Systems showing collective motion are partially described by a distribution of positions and a distribution of velocities. While models of collective motion often focus on system features governed mostly by velocity distributions, the model presented in this paper also incorporates features influenced by positional distributions. A significant feature, the size of the largest connected component of the graph induced by the particle positions and their perception range, is identified using a 1-d self-propelled particle model (SPP). Based on largest connected components, properties of the system dynamics are found that are time-invariant. A simplified macroscopic model can be defined based on this time-invariance, which may allow for simple, concise, and precise predictions of systems showing collective motion.
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