AbstractCollective behavior is an emergent property of numerous complex systems, from financial markets to cancer cells to predator-prey ecological systems. Characterizing modes of collective behavior is often done through human observation, training generative models, or other supervised learning techniques. Each of these cases requires knowledge of and a method for characterizing the macro-state(s) of the system. This presents a challenge for studying novel systems where there may be little prior knowledge. Here, we present a new unsupervised method of detecting emergent behavior in complex systems, and discerning between distinct collective behaviors. We require only metrics, d(1), d(2), defined on the set of agents, X, which measure agents’ nearness in variables of interest. We apply the method of diffusion maps to the systems (X, d(i)) to recover efficient embeddings of their interaction networks. Comparing these geometries, we formulate a measure of similarity between two networks, called the map alignment statistic (MAS). A large MAS is evidence that the two networks are codetermined in some fashion, indicating an emergent relationship between the metrics d(1) and d(2). Additionally, the form of the macro-scale organization is encoded in the covariances among the two sets of diffusion map components. Using these covariances we discern between different modes of collective behavior in a data-driven, unsupervised manner. This method is demonstrated on empirical fish schooling data. We show that our state classification subdivides the known behaviors of the school in a meaningful manner, leading to a finer description of the system’s behavior.Author summaryMany complex systems in society and nature exhibit collective behavior where individuals’ local interactions lead to system-wide organization. One challenge we face today is to identify and characterize these emergent behaviors, and here we have developed a new method for analyzing data from individuals, to detect when a given complex system is exhibiting system-wide organization. Importantly, our approach requires no prior knowledge of the fashion in which the collective behavior arises, or the macro-scale variables in which it manifests. We apply the new method to an agent-based model and empirical observations of fish schooling. While we have demonstrated the utility of our approach to biological systems, it can be applied widely to financial, medical, and technological systems for example.
Mathematical Model in relation to the Significance of Fish Schooling
