Self-propelled particle (SPP) models are increasingly used to study complex collective behaviours such as flocking and swarming in active systems. The Vicsek model is a widely used SPP model that exhibits a phase transition from an isotropic phase to a polar ordered phase under desired conditions. Recent studies indicated that such phase transition is related to the clusters formed in the system. However, a comprehensive cluster analysis under different conditions has not yet been performed on the Vicsek system. This study presents detailed cluster structure analysis of the Vicsek system and investigates the connections between clusters and the effects of key controlling variables such as noise and density on the order transition. It is shown that local clusters can emerge purely based on alignment interactions without the need for additional cohesion, and that cluster formation may induce density inhomogeneities in the system but can have a positive effect on the efficiency of alignment interaction. In addition, the clusters are assumed to be complex networks by defining arrangement interactions between two active particles as edges connecting two vertices, and quantitative analysis is performed through complex network characterization. Interestingly, as the controlling variables change, the clusters evolve into different types of complex networks, and the evolutions are further understood by analysing the connectivity of subgraphs within the cluster network structure. The cluster network analysis not only improves the understanding of order transitions in active systems, but also reveals the connection between network dynamics and the collective behaviour of active systems.
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