Vicsek model by time-interlaced compression: A dynamical computable information density

Collective behavior, both in real biological systems and in theoretical models, often displays a rich combination of different kinds of order. A clear-cut and unique definition of “phase” based on the standard concept of the order parameter may therefore be complicated, and made even trickier by the lack of thermodynamic equilibrium. Compression-based entropies have been proved useful in recent years in describing the different phases of out-of-equilibrium systems. Here, we investigate the performance of a compression-based entropy, namely, the computable information density, within the Vicsek model of collective motion. Our measure is defined through a coarse graining of the particle positions, in which the key role of velocities in the model only enters indirectly through the velocity-density coupling. We discover that such entropy is a valid tool in distinguishing the various noise regimes, including the crossover between an aligned and misaligned phase of the velocities, despite the fact that velocities are not explicitly used. Furthermore, we unveil the role of the time coordinate, through an encoding recipe, where space and time localities are both preserved on the same ground, and find that it enhances the signal, which may be particularly significant when working with partial and/or corrupted data, as is often the case in real biological experiments.

Figure 1

Mean $\langle \mathrm{\Phi }\rangle$ of the order parameter $\mathrm{\Phi }=|{\sum }_i{e}^{i{\theta }_i}/N|$ and its fluctuations ${\sigma }^2=\langle {(\mathrm{\Phi }-\langle \mathrm{\Phi }\rangle )}^2\rangle$ for several system sizes $W$, as a function of noise strength $\eta$, for intrinsic (top) and extrinsic (bottom) cases.

Figure 2

Procedure to get the Z value, i.e., the coordinate along the Z-order curve which is a one-dimensional spanning of a multidimensional space, preserving locality. Each point has a Z value which is obtained by interleaving the bits of the $x$ coordinate with the bits of the $y$ coordinate. This, in two dimensions, can be done in two different ways: Bits can be interleaved in the order $xyxyxy\cdots$ ($x$ first) or in the order $yxyxyx\cdots$ ($y$ first). The two possibilities are illustrated for two different points, using colors (green and blue) to make appreciable the choice of the interleaving order. One choice gives us the solid curve; the other gives us the dashed curve. In the example, the distance (along the Z curve) between the two points depends on the choice of the order: It is $38-7=31$ for $x$-first order and $25-11=14$ for $y$-first order.

Figure 3

Comparing STC (red curve) and ITC (green curve) schemes. STC simply places the evolving configurations one behind the other after a two-dimensional Z order spanning over the cells of a single time step. Differently, ITC interlaces bits of configurations at different times following a three-dimensional Z-order curve. Note that the numbers of 1s and 0s are identical and only the positions in the output one-dimensional string are different.

Figure 4

The trend of $Q$ for $T=1$ by varying the cell size $b$. For finite-size systems, $Q$ tends to trivial value 0 for $b\to 0$ (the few filled cells are randomly distributed) and $b\to l$ (each cell is filled with a 1); consequently, since $Q\ge 0$, there must be a maximum for some intermediate value, in this case, $b=1$. (Data are for VM simulations with intrinsic noise; parameters are $\rho =2, W=128$, and $l=64$.)

Figure 5

(a) $Q$ computed by the simple time-concatenation STC scheme (red curves) and by the Z order in space-time with the ITC scheme (green curves) compared with $Q$ computed on a single configuration (black curve $T=1$). Polarization $\mathrm{\Phi }$ is the dashed gray line. The ITC scheme gives a $Q$ higher than STC already at $T=2$, and its $Q$ increases significantly faster than STC as $T\to 64$ (insets). (b) Evolution of $Q$ variability range due to permutations. We show that STC and ITC schemes produce, not only on average but also step by step, $Q$ values that are always well separated. (Data are for VM simulations with intrinsic noise; parameters are $N=32768, W=128, l=64$, and $T=64$.)

Figure 6

Consequence for $Q$ when the sequences of $T$ length are built by skipping $\mathrm{\Delta }t$ time steps. The ITC scheme shows a decay of $Q$ with varying $\mathrm{\Delta }t$ that, as we expect, slows down as the transition is approached. Differently, the STC scheme (inset) does not show meaningful information. (Data are for VM simulations with intrinsic noise; parameters are $N=32768, W=128, l=64$, and $T=64$.)

Figure 7

Results obtained with VM simulations, both with intrinsic (top) and extrinsic (bottom) noise. Left: $\langle Q\rangle$ vs $\eta$ computed with the ITC scheme at increasing system size $W$ and time length $T$; dashed black curves refer to polarization $\mathrm{\Phi }$ computed for $W=128$. The inset shows the two peaks of $|dQ/d\eta |$ for $W=128$ located at ${\eta }_b$ and ${\eta }_c$ (and the local maximum at ${\eta }^{*}$) which delimit the phase of traveling bands. The density structures are shown in the strips below. We estimate ${\eta }_c\simeq 0.48$ and ${\eta }_b\simeq 0.34$ for intrinsic noise and ${\eta }_c\simeq 0.62$ and ${\eta }_b\simeq 0.56$ for extrinsic noise. Right: Differences compared with the STC scheme with varying noise strength.

Figure 8

(a) The effect of data corruption on $\langle Q\rangle$ vs $\eta$. (b) Dependence on $\mu$ of the difference between $\langle Q_I\rangle$ and $\langle Q_S\rangle$ for some value of noise strength $\eta$. (Data are for VM simulations with extrinsic noise; parameters are $N=32768, W=128, l=64$, and $T=64$.)