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 .
 the correlated motion of flocks is an instance of global order emerging 
 from local interactions . 
 an essential difference with analogous ferromagnetic 
 systems is that flocks are active : animals move relative to each other , 
 dynamically rearranging their interaction network . 
 the effect of this 
 off - equilibrium element is well studied theoretically , but its impact on actual 
 biological groups deserves more experimental attention . here , we introduce a 
 novel dynamical inference technique , based on the principle of maximum entropy , 
 which accodomates network rearrangements and overcomes the problem of slow 
 experimental sampling rates . 
 we use this method to infer the strength and range 
 of alignment forces from data of starling flocks . 
 we find that local bird 
 alignment happens on a much faster timescale than neighbour rearrangement . 
 accordingly , equilibrium inference , which assumes a fixed interaction network , 
 gives results consistent with dynamical inference . 
 we conclude that bird 
 orientations are in a state of local quasi - equilibrium over the interaction 
 length scale , providing firm ground for the applicability of statistical physics 
 in certain active systems . 
Materials and Methods
Flocking data
Simulated data
Spin-wave approximation
Maximum likelihood Inference
Data Availability
Supplementary Material