Anization and complexity. For example, if a particular set of states and their dependent structure correspond to a highly robust yet agile collective motion, then one can use this information theoretic inspired metrics for engineering the agent-to-agent interactions rather than focusing on the highly expensive computation strategy for an agent based model to achieve a certain degree of emergence, self-organization and complexity. We clarify this further in discussion section of manuscript. This framework can also help to study the evolution of the motion of various animal groups in SIS3 supplement nature to better understand their means to achieve energy efficiency46. The remaining of this paper is organized as follows: In the first section of results, we present our framework to extract the possible states in the collective motion and the strategy to build the corresponding energy landscape for transitions between them. To demonstrate the benefits of our approach, we first apply this strategy to quantify the energy landscape of a self-organizing model of a simulated group of agents based on local interactions among its individuals. Next, we define the missing information for the group structure. In the second section, we apply the same framework to three natural groups of swimming bacteria, flying pigeons and ants and study their energy landscapes. We define emergence, self-organization, and quantify the complexity of a collective motion based on these newly introduced metrics. For the case of bacteria, we concluded that adding chemoattractant to the environment, decreases the number of possible states for the group motion and the free energy landscape is smoother compared to the case without chemoattractant. Finally, the discussion section concludes the paper and outlines some future research directions.ResultsEstimating the free energy landscape for a collective motion based on identified spatio-temporal structural states of the group. The agents move coherently within a collective group while interactingwith their immediate neighbors and determine their overall trajectory of motion with respect to other agents. Consequently, the group’s structure evolves among various spatio-temporal structural states. We can identify and extract these states of the group moving in three-dimensional space from the individuals’ trajectories using our algorithm explained as follow (see the free energy landscape section in the Methods for more details). First, we divide the trajectories of all the individuals into equal sub-intervals of a specific lenght. Next, we compute the multivariable probability distribution Necrosulfonamide manufacturer function of the location of all the individuals in every sub-interval (Fig. 1a). We use Kantorovich metric (see equation (5) in free energy landscape section in Methods) to cluster these subinterval time series based on their similarities and closeness in the probability distribution function (Fig. 1b). Each cluster contains subintervals with similar dynamical configuration and can be interpreted as a distinct state.Scientific RepoRts | 6:27602 | DOI: 10.1038/srepwww.nature.com/scientificreports/Figure 2. Various collective patterns of a simulated model of a group of agents moving in a three dimensional space. (a) Torus: Individuals rotate around a center point within an empty space (See the simulation section in the Methods for more details about the model). (b) Swarm: Individuals show attraction and repulsion behavior between themselves and there is no ori.Anization and complexity. For example, if a particular set of states and their dependent structure correspond to a highly robust yet agile collective motion, then one can use this information theoretic inspired metrics for engineering the agent-to-agent interactions rather than focusing on the highly expensive computation strategy for an agent based model to achieve a certain degree of emergence, self-organization and complexity. We clarify this further in discussion section of manuscript. This framework can also help to study the evolution of the motion of various animal groups in nature to better understand their means to achieve energy efficiency46. The remaining of this paper is organized as follows: In the first section of results, we present our framework to extract the possible states in the collective motion and the strategy to build the corresponding energy landscape for transitions between them. To demonstrate the benefits of our approach, we first apply this strategy to quantify the energy landscape of a self-organizing model of a simulated group of agents based on local interactions among its individuals. Next, we define the missing information for the group structure. In the second section, we apply the same framework to three natural groups of swimming bacteria, flying pigeons and ants and study their energy landscapes. We define emergence, self-organization, and quantify the complexity of a collective motion based on these newly introduced metrics. For the case of bacteria, we concluded that adding chemoattractant to the environment, decreases the number of possible states for the group motion and the free energy landscape is smoother compared to the case without chemoattractant. Finally, the discussion section concludes the paper and outlines some future research directions.ResultsEstimating the free energy landscape for a collective motion based on identified spatio-temporal structural states of the group. The agents move coherently within a collective group while interactingwith their immediate neighbors and determine their overall trajectory of motion with respect to other agents. Consequently, the group’s structure evolves among various spatio-temporal structural states. We can identify and extract these states of the group moving in three-dimensional space from the individuals’ trajectories using our algorithm explained as follow (see the free energy landscape section in the Methods for more details). First, we divide the trajectories of all the individuals into equal sub-intervals of a specific lenght. Next, we compute the multivariable probability distribution function of the location of all the individuals in every sub-interval (Fig. 1a). We use Kantorovich metric (see equation (5) in free energy landscape section in Methods) to cluster these subinterval time series based on their similarities and closeness in the probability distribution function (Fig. 1b). Each cluster contains subintervals with similar dynamical configuration and can be interpreted as a distinct state.Scientific RepoRts | 6:27602 | DOI: 10.1038/srepwww.nature.com/scientificreports/Figure 2. Various collective patterns of a simulated model of a group of agents moving in a three dimensional space. (a) Torus: Individuals rotate around a center point within an empty space (See the simulation section in the Methods for more details about the model). (b) Swarm: Individuals show attraction and repulsion behavior between themselves and there is no ori.