2.1. Graph theory

We first briefly introduce basic notions in graph theory. A direct graph (in short digraph) $\mathcal{G}=(\mathcal{V}(\mathcal{G}),\mathcal{E}(\mathcal{G}))$ consists of a finite set $\mathcal{V}(\mathcal{G})=\{\beta _1,\ldots,\beta _N\}$ of vertices and a set $\mathcal{E}(\mathcal{G})\subset \mathcal{V}(\mathcal{G})\times \mathcal{V}(\mathcal{G})$ of arcs. If a pair $(\beta _j,\beta _i)$ is contained in $ \mathcal{E}(\mathcal{G})$ , $\beta _j$ is said to be a neighbour of $\beta _i$ , and we denote the neighbour set of the vertex $\beta _i$ by $\mathcal{N}_i\;:\!=\; \{j:(\beta _j,\beta _i)\in \mathcal{E}(\mathcal{G})\}$ . For given digraph $\mathcal{G}$ , we define its corresponding adjacency matrix $\chi (\mathcal{G})=(\chi _{ij})(\mathcal{G})$ as

\begin{align*} \begin{aligned} \chi _{ij}={\begin{cases} 1& \text{if}\quad j\in \mathcal{N}_i\cup \{i\},\\ 0& \text{otherwise}. \end{cases}} \end{aligned} \end{align*}

Since it is obvious that this correspondence is a one-to-one, we can say that given a matrix $A$ consisting of zeros and ones and with diagonal components all equal to one, we can also uniquely determine its corresponding digraph, which we write it as $\mathcal{G}(A)$ . A path in a digraph $\mathcal{G}$ from $\beta _{i_0}$ to $\beta _{i_p}$ is a finite sequence $\{\beta _{i_0},\ldots,\beta _{i_p}\}$ of distinct vertices such that each successive pair of vertices is an arc of $\mathcal{G}$ . The integer $p$ which is the number of its arcs is said to be the length of the path. If there is a path from $\beta _i$ to $\beta _j$ , then vertex $\beta _j$ is said to be reachable from vertex $\beta _i$ . We say $\mathcal{G}$ contains a spanning tree if there exists a vertex such that any other vertex of $\mathcal{G}$ is reachable from it. In this case, this vertex is said to be a root.

2.2. Conservation and dissipation law

In this subsection, we show that the system (1.2) has a conservation of each speed of agent and dissipation of their velocity diameter.

Lemma 2.1. (Conservation of speeds) Suppose that $(X,V)$ is a solution to the system (1.2). Then, it follows that for each $w\in \Omega$ ,

\begin{equation*}\|v_i(t,w)\|=1,\quad t\geq 0,\quad i\in [N].\end{equation*}

Proof. Once we take an inner product $v_i$ with (1.2) $_2$ , the following relation holds:

\begin{equation*}\left \langle v_j-\frac {\langle v_j,v_i\rangle v_i}{\|v_i\|^2},v_i\right \rangle =0,\quad i,j\in [N].\end{equation*}

Therefore, we have $\frac{d}{dt}\|v_i\|^2=0$ for all $i\in [N]$ , which is our desired result.

This means that the difference in velocities is determined entirely by the angle between them. However, one can also verify that for each $w\in \Omega$ , the maximal angle

\begin{equation*}A_V(t,w)\;:\!=\; \min _{i,j\in [N]}\langle v_i(t,w),v_j(t,w) \rangle \end{equation*}

is monotonically increasing in $t\geq 0$ , provided that $\displaystyle A_V(0,w)\;:\!=\; \min _{i,j\in [N]}\langle v_i^0(w),v_j^0(w) \rangle$ is strictly positive.

Lemma 2.2. (Monotonicity of $A_V$ ) Let $(X,V)$ be a solution to the system (1.2) satisfying

\begin{equation*}{A}_V(0,w)\gt 0\end{equation*}

for some $\omega \in \Omega$ . Then, the smallest inner product between heading angles $A_V(\cdot,w)$ is monotonically increasing.

Proof. For fixed $t\geq 0$ , we denote $i_t,j_t\in [N]$ the two indices satisfying

\begin{equation*}A_V(t,w)\;:\!=\; \langle v_{i_t}(t,w),v_{j_t}(t,w)\rangle .\end{equation*}

If we define a set $\mathcal{S}$ as

\begin{equation*}\mathcal {S}\;=\!:\;\{t\geq 0\,|\,A_V(t,w)\leq 0\},\end{equation*}

it follows from the condition $A_V(0,w)\gt 0$ and the continuity of $A_V(\cdot,w)$ that there exists $T_1\in (0,\infty ]$ satisfying

\begin{equation*}[0,T_1)\cap \mathcal {S}=\emptyset .\end{equation*}

Now, define $\mathcal{T}\;:\!=\; \inf \mathcal{S}$ and suppose we have $\mathcal{T}\lt \infty$ . Then, from the continuity of $A_V$ , we have

(2.1) \begin{equation} \lim _{t\to \mathcal{T}-}A_V(t,w)=0. \end{equation}

On the other hand, the locally Lipschitz function $A_V(\cdot,\omega )$ is differentiable at almost every $t\in (0,\mathcal{T})$ . Then, we use (1.2) $_2$ to obtain

\begin{align*} \frac{dA_V}{dt}&=\langle \dot{v}_{i_t},v_{j_t}\rangle +\langle \dot{v}_{j_t},v_{i_t}\rangle \\ &=\frac{1}{N}\sum _{k=1}^N\chi _{i_tk}^\sigma \phi (\|x_k-x_{i_t}\|)\left (\langle v_k,v_{j_t} \rangle -\langle v_k,v_{i_t}\rangle \langle v_{i_t},v_{j_t}\rangle \right )\\ &\quad +\frac{1}{N}\sum _{k=1}^N\chi _{j_tk}^\sigma \phi (\|x_k-x_{j_t}\|)\left (\langle v_k,v_{i_t} \rangle -\langle v_k,v_{j_t}\rangle \langle v_{i_t},v_{j_t}\rangle \right )\\ &\geq 0, \end{align*}

where we used

\begin{equation*} \begin{aligned} &1\geq \langle v_k,v_{i_t} \rangle \geq 0 \quad \text{and} \quad \langle v_k,v_{j_t} \rangle \geq \langle v_{i_t},v_{j_t}\rangle \geq 0,\\ &1\geq \langle v_k,v_{j_t} \rangle \geq 0 \quad \text{and} \quad \langle v_k,v_{i_t} \rangle \geq \langle v_{i_t},v_{j_t}\rangle \geq 0, \end{aligned} \end{equation*}

in the last inequality. Hence, $A_V(\cdot,w)$ is monotonically increasing in $[0, \mathcal{T}]$ , which leads a contradiction to (2.1). Therefore, we have $\mathcal{T}=\infty$ and obtain the monotone increasing property of $A_V(\cdot,\omega )$ .

As a direct consequence of Lemma 2.2, the velocity diameter $D_V$ is monotonically decreasing, due to the relation between $D_V$ and $A_V$ .

Corollary 2.1. (Monotonicity of $D_V$ ) Assume that $(X,V)$ is a solution to the system (1.2) with

\begin{equation*}D_V(0,w)\lt \sqrt {2}\end{equation*}

for some $\omega \in \Omega$ . Then, the velocity diameter $D_V(\cdot,w)$ is monotonically decreasing in time.

Proof. From Lemma 2.1, we have

\begin{equation*}\|v_i-v_j\|^2=2-2\langle v_i,v_j \rangle .\end{equation*}

Therefore, one can obtain the monotone decreasing property of

\begin{equation*}D_V^2=2-2A_V\end{equation*}

by using Lemma 2.2.

2.3. Matrix formulation

Now, we reorganise the system (1.2) to a matrix formulation. For this, we first fix $w\in \Omega$ and

\begin{equation*}X\;:\!=\; (x_1,\ldots,x_N)^T,\quad V\;:\!=\; (v_1,\ldots,v_N)^T,\end{equation*}

and we use the result of Lemma 2.1, that is,

\begin{equation*}\|v_i(t,w)\|=1,\quad t\gt 0,\quad i\in [N].\end{equation*}

Then, the right-hand side of (1.2) $_2$ can be rewritten as

\begin{align*} &\sum _{j=1}^{N}\chi ^{\sigma }_{ij}\phi (\|x_i-x_j\|)\left (v_j-\langle v_j,v_i\rangle v_i\right )\\ &=\sum _{j=1}^{N}\chi ^{\sigma }_{ij}\phi (\|x_i-x_j\|)\left (v_j-v_i\right )+\sum _{j=1}^{N}\chi ^{\sigma }_{ij}\phi (\|x_i-x_j\|)\left (v_i-\langle v_j,v_i\rangle v_i\right )\\ &=\sum _{j=1}^{N}\chi ^{\sigma }_{ij}\phi (\|x_i-x_j\|)\left (v_j-v_i\right )+\frac{1}{2}\sum _{j=1}^{N}\chi ^{\sigma }_{ij}\phi (\|x_i-x_j\|)\|v_i-v_j\|^2v_i. \end{align*}

On the other hand, we also consider the Laplacian matrices

\begin{equation*}\mathcal {L}_l\;:\!=\; \mathcal {D}_l-\mathcal {A}_l, \quad 1\leq l\leq N_G,\end{equation*}

where $\mathcal{A}_l\in \mathbb{R}^{N\times N}$ and $\mathcal{D}_l\in \mathbb{R}^{N\times N}$ are defined as

\begin{equation*}(\mathcal {A}_l)_{ij}\;:\!=\; \chi ^{l}_{ij}\phi (\|x_i-x_j\|),\quad \mathcal {D}_l\;:\!=\; \text{diag}(d^l_1,\ldots, d^l_N),\quad d_i^l\;:\!=\; \sum _{j=1}^{N}\chi ^{l}_{ij}\phi (\|x_i-x_j\|),\quad i,j\in [N].\end{equation*}

Then, (1.2) can be represented by the following matrix form:

(2.2) \begin{equation} \begin{cases} \displaystyle \dot{X}=V,\\ \displaystyle \dot{V}=-\frac{1}{N}\mathcal{L}_{\sigma }V+\frac{1}{N}\mathcal{R}_{\sigma }, \end{cases} \end{equation}

where the vector $\mathcal{R}_{l}$ is defined as

\begin{equation*}\mathcal {R}_{l}\;:\!=\; (r^l_1,\ldots, r^l_N)^T\quad \text {and} \quad r^l_i\;:\!=\; \frac {1}{2}\sum _{j=1}^{N}\chi ^{l}_{ij}\phi (\|x_i-x_j\|)\|v_i-v_j\|^2v_i,\quad 1\leq l\leq N_G.\end{equation*}

2.4. Matrix analysis

Now, we review some basic concepts on the matrix analysis to be used in Section 3.

In addition, the following ergodicity coefficient $\mu$ of $A\in \mathbb{R}^{N\times N}$ also plays a key role in the analysis in Section 3:

\begin{equation*}\mu (A)\;:\!=\; \min _{i,j\in [N]}\sum _{k=1}^N\min (A_{ik},A_{jk}).\end{equation*}

For instance, one can easily verify the following two facts on the ergodicity coefficient:

1. 1. $A\in \mathbb{R}^{N\times N}$ is a scrambling matrix $\iff$ $\mu (A)\gt 0$ .

2. 2. Assume that $A,B\in \mathbb{R}^{N\times N}$ . Then, one has

   \begin{equation*}A\geq B \geq 0 \Longrightarrow \mu (A)\geq \mu (B).\end{equation*}

In what follows, we offer two results concerning stochastic matrices and scrambling matrices, which will be used to derive the asymptotic flocking of (1.2). The proof of the following lemma starts by applying the ideas of [Reference Dong, Ha and Kim15]. An improvement over the result in [Reference Dong, Ha and Kim15] is that the remaining term $B$ now plays a role by $D_B$ (the diameter between its columns) rather than the Frobenius norm $\|B\|$ , which allows us to apply Lemma 2.3 recursively. Consequently, we can lower the order of $N$ by one in the sufficient framework $(\mathcal{F}5)-(\mathcal{F}6)$ , compared to the result which uses the lemma in [Reference Dong, Ha and Kim15] directly.

Lemma 2.3. Let $A\in \mathbb{R}^{N\times N}$ be a non-negative stochastic matrix and let $B,Z,W\in \mathbb{R}^{N\times d}$ be matrices satisfying

(2.3) \begin{equation} W=AZ+B. \end{equation}

Then, for every norm $\|\cdot \|$ on $\mathbb{R}^d$ , we have

\begin{equation*}D_W\leq (1-\mu (A))D_Z+D_B,\end{equation*}

where $D_W, D_Z$ and $D_B$ are defined as

\begin{align*} &B\;:\!=\; (b_1,\ldots, b_N)^T,\quad Z\;:\!=\; (z_1,\ldots, z_N)^T,\quad W\;:\!=\; (w_1,\ldots,w_N)^T,\\ &b_i\;:\!=\; (b_i^1,\ldots,b_i^d)^T,\quad z_i\;:\!=\; (z_i^1,\ldots,z_i^d)^T, \quad z_i\;:\!=\; (z_i^1,\ldots,z_i^d)^T,\\ &D_W\;:\!=\; \max _{i,k\in [N]}\|w_i-w_k\|,\quad D_Z\;:\!=\; \max _{i,k\in [N]}\|z_i-z_k\|,\quad D_B\;:\!=\; \max _{i,k\in [N]}\|b_i-b_k\|. \end{align*}

Proof. First of all, the condition that $A$ is stochastic matrix implies

(2.4) \begin{equation} \sum _{j=1}^{N}\max \{0,a_{ij}-a_{kj}\}+\sum _{j=1}^{N}\min \{0,a_{ij}-a_{kj}\}=\sum _{j=1}^{N}(a_{ij}-a_{kj})=0. \end{equation}

Then, the direct calculation from (2.3) yields

(2.5) \begin{eqnarray} \begin{aligned} \|w_i-w_k\|&=\left \|\sum _{j=1}^{N}a_{ij}z_j+b_i-\sum _{j=1}^{N}a_{kj}z_j-b_k \right \|\\ &=\left \|\sum _{j=1}^{N}(a_{ij}-a_{kj})z_j+(b_i-b_k) \right \|\\ &\leq \left \|\sum _{j=1}^{N}(a_{ij}-a_{kj})z_j \right \|+\|b_i-b_k\|\\ &=\sup _{\substack{\phi \in (\mathbb{R}^d)^*\\\|\phi \|=1}}\sum _{j=1}^{N}(a_{ij}-a_{kj})\phi (z_j) +\|b_i-b_k\|\\ &=\sup _{\substack{\phi \in (\mathbb{R}^d)^*\\\|\phi \|=1}}\left [\sum _{j=1}^{N}\max \{0,a_{ij}-a_{kj}\}\phi (z_j)+\sum _{j=1}^{N}\min \{0,a_{ij}-a_{kj}\}\phi (z_j)\right ] +\|b_i-b_k\|\\ &\leq \sup _{\substack{\phi \in (\mathbb{R}^d)^*\\\|\phi \|=1}}\left [\sum _{j=1}^{N}\max \{0,a_{ij}-a_{kj}\}\max _{n\in [N]}\phi (z_n)+\sum _{j=1}^{N}\min \{0,a_{ij}-a_{kj}\}\min _{n\in [N]}\phi (z_n)\right ]\\ &\quad +\|b_i-b_k\|. \end{aligned} \end{eqnarray}

Now, we substitute (2.4) to (2.5) to get

\begin{equation*} \begin{aligned} \|w_i-w_k\|&\leq \sup _{\substack{\phi \in (\mathbb{R}^d)^*\\\|\phi \|=1}}\sum _{j=1}^{N}\max \{0,a_{ij}-a_{kj}\}\max _{n,m\in [N]}\phi (z_n-z_m)+\|b_i-b_k\|\\ &=\sum _{j=1}^{N}\max \{0,a_{ij}-a_{kj}\}\max _{n,m\in [N]}\|z_n-z_m\|+\|b_i-b_k\|\\ &=\sum _{j=1}^{N}\left (a_{ij}-\min \{a_{ij},a_{kj}\}\right )\max _{n,m\in [N]}\|z_n-z_m\|+\|b_i-b_k\|\\ &\leq (1-\mu (A))D_Z+D_B, \end{aligned} \end{equation*}

where we used the definition of $\mu (A)$ and the fact

\begin{equation*}\sum _{j=1}^{N}a_{ij}=1\end{equation*}

in the last inequality, which implies our desired result.

Finally, we review several properties of state-transition matrix. Let $t_0\in \mathbb{R}$ and $A\;:\; [t_0,\infty )\to \mathbb{R}^{N\times N}$ be a right-continuous matrix-valued function. We consider a linear ODE governed by the following Cauchy problem:

(2.6) \begin{equation} \frac{d\mathcal{\xi }(t)}{dt}=A(t)\mathcal{\xi }(t),\quad t\gt t_0. \end{equation}

Then, the solution of (2.6) can be written as

(2.7) \begin{equation} \mathcal{\xi }(t)=\Phi (t,t_0)\mathcal{\xi }(t_0),\quad t\geq t_0, \end{equation}

where the $A$ -dependent matrix $\Phi (t,t_0)$ is said to be the state-transition matrix, which can be represented by using the Peano–Baker series [Reference Sontag31]:

(2.8) \begin{equation} \Phi (t,t_0)=I_{N}+ \sum _{k=1}^\infty \int _{t_0}^t\int _{t_0}^{s_1}\cdots \int _{t_0}^{s_{k-1}}A(s_1)\cdots A(s_{k})ds_k\cdots ds_1. \end{equation}

Now, fix $c\in \mathbb{R}$ and consider the following two Cauchy problems:

(2.9) \begin{equation} \frac{d\mathcal{\xi }(t)}{dt}=A(t)\mathcal{\xi }(t),\quad \frac{d\mathcal{\xi }(t)}{dt}=(A(t)+cI_{N})\mathcal{\xi }(t),\quad t\gt t_0. \end{equation}

If $\Phi (t,t_0)$ and $\Psi (t,t_0)$ are the state-transition matrices for the two linear ODEs in (2.9), respectively, then the authors of [Reference Dong, Ha and Kim15] obtained the following relation between $\Phi (t,t_0)$ and $\Psi (t,t_0)$ :

(2.10) \begin{equation} \Phi (t,t_0)=\exp ({-}c(t-t_0))\Psi (t,t_0),\quad \,t\gt t_0. \end{equation}

3.1. Sufficient frameworks

In this subsection, we describe suitable sufficient frameworks in terms of initial data and system parameters to guarantee the asymptotic flocking of the system (3.1). For this, motivated from the methodologies studied in [Reference Dong, Ha, Jung and Kim13], we provide

• $(\mathcal{F}1)$ There exists a probability space $(\Omega,\mathcal{F},\mathbb{P})$ and a sequence of independent and identically distributed (i.i.d) random variables $\tau _i: \Omega \to \mathbb{R},\,i\in \mathbb{N}$ such that

  \begin{equation*}\exists \,m,M\gt 0,\quad \mathbb {P}(m\leq \tau _i\leq M)=1,\quad i\in \mathbb {N}. \end{equation*}

• $(\mathcal{F}2)$ Define the sequence of random variables $\{t_n: \Omega \to \mathbb{R}\}_{n\in \mathbb{N}\cup \{0\}}$ as

  \begin{equation*}t_n=\begin {cases} 0 & n=0,\\ \sum _{i=1}^n\tau _i&n\geq 1. \end {cases} \end{equation*}
  Then for each $\omega \in \Omega$ and $n\in \mathbb{N}$ , the sample path $\sigma (\cdot,\omega )$ satisfies
  \begin{equation*}\sigma (t,\omega )=\sigma (t_n(\omega ),\omega )\in [N_G],\quad \forall \,t\in [t_n(\omega ),t_{n+1}(\omega )), \end{equation*}
  and this implies that the process $\sigma$ is right continuous. In addition, we use the following notation for simplicity:
  \begin{equation*}\sigma _{t_n}(\omega )\;:\!=\; \sigma (t_n(\omega ),\omega ),\quad \omega \in \Omega,\quad n\in \mathbb {N}\cup \{0\}. \end{equation*}

• $(\mathcal{F}3)$ $\{\sigma _{t_n}:\Omega \to [N_G]\}_{n\in \mathbb{N}\cup \{0\}}$ is the sequence of i.i.d random variables, where

  \begin{equation*}\mathcal {P}_l\;:\!=\; \mathbb {P}(\sigma _{t_n}=l)\gt 0\quad \text {for all}\quad l\in [N_G]\quad \text {and}\quad n\in \mathbb {N}\cup \{0\}.\end{equation*}

• $(\mathcal{F}4)$ The union of all admissible network topologies $\{\mathcal{G}_1,\ldots,\mathcal{G}_{N_G}\}$ contains a spanning tree with vertices $\mathcal{V}=\{\beta _1,\ldots,\beta _N\}$ .

• $(\mathcal{F}5)$ There exist $R\gt 1$ and $\bar{n}\in \mathbb{N}$ such that

  \begin{equation*} \begin{aligned} &r\;:\!=\; \frac{R}{\displaystyle \min _{1\leq l\leq N_G}\{-\log (1-\mathcal{P}_l)\}}\lt \frac{1}{M(N-1)\phi (0)},\quad \sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}}\leq 1-\frac{1}{R},\\ &\left (\frac{m\phi (0)\exp ({-}\phi (0)M\bar{n})(N-1)^{-rM\phi (0)}}{N}\right )^{N-1}\leq \log 2, \end{aligned} \end{equation*}

• $(\mathcal{F}6)$ There exist $\overline{M}\gt 0$ and a sufficiently large number $D_X^\infty \gt 0$ such that for $\mathbb{P}-$ almost every $w\in \Omega$ ,

  \begin{equation*} \begin {aligned} D_V(0,w)\lt \min \left \{\frac {N\log \overline {M}}{(N-1)\phi (0)\overline {M}C_0},\sqrt {2}\right \},\, D_X(0,w)+\overline {M}C_0D_V(0,w)\lt D_X^\infty,\\ \end {aligned}\end{equation*}
  where we set
  \begin{equation*} \begin{aligned} &C\;:\!=\; \left (\frac{m\phi (D_X^\infty )\exp ({-}\phi (0)M\bar{n})(N-1)^{-rM\phi (0)}}{N}\right )^{N-1},\\ &C_0\;:\!=\; M(N-1)\sum _{l=1}^\infty \Bigg [\big (\bar{n}+r\log l(N-1)\big )\exp \bigg ({-}\frac{C\left (l^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\Bigg ]. \end{aligned} \end{equation*}

Here, conditions $(\mathcal{F}1)-(\mathcal{F}4)$ mean that the interaction network is randomly and repeatedly determined after a period of time within an appropriate range, and the probability of each network topology being selected is the same regardless of time. Each of these network topologies may not have sufficient connectivity, but the fact that the union of graphs that can be selected contains some spanning tree allows all agents to send or receive information without being isolated in the long run. In addition, $(\mathcal{F}5)-(\mathcal{F}6)$ quantitatively represents the initial conditions to ensure the flocking phenomenon. In summary, these mean that the essential supremum of the initial velocity diameter ( $\displaystyle =\textrm{ess}\,{\rm sup}_{\omega \in \Omega }D_V(0,\omega )$ ) has to be sufficiently small.

Note that for $\sigma _t(w)\in [N_G]$ , the adjacency matrix $\chi _{ij}^{\sigma _t(w)}$ is defined by

\begin{align*} \begin{aligned} \chi _{ij}^{\sigma _t(w)}={\begin{cases} 1,\quad \text{if } (j,i)\in \mathcal{E}(\mathcal{G}_{\sigma _t(w)}),\\ 0,\quad \text{if } (j,i)\notin \mathcal{E}(\mathcal{G}_{\sigma _t(w)}), \end{cases}} \end{aligned} \end{align*}

and in particular, we have

\begin{equation*}(\beta _i,\beta _i)\in \mathcal {E}(\mathcal {G}_{k}),\quad i\in [N],\end{equation*}

which makes each vertex $\beta _i$ in the graph $\mathcal{G}_k$ ‘seems’ to have a self loop.

3.2. Asymptotic flocking dynamics

First of all, we will consider the union of the network topology on time intervals of the form $[a,b)$ , which we denote

\begin{equation*}\mathcal {G}([a,b))(w)\;:\!=\; \left (\mathcal {V},\bigcup _{t\in [a,b)} \mathcal {E}(\mathcal {G}_{\sigma (t,w)})\right ).\end{equation*}

In addition, we also define a sequence $\{n_k=n_k(\bar{n})\}_{k\in \mathbb{N}\cup \{0\}}$ as

\begin{equation*} \begin{aligned} n_k(\bar{n})\;:\!=\; k\bar{n}+\sum _{l=1}^{k}\lfloor r\log l\rfloor,\quad k\in \mathbb{N}\cup \{0\}, \end{aligned} \end{equation*}

where $\lfloor a\rfloor$ denotes the largest integer less than or equal to $a$ . Then, the following lemma provides a lower bound estimate of the probability to the random digraph

\begin{equation*}\omega \mapsto \mathcal {G}([t_{n_k},t_{n_{k+1}}))(w)\end{equation*}

to have a spanning tree for all $k\in \mathbb{N} \cup \{0\}$ .

Lemma 3.1. Let $(X,V)$ be a solution process to (3.1) satisfying $(\mathcal{F}1)-(\mathcal{F}5)$ . Then, the following probability estimate holds:

\begin{align*} &\mathbb{P}\left (\bigcup _{k=0}^{\infty }\bigg \{w\;:\;\mathcal{G}([t_{n_k},t_{n_{k+1}}))(w)\,\textrm{has a spanning tree}\bigg \}\right )\geq \exp \left ({-}\frac{R^2 \log R}{(R-1)^2} \sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}-1}\right ). \end{align*}

Proof. Since $\{\sigma _{t_n}\}_{n\in \mathbb{N}\cup \{0\}}$ is a i.i.d sequence of random variables, we have

\begin{align*} &\mathbb{P}\left (w\;:\;\mathcal{G}([t_{n_k},t_{n_{k+1}}))(w)\,\text{does not contain a spanning tree}\right )\\[4pt] &\leq \mathbb{P}\left (w\;:\;\exists \,l\in \{1,\ldots,N_G\}\quad \text{such that}\,\,\sigma _{t_{n_k+i}}(w)\neq l\,\,\text{for all}\,\,0\leq i\lt n_{k+1}-n_{k}\right )\\[4pt] &\leq \sum _{l=1}^{N_G}\mathbb{P}\left (w\;:\;\sigma _{t_{n_k+i}}(w)\neq l \,\,\text{for all} \,\,0\leq i\lt n_{k+1}-n_{k}\right )\\[4pt] &=\sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{n_{k+1}-n_{k}}=\sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}+\lfloor r\log (k+1)\rfloor }\leq \sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}}\leq 1-\frac{1}{R}. \end{align*}

Then, the probability to contain a spanning tree is

(3.2) \begin{equation} \begin{aligned} &\mathbb{P}\left (\bigcup _{k=0}^{\infty }\bigg \{w\;:\;\mathcal{G}([t_{n_k},t_{n_{k+1}}))(w)\,\text{has a spanning tree}\bigg \}\right )\\[5pt] &\quad \geq \prod _{k=0}^\infty \left (1-\sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}+\lfloor r\log (k+1)\rfloor }\right )\\[5pt] &\quad \geq \prod _{k=0}^\infty \exp \left ({-\frac{R\log R }{R-1}\sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}+\lfloor r\log (k+1)\rfloor }}\right )\\[5pt] &\quad =\exp \left ({-}\frac{R \log R}{R-1} \sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}}\sum _{k=0}^\infty (1-\mathcal{P}_l)^{\lfloor r\log (k+1)\rfloor }\right ), \end{aligned} \end{equation}

where we used the following relation for $x\in [0,1-\frac{1}{R}]$ :

\begin{align*} 1-x\geq R^{-\frac{Rx}{R-1}}=\exp \left ({-}\frac{R\log R}{R-1}x \right ). \end{align*}

In addition, since we have

\begin{align*} 1\lt R=\min _{1\leq l\leq N_G}\{-r\log (1-\mathcal{P}_l)\}=-r\max _{1\leq l\leq N_G}\log (1-\mathcal{P}_l), \end{align*}

the convergence of $p$ -series yields

(3.3) \begin{equation} \begin{aligned} \sum _{k=0}^\infty (1-\mathcal{P}_l)^{\lfloor r\log (k+1)\rfloor }&\leq \sum _{k=0}^\infty (1-\mathcal{P}_l)^{ r\log (k+1)-1}\\ &=\frac{1}{1-\mathcal{P}_l}\sum _{k=0}^\infty \frac{1}{(k+1)^{-r\log (1-\mathcal{P}_l)}}\\ &\lt \frac{1}{1-\mathcal{P}_l}\cdot \frac{-r\log (1-\mathcal{P}_l)}{-r\log (1-\mathcal{P}_l)-1}\\ &\leq \frac{1}{1-\mathcal{P}_l}\cdot \frac{R}{R-1}, \end{aligned} \end{equation}

where we used

\begin{equation*} \begin{aligned} &(1-\mathcal{P}_l)^{ r\log (k+1)}=(1-\mathcal{P}_l)^{ r\log (k+1)}=\exp \left (r\log (k+1)\log (1-\mathcal{P}_l) \right )=(k+1)^{r\log (1-\mathcal{P}_l)},\\ &\sum _{k=0}^\infty \frac{1}{(k+1)^{-r\log (1-\mathcal{P}_l)}}\lt 1+\int _{1}^{\infty }\frac{1}{x^{-r\log (1-\mathcal{P}_l)}} dx\\ &\quad =1+\frac{1}{-r\log (1-\mathcal{P}_l)-1}{\left [-\frac{1}{x^{-r\log (1-\mathcal{P}_l)-1}} \right ]}_1^{\infty }\\ &\quad =1+\frac{1}{-r\log (1-\mathcal{P}_l)-1}, \end{aligned} \end{equation*}

in the equality and the second inequality, respectively. Finally, we apply the inequality (3.3) to (3.2) to obtain the desired result:

\begin{align*} \mathbb{P}\left (\bigcup _{k=0}^{\infty }\bigg \{w\;:\;\mathcal{G}([t_{n_k},t_{n_{k+1}}))(w)\,\text{has a spanning tree}\bigg \}\right )\geq \exp \left ({-}\frac{R^2 \log R}{(R-1)^2} \sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}-1}\right ). \end{align*}

Now, we recall the matrix formulation of (3.1). According to (2.2), the matrix formulation of (3.1) was given as

(3.4) \begin{equation} \begin{cases} \displaystyle \dot{X}^{(1)}=V^{(1)},\\ \displaystyle \dot{V}^{(1)}=-\frac{1}{N}\mathcal{L}_{\sigma }({X}^{(1)})V^{(1)}+\frac{1}{N}\mathcal{R}_{\sigma }(X^{(1)},V^{(1)}). \end{cases} \end{equation}

Here, we take into account the homogeneous part of (3.4), which becomes

(3.5) \begin{equation} \dot{V}^{(2)}=-\frac{1}{N}\mathcal{L}_{\sigma }(X^{(1)})V^{(2)}. \end{equation}

If we denote the state-transition matrix corresponding to (3.5) as $\overline{\Phi }$ , then it follows from (2.7) that for $a\geq b\geq 0$ ,

(3.6) \begin{equation} V^{(2)}(a)=\overline{\Phi }(a,b)V^{(2)}(b). \end{equation}

Then, we have the following explicit form of the solution to (3.4):

(3.7) \begin{equation} V^{(1)}(a)=\overline{\Phi }(a,b)V^{(1)}(b)+\frac{1}{N}\int _{b}^{a}\overline{\Phi }(a,s)\mathcal{R}_{\sigma _s}(X^{(1)}(s),V^{(1)}(s))ds. \end{equation}

In the following lemma, we present a lower bound estimate of the ergodicity coefficient for the state-transition matrix $\overline{\Phi }$ when the sample path $(X,V)(\omega )$ satisfies

(3.8) \begin{equation} \mathcal{G}([t_{n_k},t_{n_{k+1}}))(w)\, \text{has a spanning tree for every}\, k\in \mathbb{N} \cup \{0\}, \end{equation}

to apply Lemma 2.3 to (3.7). For this, we assume some a priori conditions for a moment; there exists a non-negative number $D_X^\infty \geq 0$ and a positive number $\overline{M}\gt 1$ such that

(3.9) \begin{equation} \begin{aligned} &\circ \,\sup _{t\in \mathbb{R}_+}D_X(t,w)\leq D_X^\infty \lt \infty,\\ &\circ \, \frac{(N-1)^2 \phi (0)M}{N} \sup _{k\in \mathbb{N}}\sum _{l=1}^k\big (\bar{n}+r\log (N-1)+r\log l\big )D_V(t_{n_{(k-1)(N-1)}})\leq \log \overline{M}. \end{aligned} \end{equation}

Then, the following lemma allows to analyse the flocking dynamics of (3.1).

Proof. (1) We employ the method used in [Reference Dong, Ha, Jung and Kim13]. First, for $k\in \mathbb{N}\cup \{0\}$ and $q\in [N-1]$ , we denote $\{{ k}_{q_a}\}_{a=1}^{N_q+1}$ the unique integer-valued increasing sequence such that

\begin{equation*} \begin{aligned} &{n_{({ k}-1)(N-1)+q-1}}=k_{q_1}\lt \cdots \lt k_{q_{N_q+1}}=n_{({ k}-1)(N-1)+q},\\ &\sigma _t=\sigma _{t_{k_{q_a}}}\neq \sigma _{t_{k_{q_{a+1}}}},\quad \forall \,t\in [t_{k_{q_a}},t_{k_{q_{a+1}}}),\quad a\in [N_q], \end{aligned} \end{equation*}

and we use the following notation for simplicity:

\begin{equation*}g_{q_a}\;:\!=\; \sigma _{t_{k_{q_a}}}\in [N_G],\quad \forall \,a\in [N_q].\end{equation*}

Now, we apply (2.6)–(2.10) to write the state-transition matrix for (3.5)–(3.6) in terms of Laplacian matrix $\mathcal{L}_l$ . Since $\phi$ is monotonically decreasing and $\chi ^{l}_{ij}\in \{0,1\}$ , the condition (3.9) $_1$ implies

\begin{equation*}\phi (\|x_i-x_j\|)\geq \phi (D_X^\infty ) \quad \text {and}\quad 0\leq d_i^l\leq N\phi (0),\quad i,j\in [N]. \end{equation*}

Then, we have

(3.10) \begin{equation} -\frac{1}{N}\mathcal{L}_{g_{q_a}}=\frac{1}{N}\left (\mathcal{A}_{g_{q_a}}-\mathcal{D}_{g_{q_a}}\right )\geq \frac{1}{N}\underline{\mathcal{A}}_{g_{q_a}}-\phi (0)I_N, \end{equation}

where each $(i,j)$ -th entry of the matrix $\underline{\mathcal{A}}_{g_{q_a}}$ is given as

\begin{align*} (\underline{\mathcal{A}}_{g_{q_a}})_{ij} \;:\!=\; \begin{cases} \displaystyle \chi ^{g_{q_a}}_{ij}\phi (D_X^\infty ),\quad &i\neq j,\\ \displaystyle \phi (0),\quad &i=j. \end{cases} \end{align*}

Thus, if $\overline{\Psi }(t_{k_{q_{a+1}}},t_{k_{q_{a}}})$ is the state-transition matrix corresponding to $-\frac{1}{N}\mathcal{L}_{g_{q_a}}+\phi (0)I_N$ , the relation (2.10) yields

\begin{equation*}\overline {\Phi }(t_{k_{q_{a+1}}},t_{k_{q_{a}}})=\exp ({-}\phi (0)(t_{k_{q_{a+1}}}-t_{k_{q_{a}}}))\overline {\Psi }(t_{k_{q_{a+1}}},t_{k_{q_{a}}}),\end{equation*}

and we apply $(\mathcal{F}1)$ and (3.10) to the Peano–Baker series representation for $\overline{\Psi }(t_{k_{q_{a+1}}},t_{k_{q_{a}}})$ to obtain

(3.11) \begin{eqnarray} \begin{aligned} \overline{\Psi }(t_{k_{q_{a+1}}},t_{k_{q_a}})&= I_N+\sum _{k=1}^\infty \int _{t_{k_{q_a}}}^{t_{k_{q_{a+1}}}}\int _{t_{k_{q_a}}}^{s_1}\cdots \int _{t_{k_{q_a}}}^{s_{k-1}}\prod _{b=1}^k \left ({-}\frac{1}{N}\mathcal{L}_{g_{q_a}}(s_b)+\phi (0)I_N \right )ds_k\cdots ds_1\\ &\geq I_N+\sum _{k=1}^\infty \int _{t_{k_{q_a}}}^{t_{k_{q_{a+1}}}}\int _{t_{k_{q_a}}}^{s_1}\cdots \int _{t_{k_{q_a}}}^{s_{k-1}} \left (\frac{1}{N}\underline{\mathcal{A}}_{g_{q_a}} \right )^kds_k\cdots ds_1\\ &= I_N+\sum _{k=1}^\infty \frac{1}{k!}(t_{k_{q_{a+1}}}-t_{k_{q_{a}}})^k\left (\frac{1}{N}\underline{\mathcal{A}}_{g_{q_a}} \right )^k\\ &=\sum _{k=0}^\infty \frac{1}{k!}(t_{k_{q_{a+1}}}-t_{k_{q_{a}}})^k\left (\frac{1}{N}\underline{\mathcal{A}}_{g_{q_a}} \right )^k\\ &= \exp \left (\frac{1}{N}(t_{k_{q_{a+1}}}-t_{k_{q_{a}}})\underline{\mathcal{A}}_{g_{q_a}}\right )\\ &\geq \exp \left (\frac{m}{N}\underline{\mathcal{A}}_{g_{q_a}}\right ). \end{aligned} \end{eqnarray}

Therefore, the state-transition matrix $\overline{\Phi }(t_{k_{q_{N_q+1}}},t_{k_{q_{1}}})=\overline{\Phi }(t_{n_{(k-1)(N-1)}+q},t_{n_{(k-1)(N-1)}+q-1})$ has a following lower bound:

(3.12) \begin{equation} \begin{aligned} \overline{\Phi }(t_{k_{q_{N_q+1}}},t_{k_{q_{1}}})&=\prod _{a=1}^{N_q}\overline{\Phi }(t_{k_{q_{a+1}}},t_{k_{q_a}})\\ &\geq \exp ({-}\phi (0)(t_{k_{q_{N_q+1}}}-t_{k_{q_{1}}}))\prod _{a=1}^{N_q}\exp \left (\frac{m}{N}\underline{\mathcal{A}}_{g_{q_a}}\right )\\ &\geq \frac{m}{N}\exp ({-}\phi (0)(t_{k_{q_{N_q+1}}}-t_{k_{q_{1}}}))\sum _{a=1}^{N_q}\underline{\mathcal{A}}_{g_{q_a}}\\ &\geq \frac{m}{N}\phi (D_X^\infty )\exp ({-}\phi (0)(t_{k_{q_{N_q+1}}}-t_{k_{q_{1}}}))A_q, \end{aligned} \end{equation}

where $A_q$ is the adjacency matrix of $\mathcal{G}([t_{k_{q_1}},t_{k_{q_{N_q+1}}}))$ for each $q\in [N-1]$ . Accordingly, we multiply (3.12) for $q=1,2,\ldots,N-1$ and obtain

(3.13) \begin{equation} \begin{aligned} &\overline{\Phi }(t_{n_{k(N-1)}},t_{n_{(k-1)(N-1)}})\\ &=\prod _{q=1}^{N-1}\overline{\Phi }(t_{n_{(k-1)(N-1)}+q},t_{n_{(k-1)(N-1)}+q-1})\\ &\geq \left (\frac{m}{N}\right )^{N-1}\phi (D_X^\infty )^{N-1}\exp \left ({-}\phi (0)(t_{n_{k(N-1)}}-t_{n_{(k-1)(N-1)}})\right )\prod _{q=1}^{N-1}A_q. \end{aligned} \end{equation}

Since $\mathcal{G}(A_q)$ contains a spanning tree for each $q\in [N-1]$ , Lemma 2.4 implies that $\prod _{q=1}^{N-1}A_q$ is a scrambling matrix and, in particular,

\begin{equation*}\mu \left (\prod _{q=1}^{N-1}A_q\right )\geq 1.\end{equation*}

Therefore, we apply $A\geq B\geq 0 \Longrightarrow \mu (A)\geq \mu (B)$ to (3.13) to obtain

\begin{equation*}\mu (\overline {\Phi }(t_{n_{k(N-1)}},t_{n_{(k-1)(N-1)}}))\geq \left (\frac {m}{N}\right )^{N-1}\phi (D_X^\infty )^{N-1}\exp \left ({-}\phi (0)\left (t_{n_{k(N-1)}}-t_{n_{(k-1)(N-1)}}\right )\right ).\end{equation*}

(2) Observe that the constant vector $V^{(2)}=[1,\ldots,1]^T$ can be a special solution to (3.5) $_2$ whatever $X^{(1)}$ is. Then, by (3.6), one has

\begin{equation*}[1,\ldots,1]^T=\overline {\Phi }(T_1,T_2)[1,\ldots,1]^T,\quad \forall \,T_1\geq T_2\geq 0.\end{equation*}

Finally, we combine the above relation and the non-negativity of $\overline{\Psi }(T_1,T_2)$ obtained from the Peano–Baker series representation as (3.11) to see that

\begin{equation*}\overline {\Phi }(T_1,T_2)=\exp ({-}\phi (0)(T_1-T_2))\overline {\Psi }(T_1,T_2)\geq 0\end{equation*}

is a stochastic matrix, which is our desired result.

Then, we apply Lemma 3.2 to (3.1) $_2$ to obtain the velocity alignment of the system (3.1) under a priori assumptions for some well-prepared initial data.

Lemma 3.3. (Velocity alignment) Let $w\in \Omega$ be an event satisfying (3.8), and assume the sample path $(X,V)(\omega )$ of the system (3.1) satisfies $(\mathcal{F}1)-(\mathcal{F}5)$ and (3.9) $_1$ . If we further assume

\begin{equation*} D_V(0,w)\lt \sqrt {2},\end{equation*}

the following inequality holds for all $t\in [t_{n_{k(N-1)}},t_{n_{(k+1)(N-1)}})$ and $k\in \mathbb{N}\cup \{0\}$ :

\begin{align*} D_V(t,w)\leq \overline{M}D_V(0,w)\exp \Bigg ({-}\frac{C\left ((k+1)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\Bigg ), \end{align*}

where $C$ is the constant defined in $(\mathcal{F}6)$ , that is,

\begin{equation*}C\;:\!=\; \left (\frac {m\phi (D_X^\infty )\exp ({-}\phi (0)M\bar {n})(N-1)^{-rM\phi (0)}}{N}\right )^{N-1}.\end{equation*}

Proof. In this proof, we suppress $w$ -dependence to simplify the notation. For every $t\in [t_{n_{(k-1)(N-1)}},t_{n_{k(N-1)}})$ , we apply the explicit formula (3.7) to obtain

\begin{equation*}V(t_{n_{k(N-1)}})=\overline {\Phi }(t_{n_{k(N-1)}},t_{n_{(k-1)(N-1)}})V(t_{n_{(k-1)(N-1)}})+\frac {1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}\overline {\Phi }(t_{n_{k(N-1)}},s)\mathcal {R}_{\sigma _s}(s)ds.\end{equation*}

Since $\overline{\Phi }(t_{n_{k(N-1)}},t_{n_{(k-1)(N-1)}})$ is a stochastic matrix ( $\because$ Lemma 3.2), we apply Lemmas 2.3 and 3.2 to get the following estimate for $D_V$ :

\begin{equation*} \begin {aligned} &D_V(t_{n_{k(N-1)}})\\ &\,\,\leq \left (1-\mu (\overline {\Phi }(t_{n_{k(N-1)}},t_{n_{(k-1)(N-1)}}))\right )D_V(t_{n_{(k-1)(N-1)}})+D_B\\ &\,\,\leq \left (1-\left (\frac {m}{N}\right )^{N-1}\phi (D_X^\infty )^{N-1}\exp \left ({-}\phi (0)\left (t_{n_{k(N-1)}}-t_{n_{(k-1)(N-1)}}\right )\right )\right )D_V(t_{n_{(k-1)(N-1)}})+D_B\\ &\,\,\;=\!:\;\mathcal {I}+D_B, \end {aligned} \end{equation*}

where the matrix $B$ is defined as

\begin{equation*}B\;:\!=\; \frac {1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}\overline {\Phi }(t_{n_{k(N-1)}},s)\mathcal {R}_{\sigma _s}(s)ds. \end{equation*}

In what follows, we estimate $\mathcal{I}$ and $D_B$ one by one.

$\bullet$ (Estimate of $\mathcal{I}$ ): By using $(\mathcal{F}1)$ and the definition of $n_k(\bar{n})$ , we have

\begin{align*} \mathcal{I}&\leq \left (1-\left (\frac{m\phi (D_X^\infty )}{N}\right )^{N-1}\exp \bigg ({-}\phi (0)M\bigg ((N-1)\bar{n}+\sum _{l=(k-1)(N-1)+1}^{k(N-1)}\lfloor r\log l\rfloor \bigg )\bigg )\right )\\ &\qquad \times D_V(t_{n_{(k-1)(N-1)}}). \end{align*}

$\bullet$ (Estimate of $D_B$ ): Since $D_B$ is the maximum distance between row vectors of $B$ , one can easily verify that

\begin{equation*}D_B\leq \frac {1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}D_{\overline {\Phi }(t_{n_{k(N-1)}},s)\mathcal {R}_{\sigma _s}(s)}ds. \end{equation*}

Then, since $\overline{\Phi }(t_{n_{k(N-1)}},s)$ is stochastic, we apply Lemma 2.3 to the integrand $D_{\overline{\Phi }(t_{n_{k(N-1)}},s)\mathcal{R}_{\sigma _s}(s)}$ to obtain

\begin{equation*} \begin{aligned} D_B&\leq \frac{1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}D_{\overline{\Phi }(t_{n_{k(N-1)}},s)\mathcal{R}_{\sigma _s}(s)}ds\\[5pt] &\leq \frac{1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}(1-\mu (\overline{\Phi }(t_{n_{k(N-1)}},s)))D_{\mathcal{R}_{\sigma _s}(s)}ds\\[5pt] &\leq \frac{1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}D_{\mathcal{R}_{\sigma _s}(s)}ds. \end{aligned} \end{equation*}

Hence, it suffices to find an upper bound for $D_{\mathcal{R}_{\sigma _s}(s)}$ , where the matrix $\mathcal{R}_{\sigma _s}(s)$ is given by

\begin{align*} \mathcal{R}_{{\sigma _s}}\;:\!=\; (r^{\sigma _s}_1,\ldots, r^{\sigma _s}_N)^T\quad \text{and} \quad r^{{\sigma _s}}_i\;:\!=\; \frac{1}{2}\sum _{j=1}^{N}\chi ^{{\sigma _s}}_{ij}\phi (\|x_i-x_j\|)\|v_i-v_j\|^2v_i. \end{align*}

To do this, we use Lemma 2.1, $\phi (\cdot )\leq \phi (0)$ , $\chi ^{\sigma _s}_{ij}\in \{0,1\}$ and Corollary 2.1 that for $s\in [t_{n_{(k-1)(N-1)}},t_{n_{k(N-1)}})$ ,

\begin{align*} \|r^{\sigma _s}_i\|&\leq \frac{1}{2}\sum _{j=1}^{N}\chi ^{\sigma _s}_{ij}\phi (\|x_i-x_j\|)\|v_i-v_j\|^2v_i\leq \frac{1}{2}\sum _{j=1}^{N}\phi (0)\|v_i-v_j\|^2\|v_i\|\\[5pt] &\leq \frac{1}{2}\sum _{j=1}^{N}\phi (0)\|v_i-v_j\|^2=\frac{1}{2}\sum _{\substack{1\leq j\leq N\\j\neq i}}\phi (0)\|v_i-v_j\|^2\\[5pt] &\leq \frac{1}{2}\sum _{\substack{1\leq j\leq N\\j\neq i}}\phi (0)D_V^2(s)\\[5pt] &\leq \frac{1}{2}\sum _{\substack{1\leq j\leq N\\j\neq i}}\phi (0)D_V(t_{n_{(k-1)(N-1)}})^2\\[5pt] &= \frac{(N-1)}{2}\phi (0)D_V(t_{n_{(k-1)(N-1)}})^2. \end{align*}

Therefore, the diameter $D_B$ has the following upper bound:

\begin{equation*} \begin {aligned} D_B &\leq \frac {1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}D_{\mathcal {R}_{\sigma _s}(s)}ds\\ &\leq \frac {1}{N}\int _{t_{n_{(k-1)(N-1)}}}^{t_{n_{k(N-1)}}}(N-1)\phi (0)D_V(t_{n_{(k-1)(N-1)}})^2ds \\ &= \frac {N-1}{N}\phi (0)(t_{n_{k(N-1)}}-t_{n_{(k-1)(N-1)}})D_V(t_{n_{(k-1)(N-1)}})^2\\ &\leq \frac {N-1}{N}\phi (0)M\left ((N-1)\bar {n}+\sum _{l=(k-1)(N-1)+1}^{k(N-1)}\lfloor r\log l\rfloor \right )D_V(t_{n_{(k-1)(N-1)}})^2. \end {aligned} \end{equation*}

Thus, we combine two estimates of $\mathcal{I}$ and $D_B$ to obtain

\begin{align*} &D_V(t_{n_{k(N-1)}})\\ &\leq \left (1-\left (\frac{m\phi (D_X^\infty )}{N}\right )^{N-1}\exp \bigg ({-}\phi (0)M\bigg ((N-1)\bar{n}+\sum _{l=(k-1)(N-1)+1}^{k(N-1)}\lfloor r\log l\rfloor \bigg )\bigg )\right )\\ &\quad \times D_V(t_{n_{(k-1)(N-1)}})\\ &\quad +\frac{N-1}{N}\phi (0)M\left ((N-1)\bar{n}+\sum _{l=(k-1)(N-1)+1}^{k(N-1)}\lfloor r\log l\rfloor \right )D_V(t_{n_{(k-1)(N-1)}})^2\\ &\leq \left (1-\left (\frac{m\phi (D_X^\infty )}{N}\right )^{N-1}\exp \bigg ({-}\phi (0)M\bigg ((N-1)\bar{n}+\sum _{l=(k-1)(N-1)+1}^{k(N-1)}r\log (k(N-1))\bigg )\bigg )\right )\\ &\qquad \times D_V(t_{n_{(k-1)(N-1)}})\\ &\quad +\frac{N-1}{N}\phi (0)M\left ((N-1)\bar{n}+\sum _{l=(k-1)(N-1)+1}^{k(N-1)} r\log (k(N-1))\right )D_V(t_{n_{(k-1)(N-1)}})^2\\ &= \left (1-\left (\frac{m\phi (D_X^\infty )}{N}\right )^{N-1}\exp \bigg ({-}\phi (0)M(N-1)\bigg (\bar{n}+r\log (k(N-1))\bigg )\bigg )\right ) D_V(t_{n_{(k-1)(N-1)}})\\ &\quad +\frac{(N-1)}{N}\phi (0)M\left ((N-1)\bigg (\bar{n}+r\log (k(N-1))\bigg )\right )D_V(t_{n_{(k-1)(N-1)}})^2\\ &= \left (1-\left (\frac{m\phi (D_X^\infty )\exp ({-}\phi (0)M\bar{n})}{N}\right )^{N-1}(k(N-1))^{-rM(N-1)\phi (0)}\right )D_V(t_{n_{(k-1)(N-1)}})\\ &\quad +\frac{(N-1)^2}{N}\phi (0)M\bigg (\bar{n}+r\log (k(N-1))\bigg )D_V(t_{n_{(k-1)(N-1)}})^2\\ &=\Bigg [1+\frac{(N-1)^2}{N}\phi (0)M\bigg (\bar{n}+r\log (N-1)+r\log k\bigg )D_V(t_{n_{(k-1)(N-1)}})\\ &\quad -\left (\frac{m\phi (D_X^\infty )\exp ({-}\phi (0)M\bar{n})(N-1)^{-rM\phi (0)}}{N}\right )^{N-1}k^{-rM(N-1)\phi (0)}\Bigg ]D_V(t_{n_{(k-1)(N-1)}})\\ &\leq \exp \left (\frac{(N-1)^2}{N}\phi (0)M\bigg (\bar{n}+r\log (N-1)+r\log k\bigg )D_V(t_{n_{(k-1)(N-1)}})\right )\\ &\quad \times \exp \left ({-}Ck^{-rM(N-1)\phi (0)}\right )D_V(t_{n_{(k-1)(N-1)}}), \end{align*}

where we used the following relation in the last inequality:

\begin{equation*}(1+\alpha )-x \leq \exp (\alpha -x),\quad \forall x\geq 0.\end{equation*}

By iterating the above process, we apply the a priori condition (3.9) $_2$ to obtain the following inequality for $t\in [t_{n_{k(N-1)}},t_{n_{(k+1)(N-1)}})$ :

\begin{align*} D_V(t)&\leq D_V(0)\exp \left (\frac{(N-1)^2}{N}\phi (0)M\sum _{l=1}^k\big (\bar{n}+r\log (N-1)+r\log l\big )D_V(t_{n_{(k-1)(N-1)}})\right )\\ &\quad \times \exp \left ({-}C\sum _{l=1}^k l^{-rM(N-1)\phi (0)}\right )\\ &\leq \overline{M}D_V(0)\exp \Bigg ({-}C\frac{(k+1)^{1-rM(N-1)\phi (0)}-1}{1-rM(N-1)\phi (0)}\Bigg ), \end{align*}

where we used

\begin{equation*}\sum _{l=1}^k l^{-rM(N-1)\phi (0)}\geq \int _{1}^{k+1} x^{-rM(N-1)\phi (0)} dx=\frac {(k+1)^{1-rM(N-1)\phi (0)}-1}{1-rM(N-1)\phi (0)} \end{equation*}

in the last inequality, which completes the proof.

From Lemma 3.3, we can estimate the decay rate of velocity diameter $D_V$ in terms of $\overline{M},m,M,r,N$ . Therefore, we can determine a suitable sufficient condition in terms of initial data and system parameters to make the assumption (3.9) $_2$ imply the assumption (3.9) $_1$ .

Lemma 3.4. (Group formation) Let $w\in \Omega$ be an event satisfying (3.8), and assume the sample path $(X,V)(\omega )$ of the system (3.1) satisfies $(\mathcal{F}1)-(\mathcal{F}5)$ and (3.9) $_2$ . If we further assume

(3.14) \begin{equation} \begin{aligned} &D_V(0,w)\lt \sqrt{2},\\ &D_X(0,w)+\overline{M}C_0D_V(0,w)\lt D_X^\infty,\\ \end{aligned} \end{equation}

the first a priori assumption (3.9) $_1$ also holds, that is,

\begin{equation*}\sup _{t\in \mathbb {R}_+}D_X(t,w)\leq D_X^\infty,\end{equation*}

where $C_0$ is the constant defined in $(\mathcal{F}6)$ .

Proof. First, we define $S$ as

\begin{equation*}S\;=\!:\;\{t\gt 0\,|\,D_X(s,w)\leq D_X^\infty,\, \forall \,s\in [0,t)\},\end{equation*}

and claim:

\begin{equation*} t^*\;:\!=\; \sup S=+\infty . \end{equation*}

To see this, suppose that the contrary holds, that is, $t^*\lt +\infty$ . Since $D_X$ is continuous in $t$ and (3.14) implies $D_X(0,\omega )\lt D_X^\infty$ , we have $t^*\gt 0$ and

\begin{equation*}D_X(t^*-,w)=D_X^\infty .\end{equation*}

Then, we integrate the result of Lemma 3.3 on $t\in [0,t^*]$ and apply (3.14) to obtain

\begin{align*} D_X(t,w)&\leq D_X(0,w)+ \int _0^{t^*} D_V(s,w) ds\\[5pt] &\leq D_X(0,w)+ \int _0^{\infty } D_V(s,w) ds\\[5pt] &\leq D_X(0,w)+\overline{M}D_V(0,w)\sum _{k=0}^\infty \Bigg [(t_{n_{(k+1)(N-1)}}-t_{n_{k(N-1)}})\\[5pt] &\qquad \times \exp \bigg ({-}\frac{C\left ((k+1)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\bigg )\Bigg ]\\[5pt] &\leq D_X(0,w)+\overline{M}M(N-1)D_V(0,w)\sum _{k=0}^\infty \Bigg [\big (\bar{n}+r\log (k+1)(N-1)\big )\\[5pt] &\qquad \times \exp \bigg ({-}\frac{C\left ((k+1)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\bigg )\Bigg ]\\[5pt] &=D_X(0,w)+\overline{M}C_0D_V(0,w)\lt D_X^\infty . \end{align*}

By taking $t\to t^*$ , this inequality yields

\begin{equation*}D_X(t^*-,w)=\lim _{t \to t^*-}D_X(t,\omega )\leq D_X(0,w)+\overline {M}C_0D_V(0,w)\lt D_X^\infty,\end{equation*}

which contradicts to $D_X(t^*-,w)=D_X^\infty$ . Therefore, we can conclude $t^*=+\infty$ , which is our desired result.

Finally, we show that the condition $(\mathcal{F}6)$ implies the assumption (3.9) $_2$ , so that the asymptotic flocking occurs for $\omega \in \Omega$ satisfying (3.8).

Lemma 3.5. Let $w\in \Omega$ be an event satisfying (3.8), and assume the sample path $(X,V)(\omega )$ of the system (3.1) satisfies $(\mathcal{F}1)-(\mathcal{F}6)$ . Then, the assumption (3.9) $_2$ holds, that is,

\begin{equation*}\frac {(N-1)^2\phi (0)M}{N} \cdot \sup _{k\in \mathbb {N}}\Bigg [\sum _{l=1}^k\big (\bar {n}+r\log l(N-1)\big )D_V(t_{n_{(l-1)(N-1)}})\Bigg ]\leq \log \overline {M}.\end{equation*}

Proof. First, define $\mathcal{S}$ as a subset of $\mathbb{N}$ satisfying

\begin{align*} \mathcal{S}\;=\!:\;&\Bigg \{k\,\Bigg |\,\frac{(N-1)^2\phi (0)M}{N}\sum _{l=1}^k\big (\bar{n}+r\log l(N-1)\big )D_V(t_{n_{(l-1)(N-1)}})\leq \log \overline{M}\Bigg \}. \end{align*}

Since $(\mathcal{F}6)$ immediately implies $1\in \mathcal{S}$ , we can define $s^*\;=\!:\;\sup \mathcal{S}\in \mathbb{N}\cup \{\infty \}$ . Then, we claim

\begin{equation*} s^*=+\infty . \end{equation*}

To see this, suppose we have $s^*\lt +\infty$ . Then,

(3.15) \begin{equation} \mathcal{J}\;:\!=\; \frac{(N-1)^2\phi (0)M}{N}\sum _{l=1}^{s^*+1}\big (\bar{n}+r\log l(N-1)\big )D_V(t_{n_{(l-1)(N-1)}})\gt \log \overline{M}. \end{equation}

On the other hand, we can apply Corollary 2.1, Lemma 3.3 and $(\mathcal{F}6)$ to get

(3.16) \begin{equation} \begin{aligned} \mathcal{J}&\leq \frac{(N-1)^2\phi (0)M\overline{M}D_V(0)}{N}\\ &\quad \times \Bigg (\sum _{l=1}^{s^*}\big (\bar{n}+r\log l(N-1)\big )\exp \bigg ({-}\frac{C\left (l^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\\ &\quad +\big (\bar{n}+r\log (s^*+1)(N-1) \big )\exp \bigg ({-}\frac{C\left ((s^*)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\Bigg ). \end{aligned} \end{equation}

By using $1-rM(N-1)\phi (0)\in (0,1)$ and $C\leq \log 2$ in $(\mathcal{F}5)$ , we have

(3.17) \begin{equation} \begin{aligned} &\sum _{l=s^*+1}^{\infty }\big (\bar{n}+r\log l(N-1)\big )\exp \bigg ({-}\frac{C\left (l^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\\ &\quad \geq \big (\bar{n}+r\log (s^*+1)(N-1)\big )\exp \bigg ({-}\frac{C\left ((s^*)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\\ &\qquad \times \sum _{l=s^*+1}^{\infty }\exp \bigg ({-}\frac{C\left (l^{1-rM(N-1)\phi (0)}-(s^*)^{1-rM(N-1)\phi (0)}\right )}{1-rM(N-1)\phi (0)}\bigg )\\ &\quad \geq \big (\bar{n}+r\log (s^*+1)(N-1)\big )\exp \bigg (-\frac{C\left ((s^*)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\\ &\qquad \times \sum _{l=s^*+1}^{\infty }\exp (-C(l-s^*))\\ &\quad \geq \big (\bar{n}+r\log (s^*+1)(N-1)\big )\exp \bigg (-\frac{C\left ((s^*)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\times \sum _{l=1}^{\infty }2^{-l}\\ &\quad =\big (\bar{n}+r\log (s^*+1)(N-1)\big )\exp \bigg (-\frac{C\left ((s^*)^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg ). \end{aligned} \end{equation}

Therefore, we combine (3.16) and (3.17) to obtain

\begin{align*} \mathcal{J} &\leq \frac{(N-1)^2\phi (0)M\overline{M}D_V(0)}{N}\\ &\quad \times \Bigg [\sum _{l=1}^{\infty }\big (\bar{n}+r\log l(N-1)\big )\exp \bigg (-\frac{C\left (l^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\Bigg ]\\ &\lt \log \overline{M}, \end{align*}

which leads a contradiction to (3.15). This implies that $s^*$ must be infinity, which means the assumption (3.9) $_2$ holds.

Now, we are ready to state our main result. By combining Lemmas 3.1–3.5, we can deduce the following result.

Theorem 3.1. (Probability of asymptotic flocking) Suppose that $(X,V)$ is a solution process of the system (3.1) satisfying $(\mathcal{F}1)-(\mathcal{F}6)$ . Then, we have

\begin{equation*}\mathbb {P}\left (w\in \Omega :\,(X,V)(\omega )\,\textit {exhibits asymptotic flocking}\right )\geq \exp \left (-\frac {R^2 \log R}{(R-1)^2} \sum _{l=1}^{N_G}(1-\mathcal {P}_l)^{\bar {n}-1}\right ).\end{equation*}

To check whether this result is meaningful, we can compare the expected behaviour of the trivial solution with the result in Theorem 3.1. On the one hand, one can easily verify that the solution of (3.1) becomes the uniform linear motion of all agents with the same velocities when the event $\omega$ satisfies $D_V(0,\omega )=0$ . On the other hand, the following corollary shows that the probability to exhibit asymptotic flocking converges to $1$ when $\displaystyle \sup _{\omega \in \Omega }D_V(0,\omega )$ converges to $0$ , which implies that the result in Theorem 3.1 is consistent with the uniform linear motion of the trivial solution.

Corollary 3.1. Suppose that $(X^{(n)},V^{(n)})$ is a sequence of the solution process of the system (3.1) satisfying $(\mathcal{F}1)-(\mathcal{F}5)$ and

(3.18) \begin{equation} \sup _{n\in \mathbb{N}}\textrm{ess}\,{\rm sup}_{\omega \in \Omega }D_{X^{(n)}}(0,\omega )\lt \infty,\quad \lim _{n\to \infty }\textrm{ess}\,{\rm sup}_{\omega \in \Omega }D_{V^{(n)}}(0,\omega )=0. \end{equation}

Then, we have

\begin{equation*}\lim _{n\to \infty }\mathbb {P}\left (w\in \Omega :\,(X^{(n)},V^{(n)})(\omega )\,{\rm exhibits\,asymptotic\,flocking}\right )=1.\end{equation*}

Proof. Note that the initial velocity diameter $D_V(0,\omega )$ only affects to $(\mathcal{F}6)$ . To meet the condition $(\mathcal{F}6)$ , we set

\begin{equation*}\overline {M}=e,\quad D_{X}^\infty =\sup _{n\in \mathbb {N}}\textrm {ess}\,{\rm sup}_{\omega \in \Omega }D_{X^{(n)}}(0,\omega )+\frac {1}{\phi (0)}. \end{equation*}

Then, $(\mathcal{F}6)$ holds true for $(X^{(n)},V^{(n)})$ if

(3.19) \begin{equation} \sup _{\omega \in \Omega }D_{V^{(n)}}(0,\omega )\lt \min \left \{\frac{1}{e\phi (0)C_0},\sqrt{2} \right \}, \end{equation}

where $C_0$ is the number determined by $\bar{n}$ :

\begin{equation*} \begin{aligned} &C=\left (\frac{m\phi (D_X^\infty )\exp (-\phi (0)M\bar{n})(N-1)^{-rM\phi (0)}}{N}\right )^{N-1},\\ &C_0=M(N-1)\sum _{l=1}^\infty \Bigg [\big (\bar{n}+r\log l(N-1)\big )\exp \bigg (-\frac{C\left (l^{1-rM(N-1)\phi (0)}-1\right )}{1-rM(N-1)\phi (0)}\bigg )\Bigg ]. \end{aligned} \end{equation*}

In fact, every sufficiently large $\bar{n}$ is allowed in the condition $(\mathcal{F}5)$ , and $C_0=C_0(m,M,\phi,D_X^\infty,\bar{n})$ can be considered as an increasing function with respect to $\bar{n}$ . By using the condition (3.18), one can see that for every sufficiently large $\bar{n}$ , there exists $n_0\in \mathbb{N}$ such that (3.19) holds for all $n\geq n_0$ . Therefore, we apply Theorem 3.1 to get

\begin{equation*} \begin{aligned} &\liminf _{n\to \infty }\mathbb{P}\left (w\in \Omega :\,(X^{(n)},V^{(n)})(\omega )\,\text{exhibits asymptotic flocking}\right )\\ &\geq \lim _{\bar{n}\to \infty } \exp \left (-\frac{R^2 \log R}{(R-1)^2} \sum _{l=1}^{N_G}(1-\mathcal{P}_l)^{\bar{n}-1}\right )\\ &=1, \end{aligned} \end{equation*}

which implies our desired result.