stereoscopic experiments on natural flocks of european 
 starlings were performed in the field in rome using three high speed machine 
 vision cameras shooting at 170 fps . the stereoscopic video acquisitions were 
 then processed using a novel purpose - built three - dimensional tracking algorithm 
 based on a recursive global optimization method . this algorithm is extremely powerful , allowing for the 
 reconstruction of full 3d trajectories of all individuals in groups of several 
 hundreds individuals . we collected 3d data from 12 flocking events with sizes 
 ranging from 50 to 600 individuals , and lasting from 2s to 6s ( for details on 
 the experiments and the dataset see table s1 and [ 10 , 
 29 ] ) . to avoid interference from 
 birds flapping , which occurs at frequency  10 hz , we subsampled all the 
 3d sequences so that two snapshots are separated by dt 
 = 0.1 s. the instantaneous flight orientations were estimated by 
 si(t)=[ri(t+dt)ri(t)]/ri(t+dt)ri(t). 
 to avoid overlap between two subsequent evaluations of si(t ) , 
 we used dt = 2dt = 0.2 s. the lower 
 sampling rates of fig . 1c , were obtained by 
 taking dt = 0.2 , 0.3 , and 0.4 s. data were simulated in three dimensions with the continuous vicsek model 
 of eq . the positions 
 ri 
 of individuals are updated according to dri / dt=0si , 
 with 0 = 1 . the simulations were set in a 8 
  8  8 box with periodic boundary conditions , and 
 n = 512 birds , so that density is exactly 1 . we set 
 2t=0.15 
 to obtain a polarization p  0.99 similar to natural 
 flocks . 1 was integrated 
 using euler s method with a simulation step 
 dtsim = 0.01 that is much smaller than any other 
 time scale in the system . the interaction range 
 nc varied from 7 to 25 , and 
 the interaction strength was picked so that 
 jnc = 1.5 , hence 
 relax = 
 ( jnc ) 
 ~ 0.7 . the flocks were first brought to a steady state before taking 
 snapshots for analysis . the polarization p quantifies the level of order in the 
 system . when p  1 , we can expand each 
 si 
 around the common direction of flight n(1/np)isi . this expansion gives si=i+1i2ni+(1i2/2)n , 
 with ni=0 . 
 at leading order in i1 , 
 eq . similarly , the equilibrium distribution ( eq . 8) can be expanded into ( 10)p()=1ze(j / t)ijijij . since this distribution is gaussian , z can be calculated 
 analytically and reads : z=(2t / j)(n1)k>0k1 , 
 where k are the eigenvalues of the matrix 
 ij . the equilibrium inference is performed by maximising 
 the likelihood of the data given by eq . 10 over the parameters 
 nc and 
 ( j / t ) ( see si for detailed 
 formulas ) . the dynamical inference based on euler s rule is 
 implemented by maximising the likelihood p({i(t+dt)}|{i(t ) } ) 
 calculated from euler s formula ( eq . 5 , rewritten as 
 (t+dt)=ejdt(t)+ , 
 where  
 is a zero - mean gaussian vector of covariance =4t0dtdu ejueju = x1 . 
 the conditional likelihood p({i(t+dt)}|{i(t ) } ) 
 now reads ( 12)det(x)(2)ne12[(t+dt)ejdt(t)]x[(t+dt)ejdt(t ) ] . 
 depending on whether one uses euler s or exact integration rules , eq . 11 or 12 is maximised over 
 j , t and 
 nc ( see si for detailed 
 formulas ) . in all three inference procedures , the parameters are learned for each 
 time t. then the median and the associated standard error are 
 calculated for each flocking event . the data that support the plots within this paper and other findings of 
 this study are available from the corresponding author upon request . stereoscopic experiments on natural flocks of european 
 starlings were performed in the field in rome using three high speed machine 
 vision cameras shooting at 170 fps . the stereoscopic video acquisitions were 
 then processed using a novel purpose - built three - dimensional tracking algorithm 
 based on a recursive global optimization method . this algorithm is extremely powerful , allowing for the 
 reconstruction of full 3d trajectories of all individuals in groups of several 
 hundreds individuals . we collected 3d data from 12 flocking events with sizes 
 ranging from 50 to 600 individuals , and lasting from 2s to 6s ( for details on 
 the experiments and the dataset see table s1 and [ 10 , 
 29 ] ) . to avoid interference from 
 birds flapping , which occurs at frequency  10 hz , we subsampled all the 
 3d sequences so that two snapshots are separated by dt 
 = 0.1 s. the instantaneous flight orientations were estimated by 
 si(t)=[ri(t+dt)ri(t)]/ri(t+dt)ri(t). 
 to avoid overlap between two subsequent evaluations of si(t ) , 
 we used dt = 2dt = 0.2 s. the lower 
 sampling rates of fig . 1c , were obtained by 
 taking dt = 0.2 , 0.3 , and 0.4 s. the positions 
 ri 
 of individuals are updated according to dri / dt=0si , 
 with 0 = 1 . the simulations were set in a 8 
  8  8 box with periodic boundary conditions , and 
 n = 512 birds , so that density is exactly 1 . we set 
 2t=0.15 
 to obtain a polarization p  0.99 similar to natural 
 flocks . 1 was integrated 
 using euler s method with a simulation step 
 dtsim = 0.01 that is much smaller than any other 
 time scale in the system . the interaction range 
 nc varied from 7 to 25 , and 
 the interaction strength was picked so that 
 jnc = 1.5 , hence 
 relax = 
 ( jnc ) 
 ~ 0.7 . the flocks were first brought to a steady state before taking 
 snapshots for analysis . the polarization p quantifies the level of order in the 
 system . when p  1 , we can expand each 
 si 
 around the common direction of flight n(1/np)isi . this expansion gives si=i+1i2ni+(1i2/2)n , 
 with ni=0 . 
 at leading order in i1 , 
 eq . similarly , the equilibrium distribution ( eq . 8) can be expanded into ( 10)p()=1ze(j / t)ijijij . since this distribution is gaussian , z can be calculated 
 analytically and reads : z=(2t / j)(n1)k>0k1 , 
 where k are the eigenvalues of the matrix 
 ij . the equilibrium inference is performed by maximising 
 the likelihood of the data given by eq . 
 10 over the parameters 
 nc and 
 ( j / t ) ( see si for detailed 
 formulas ) . the dynamical inference based on euler s rule is 
 implemented by maximising the likelihood p({i(t+dt)}|{i(t ) } ) 
 calculated from euler s formula ( eq . 5 , rewritten as 
 (t+dt)=ejdt(t)+ , 
 where  
 is a zero - mean gaussian vector of covariance =4t0dtdu ejueju = x1 . 
 the conditional likelihood p({i(t+dt)}|{i(t ) } ) 
 now reads ( 12)det(x)(2)ne12[(t+dt)ejdt(t)]x[(t+dt)ejdt(t ) ] . 
 depending on whether one uses euler s or exact integration rules , eq . 11 or 12 is maximised over 
 j , t and 
 nc ( see si for detailed 
 formulas ) . in all three inference procedures , the parameters are learned for each 
 time t. then the median and the associated standard error are 
 calculated for each flocking event . the data that support the plots within this paper and other findings of 
 this study are available from the corresponding author upon request .