Fig. 5: Peak susceptibility at a critical group size.

In the left panel, scatter points show the group response to external forces in experiments for F_ext in the 0.30 ± 0.05 mN range for the four cargo sizes (with diameters of 0.3, 1, 2, and 4 cm) and average number of ants, N = 3 for the small cargo, N = 18 ants for the intermediate cargo, N = 34 ants for the large cargo, and N = 54 ants for the largest cargo. The error bars are the standard error of the mean. The continuous curves are the normalized response curves obtained from the numerical simulations for group sizes N = 2–70, averaged over 100 runs each for a range of forces between 0.25 and 0.35 mN. The response curve for F_ext = 0.30 mN is the dashed black curve in the figure. The inset plots show the angular velocity distribution of the cargo in the disordered state (cargo size: small), at the transition zone (cargo size: intermediate), and in the ordered state (cargo size: very large), all empirically derived from experiments in the absence of any external force from the robot (also shown in Fig. 2). On the right, snapshots of cargo trajectories from experiments are displayed for the three group sizes before, during, and after the application of the external force. The trajectories exhibit fluctuations before and after for the smallest group size (A), suggesting no consensus or persistence. For the intermediate group size (B), the group demonstrates consensus by switching its response to the external force and persisting after the delay time has elapsed. In contrast, the largest group (C) resists the force and continues moving along its original direction of motion. The time stamps, denoted as \({t}^{{\prime} }\), represent the time of force application, while Δt signifies the duration of force application. Additionally, τ denotes the delay time for measuring group persistence. A fixed time delay of τ = 5 s is utilized in both experiments and simulations, and the maximal response is used to normalize values in both cases. A particular instance of the angular position of the cargo, θ(t)_{without force}, in the absence of the external force, is shown in cyan. The angular data for all the cargo sizes has been rotated to reflect that the external force always points in the same direction. Source data are provided in the Source data file.