Commencement Program, June (1939)

https://red.mnstate.edu/commencement/1056/thumbnail.jp

Relationship between agent speed and order in constant-speed agent based simulation model with 150 agents.

<p>(<i>A</i>) Density plot of agent speed as function of rotation <i>O_r</i> and polarization <i>O_p</i>, revealing a bistable regime between the milling and the polar states for high speeds. (<i>B</i>) Normalized probability plot of polarization <i>O_p</i> as function of agent speed. (<i>C</i>) Normalized probability plot of rotation <i>O_r</i> as function of agent speed. The last two plots illustrate the bifurcation that occurs as the speed is increased, where the system transitions from a swarm state and is found either in a highly polarized state or in a milling state. (See <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002915#s3" target="_blank">Methods</a> for simulation details).</p

Structural properties.

<p>(<i>A</i>) Density plots of packing fraction and average individual speed (averaged per frame) as functions of rotation <i>O_p</i> and polarization <i>O_r</i> for 150 fish. (<i>B</i>) The plot illustrates the correlation between individual speed and local polarization estimated in two ways from the underlying density maps (the example shown in the background is for 150 fish). The stapled curves are produced by averaging across individual speeds for each value of the order parameter; the solid curves from averaging across the order parameter values for each individual speed. The local polarization of an individual fish is defined as the polarization Op restricted to the area inside a circle with radius 15.6 cm (approximately 3 body lengths) centered at the individual fish. (<i>C</i>) Average individual speed at different radial positions in the milling state. (<i>D</i>) Average rotational order parameter at the same positions. The radial division of the milling state in (<i>C</i>) and (<i>D</i>) is constructed by centering six shells outside each other, where the outermost shell has a radius defined by the distance from the group's center of mass to the median distance of the five most peripheral fish (see illustration in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002915#pcbi.1002915.s011" target="_blank">Fig. S11</a>). The width of each shell is the radius of the outer shell divided by six. The averages are calculated for each shell, where the outer shell even includes peripheral fish.</p

Time-lapse examples demonstrating transition mechanisms.

<p>(<i>A</i>) Transition initiated by interaction with the tank wall. In the first picture, the fish is in a milling state, indicated by the blue arrow, and the lower part of the group is close to the tank wall (grey line). In the second picture, the interaction with the wall has caused a local increase in density, marked by the blue region, and a few individuals have started to turn opposite the milling direction. This cascades, and in the third picture the flow of the mill is interrupted as a large proportion is breaking away from the milling direction. The result is total unraveling of the milling state and transition into the polar state, seen in the last picture. (<i>B</i>) Transition from the polar state to the milling state, initiated by individuals in the shoal. In the first picture, the group is in a polar state, signified by the red arrow. A few individuals in the front, visible in the red region, have started to turn downwards. This leads the group as a whole into a sharp right turn, and as picture two demonstrates, the group is forced into a shape with larger curvature. Now, when the individuals in front of the group can spot the back of the group, they continue the turning and start following the back, as seen in the third picture. In the final picture, as the front individuals catches up with the tail, the loop closes and the transition into the milling state is complete.</p

Dynamical states of schooling fish.

<p>(<i>A</i>) Snapshots of a group of 150 golden shiners swimming in a shallow tank. The different images (thresholded for clarity) demonstrate the typical configurations displayed by the fish school: swarm state (S), polarized state (P) and milling state (M). (<i>B</i>) Extracts of time series of order parameters for groups of 30, 70, 150, and 300 golden shiners. Polarization <i>O_p</i> (in blue) measures how aligned the fish are, while rotation <i>O_r</i> (in red) measures the degree of rotation around the center of mass of the fish shoal.</p

Transition patterns for 150 fish.

<p>(<i>A</i>) Density plot of the smallest distance from the center of mass of the fish shoal to the tank boundary as a function of rotation and polarization (d_min = 26 and d_max = 52 cm). The overlaid arrows are the averaged trajectories of all transitions in the rotation-polarization phase space. (<i>B–D</i>) Density plots of transitions from polarized to milling state (<i>B</i>), from polarized to swarm state (C) and from milling to swarm state (D). Overlaid the density plots are the corresponding velocity fields of the transition data (in the rotation-polarization phase space). Plots of the reverse transitions and group sizes 30, 70 and 300 are provided in SI.</p

Statistics of state transitions.

<p>(<i>A</i>) Fraction of time spent in the different dynamical states shown for each group size. The error bars are showing the standard deviation measured across replicates. The group of 30 fish is predominantly in the polarized state, but less time is spent in this state with increasing group size (GLM: F1,22 = 36.21, P = 4.6e-06). As the group size increases, the groups gradually spend more time in the milling state (GLM: F1,22 = 19.31, P = 0.00023). The amount of time spent in the transition regime is high, but constant, for all group sizes (GLM: F1,22 = 0.67, P = 0.42). Across all group sizes (GLM: F1,22 = 0.053, P = 0.82) little time is spent in the swarm state. (<i>B</i>) Fraction of transitions from one state to another for the different groups. The error bars are showing the standard deviation measured across replicates. For all group sizes, the polar state predominantly transitions into the swarm state compared to the milling state (GLMM: F1,23 = 58.77, P<0.0001). The swarm state is dominated by transitions into the polarized state (GLMM: F1,23 = 55.69, P<0.0001). Although these transitions are consistent across group sizes, there is a significant interaction between group size and the frequencies of transitioning from the milling state to the swarm and polarized states (GLMM: F1,22 = 13.30, P = 0.0014). While the milling state tended to transition into the polar state, for 300 fish there was a roughly equal probability of transitioning to the polar or swarm states. (<i>C</i>) Rank plots showing the probability of being in a state longer than time T_s before moving into a different regime. There was no significant difference between group sizes in the persistence of the polar state (GLMM: F1,22 = 0.58, P = 0.45), although group size increased the persistence of the transition (F1,22 = 24.52, P = 1e-04) and milling states (F1,22 = 17.54, P = 4.0e-04), and to a lesser degree, the swarm state (F1,22 = 5.31, P = 0.031).</p

Density plots of polarization vs. rotation from experiments.

<p>The data shown are averaged over all replicates for each of the groups of 30, 70, 150, and 300 golden shiners. The order parameter space is divided into four regions—swarm (S), polarized (P), milling (M), and transition (T)—each being characterized by the dominant dynamical state of the fish school in that particular region. Different values of p_min and p_max were used for each group size to emphasize the density patterns and regions with no data are colored black. The insert in the 30 fish plot shows the density plot from an experiment with 30 fish and the tank area reduced to one tenth of the original.</p

Electronic supplementary text and figures from Counteracting estimation bias and social influence to improve the wisdom of crowds

Provides additional detail on the methods used in this study, and supplementary results and figure

Data from Counteracting estimation bias and social influence to improve the wisdom of crowds

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