Collective Dynamics in the Vicsek and Vectorial Network Models Beyond Uniform Additive Noise

Abstract

In this work, we analyze the coordination of interacting individuals in two nonlinear dynamical models that are subject to a new form of noise. Specifically, we propose extensions both to the classical Vicsek model, whereby each individual averages the orientation of its geographically proximal neighbors, and to the vectorial network model, in which the selection of neighbors is random and independent of the group geometric configuration. In the traditional forms of these models, the update rule for the individuals’ orientations is affected by additive uniform noise. Motivated by biological groups in which individuals’ turn rates exhibit sporadic and large changes, we extend the uniform additive noise model to a turn rate stochastic process. Through comprehensive numerical simulations, we demonstrate the impact of such occasional large deviations (intensity and frequency), along with the role of the neighbors’ selection process, on the coordination of the group. In addition, we establish a closed-form expression for the group polarization for the vectorial network model in the vicinity of an ordered state.

Appendix

In this Appendix, we derive a closed-form expression for the polarization in the VNM-$\omega$ in the vicinity of the ordered state, that is, when the noise scaling factor $\eta$ is small.

1.1 Linear Approximation of the VNM-$\omega$

We linearize (5) in the neighborhood of the synchronized state $\theta _0$ by defining $\psi _i(k) = \theta _i(k) - \theta _0$ to obtain the linear stochastic dynamics of the individuals’ orientation disagreements, that is,

$$\begin{aligned} \psi _i(k+1)= & {} \frac{1}{K} \sum \limits _{j=1}^{K} \psi _{i_j}(k) + \eta \omega _i(k), \end{aligned}$$

(9a)

$$\begin{aligned} \omega _i(k+1)= & {} \omega _i(k)( 1 -\alpha ) + \varepsilon _i + \gamma \tau _i. \end{aligned}$$

(9b)

For convenience, we define $\psi (k) = \left[ \psi _1(k), \ldots , \psi _{N}(k)\right] ^\mathrm {T}$, and similarly, we write the intrinsic noise vector as $\omega (k) = \left[ \omega _1(k),\ldots ,\omega _{N}(k)\right] ^\mathrm {T}$, where $\mathrm {T}$ indicates matrix transposition. Using this compact notation, the linear stochastic system in (9) can be rewritten as

$$\begin{aligned} \psi (k+1)= & {} W \psi (k) + \eta \omega (k), \end{aligned}$$

(10a)

$$\begin{aligned} \omega (k+1)= & {} \omega (k)( 1 -\alpha ) + \varepsilon + \gamma \tau , \end{aligned}$$

(10b)

where $\varepsilon = \left[ \varepsilon _1,\ldots ,\varepsilon _N\right] ^\mathrm {T}$ and $\tau = \left[ \tau _1,\ldots ,\tau _N\right] ^\mathrm {T}$ are $N\times 1$ vectors.

W is a random variable, which is defined so that its rows are i.i.d. vectors with K randomly selected entries taking value 1 / K, while any other entry equals 0. This corresponds to a specific instance of the numerosity-constrained network construction defined in Abaid and Porfiri (2011), with numerosity equal to $K-1$ and weight 1 / K; in the context of numerosity-constrained networks, $K(I_N - W)$ is the graph Laplacian of the network. While linear noisy consensus over numerosity-constrained networks is addressed in Abaid et al. (2012b), the analysis presented therein pertains to a uniform additive noise, similar to the traditional VNM. In what follows, we extend the analysis to the stochastic turn rate model in (10b).

1.2 Steady-State Mean Square Deviation

We introduce the disagreement vector $\xi (k)= R \psi (k)$ through the orthogonal projection $R =I_N-\frac{1}{N} 1_N 1_N^\mathrm {T}$. Next, we multiply both sides of (10a) by R to obtain

$$\begin{aligned} \xi (k+1)= & {} R W \xi (k) + \eta R \omega (k), \end{aligned}$$

(11a)

$$\begin{aligned} \omega (k+1)= & {} ( 1 -\alpha )\omega (k) + \varepsilon + \gamma \tau . \end{aligned}$$

(11b)

For any time step k, we define $\delta (k) = \mathbf {E}\left[ \Vert \xi (k)\Vert ^2\right]$ as the expected value of the disagreement norm. In line with Porfiri (2014), Abaid et al. (2012b), Abaid and Porfiri (2011, 2012) and Porfiri (2011), we consider the autocorrelation matrix $\varXi (k) = \mathbf {E}\left[ \xi (k)\xi (k)^\mathrm {T}\right]$ to investigate the mean square behavior of the stochastic system (11). Notably, the trace of this matrix corresponds to the expected value of the disagreement norm, that is, $\delta (k)$. In the absence of noise, the consensus protocol in (10a) is said to be asymptotically mean square consentable if the disagreement dynamics in (11a), with $\eta =0$, is mean square stable, which is equivalent to $\lim _{k\rightarrow \infty }\delta (k) = 0$ for any initial condition $\xi (0)$. In the presence of noise, the disagreement norm does not approach zero, and we define the steady-state mean square deviation as $\delta _\infty = \lim _{k \rightarrow \infty }\delta (k)$.

Using well-known properties of the Kronecker algebra (Bernstein 2009), we obtain the following recursive relation for the autocorrelation matrix:

$$\begin{aligned} \mathrm {vec}\left( {\varXi }(k+1)\right)= & {} G \mathrm {vec}\left( {\varXi }(k)\right) + \eta \left( R\otimes R\mathbf {E}[W]\right) \mathrm {vec}\left( \mathbf {E}\left[ \xi (k)\omega (k)^\mathrm {T}\right] \right) \nonumber \\&+\, \eta \left( R\mathbf {E}[W] \otimes R\right) \mathrm {vec}\left( \mathbf {E}\left[ \omega (k)\xi (k)^\mathrm {T}\right] \right) \nonumber \\&+\, \eta ^2 R\otimes R \mathrm {vec}\left( \mathbf {E}\left[ \omega (k)\omega (k)^\mathrm {T}\right] \right) , \end{aligned}$$

(12)

where $\otimes$ is the Kronecker product, $\mathrm {vec}(\cdot )$ indicates vectorization, and $G= (R\otimes R) \ \mathbf {E}\left[ W \otimes W\right]$ is a ${N^2 \times N^2 }$ matrix.

Adapting results from Abaid and Porfiri (2011), it can be verified that

$$\begin{aligned}&\mathbf {E}[W] = I_N -\frac{N(K-1)}{K(N-1)} R, \\&R\mathbf {E}[W] = \mathbf {E}[W] R = \left( 1- \frac{N(K-1)}{K(N-1)}\right) R, \\&R\otimes R\mathbf {E}[W] = \mathbf {E}[W] R \otimes R = \left( 1- \frac{N(K-1)}{K(N-1)}\right) R\otimes R, \\&G=\left( 1-\frac{N(K-1)}{K(N-1)}\right) ^2(R\otimes R)+(I_N\otimes R)F, \end{aligned}$$

where the lengthy expression for the matrix F can be retrieved from Proposition 2 in Abaid and Porfiri (2011).

The evolution of the disagreement autocorrelation matrix is then obtained by iterating (12) from time 0 to time k , and using the fact that W and $\omega (s)$ are independent random variables, for all s, to find

$$\begin{aligned}&\mathrm {vec}\left( {\varXi }(k+1)\right) \nonumber \\&\quad = G^{k+1} \mathrm {vec}\left( {\varXi }(0)\right) \nonumber \\&\qquad +\, \eta \left( 1- \frac{N(K-1)}{K(N-1)}\right) \sum _{s=0}^{k-1} G^{k-s-1} R\otimes R \Big \{ \mathrm {vec}\left( \mathbf {E}\left[ \xi (s)\omega (s)^\mathrm {T}\right] \right) \nonumber \\&\qquad +\, \mathrm {vec}\left( \mathbf {E}\left[ \omega (s)\xi (s)^\mathrm {T}\right] \right) \Big \} + \eta ^2 \sum _{s=0}^{k} G^{k-s} R\otimes R \ \mathrm {vec} \left( \mathbf {E}\left[ \omega (s)\omega (s)^\mathrm {T}\right] \right) ,\qquad \end{aligned}$$

(13)

where, for consistency with the main text, we have set the initial value of the turn rate to zero, that is, $\omega (0) = {0}_N$.

From (11a) and (11b), we establish the following recursive relations:

$$\begin{aligned} \mathrm {vec}\left( \mathbf {E}\left[ \xi (s)\omega (s)^\mathrm {T}\right] \right)= & {} (1-\alpha ) (I_N \otimes R) \Big \{ \mathrm {vec}\left( \mathbf {E}\left[ \xi (s-1)\omega (s-1)^\mathrm {T}\right] \right) \nonumber \\&+\, \eta \mathrm {vec}\left( \mathbf {E}\left[ \omega (s-1) \omega (s-1)^\mathrm {T}\right] \right) \Big \} , \end{aligned}$$

(14a)

$$\begin{aligned} \mathrm {vec}\left( \mathbf {E}\left[ \omega (s)\xi (s)^\mathrm {T}\right] \right)= & {} (1-\alpha ) (R \otimes I_N) \Big \{ \mathrm {vec}\left( \mathbf {E}\left[ \omega (s-1)\xi (s-1)^\mathrm {T}\right] \right) \nonumber \\&+\, \eta \mathrm {vec}\left( \mathbf {E}\left[ \omega (s-1) \omega (s-1)^\mathrm {T}\right] \right) \Big \}. \end{aligned}$$

(14b)

Using the independence between $\varepsilon$ and $\tau$, one can also verify from (11b) that the evolution of the noise autocorrelation matrix satisfies the following relation:

$$\begin{aligned} \mathbf {E}\left[ \omega (s)\omega (s)^\mathrm {T}\right] = \frac{1 - (1-\alpha )^{2s}}{1-(1-\alpha )^2}\left( 1 + \lambda \gamma ^2\right) I_N, \end{aligned}$$

(15)

where for $i=1,\ldots ,N$, using the independence between $\varepsilon _i$ and $\tau _i$, one can verify that the variance of the zero-mean random variable $\varepsilon _i+\gamma \tau _i$ equals $1 + \lambda \gamma ^2$. By replacing (14) and (15) into (13) and iterating over k, we find

$$\begin{aligned}&\mathrm {vec}\left( {\varXi }(k+1)\right) \nonumber \\&\quad = G^{k+1} \mathrm {vec}\left( {\varXi }(0)\right) \nonumber \\&\qquad +\, 2\eta ^2 \frac{1+\lambda \gamma ^2}{1-(1-\alpha )^2}\Biggl \{ \frac{\varGamma }{1-\varGamma } \left( \sum _{s=0}^{k-1} G^{s} - \varGamma ^k \sum _{s=0}^{k-1} {\varGamma }^{-s}G^s \right) \nonumber \\&\qquad - \, (1-\alpha )^{2k}\sum _{s=0}^{k-1} (1-\alpha )^{-2s}{G}^s\frac{1-\left( \frac{\varGamma }{(1-\alpha )^2}\right) ^{s}}{1-\frac{\varGamma }{(1-\alpha )^2}} \Biggl \}\mathrm {vec}\left( R \right) \nonumber \\&\qquad +\, \eta ^2 \frac{1+\lambda \gamma ^2}{1-(1-\alpha )^2} \left( \sum _{s=0}^{k}G^{s} - (1-\alpha )^{2k} \sum _{s=0}^{k} (1-\alpha )^{-2s}{G}^s\right) \mathrm {vec} \left( R\right) , \end{aligned}$$

(16)

where $\varGamma = (1-\alpha )\left( 1- \frac{N(K-1)}{K(N-1)}\right)$.

The convergence of the series in (16) depends on the spectral radius of G, which as shown in Abaid and Porfiri (2011) corresponds to the asymptotic convergence factor of the linear consensus protocol in the absence of noise. This quantity can be exactly calculated using Theorem 2 in Abaid and Porfiri (2011), which yields

$$\begin{aligned} r_a= 1 - \frac{\left( K-1\right) \left( N + K(N^2 - N - 1)\right) }{K^2(N-1)^2}. \end{aligned}$$

(17)

Thus, when the consensus protocol in (11) satisfies $r_a < (1-\alpha )^{2} < 1$, the series in (16) admits a finite limit, that is,

$$\begin{aligned} \lim \limits _{k\rightarrow \infty }\mathrm {vec}\left( {\varXi }(k)\right) = \eta ^2 \mu \, \left( I_N\otimes I_N - G \right) ^{-1} \mathrm {vec}(R), \end{aligned}$$

(18)

where $\mu$ is defined in (8). We right multiply (18) by $\mathrm {vec}\left( I_N\right) ^\mathrm {T}$ in order to obtain the steady-state mean square deviation, that is,

$$\begin{aligned} \delta _{\infty } = \eta ^2 \mu \, \mathrm {vec}(I_N)^{\mathrm {T}} \left[ I_N \otimes I_N -G\right] ^{-1} \mathrm {vec}(R). \end{aligned}$$

(19)

The final computation of $\delta _\infty$ in (19) can be performed by employing Proposition 2 in Abaid et al. (2012b) and the related arguments. Thus, we find that when $r_a < (1-\alpha )^2 < 1$, the steady-state mean square deviation in (19) reduces to

$$\begin{aligned} \delta _\infty =\eta ^2 \frac{\mu (N-1)}{1-r_a}, \end{aligned}$$

(20)

where $\mu$ and $r_a$ depend on K and N through (8) and (17).