Current research methods directed towards measuring the influence of specific agents on the dynamics of a large-scale multi-agent system (MAS) rely largely on the notion of controllability of the full-order system, or on the comparison of agent dynamics via a user-defined macroscopic system property. However, it is known that several large-scale multi-agent systems tend to self-organize, and their dynamics often reside on a low-dimensional manifold. The proposed framework uses this fact to measure an agent's influence on the macroscopic dynamics. First, the minimum embedding dimension that can encapsulate the low-dimensional manifold associated with the self-organized dynamics is identified using a modification of the method of false neighbors. Second, the full-order dynamics are projected onto the local low-dimensional manifold using Krylov subspace-inspired model order reduction techniques. Finally, an existing controllability-based metric is applied to the local reduced-order representation to measure an agent's influence on the self-organized dynamics. With this technique, one can identify regions of the state space where an agent has significant local influence on the dynamics of the self-organizing MAS. The proposed technique is demonstrated by applying it to the problem of vehicle cluster formation in traffic, a prototypical self-organizing system. As a result, it is now possible to identify regions of the roadway where an individual driver has the ability to influence the dynamics of a self-organized traffic jam.