It introduces LPVM and LVVM variants and shows that homogeneous rules yield ER-like networks while heterogeneity drives BA-like, scale-free structures; The final collective order is governed by the initial average degree and follows a universal stretched-exponential relation $v_a = 1 - a exp(-b (avgd)^c)$ across all models; The LVVM variant demonstrates the strongest robustness to system size and agent heterogeneity, offering design insights for robust multi-agent coordination.

Abstract

Collective motion is a fundamental phenomenon in biological swarms. As a framework for studying synchronization in motions, the Vicsek model is simple and efficient, assuming isotropic interactions with a complete field of view. Drawing inspiration from natural swarms, we incorporate realistic constraints into the model. By analysing the interaction structures from the complex network perspective, we demonstrate that models with the homogeneous interaction rules naturally form Erdos-Renyi networks, whereas the introduction of heterogeneity leads to Barabasi-Albert networks. Furthermore, we discover that the model's synchronization is fundamentally governed by the average degree of the interaction network. Through a comparative analysis across these topologies, we identify a stretched-exponential relationship between the average degree and the synchronization metrics.

Figures (11)

• Figure 3: Degree distributions of the interaction networks. (a) SVM and (b) RVM display approximately normal degree distributions, fitted by Gaussian curves. In contrast, (c) LPVM and (d) LVVM exhibit heavy-tailed power-law distributions $P(d_i) \sim d_i^{-\delta}$ on log-log scales, characteristic of heterogeneous scale-free networks. The insets illustrate typical network snapshots. Parameters: $\langle d \rangle = 5.0$, $N = 100$, domain size $10 \times 10$.
• Figure 4: Impact of initial connectivity on synchronization and network evolution. Top row: The stationary synchronization order parameter $v_a$ as a function of the initial average degree $\langle d \rangle_i$. Bottom row: The final average degree $\langle d \rangle_f$ versus the initial value $\langle d \rangle_i$. Columns correspond to different swarm sizes $N = 50, 100, 250, 500$.
• Figure 5: Final synchronization level $v_a$ as a function of the initial average degree ${\left\langle d \right\rangle _{i}}$ for LPVM (red curves) and LVVM (blue curves). Columns correspond to different system sizes $N$, rows to different fractions of heterogeneous agents $\gamma$, and line styles to different sensing factors $\sigma$.
• Figure 6: Semi-log plots of $1-v_a$ versus the initial average degree ${\left\langle d \right\rangle _{i}}$ for the four models, where symbols denote simulation results and solid lines the corresponding stretched-exponential fits.
• Figure 7: Final synchronization level $v_a$ for the four models SVM, RVM, LPVM and LVVM. The left panel shows the dependence of $v_a$ on the system size $N$. The right panel shows the dependence of $v_a$ on the initial average degree $\langle d\rangle_{i}$ for each model, with symbols indicating $N = 50, 100, 250, 500$.