# Event-triggered privacy preserving consensus control with edge-based
additive noise

Limei Liang, Ruiqi Ding, Shuai Liu Manuscript created March, 2023.
_(Corresponding author: Shuai Liu.)_ Limei Liang and Shuai Liu are with the
School of Control Science and Engineering, Shandong University, Jinan 250061,
China (E-mail<EMAIL_ADDRESS>liushuai@sdu.edu.cn). Ruiqi Ding is
with the School of Mathematics, Shandong University, Jinan 250100, China
(E-mail:202022171217@mail.sdu.edu.cn).

###### Abstract

In this article, we investigate the distributed privacy preserving weighted
consensus control problem for linear continuous-time multi-agent systems under
the event-triggering communication mode. A novel event-triggered privacy
preserving consensus scheme is proposed, which can be divided into three
phases. First, for each agent, an event-triggered mechanism is designed to
determine whether the current state is transmitted to the corresponding
neighbor agents, which avoids the frequent real-time communication. Then, to
protect the privacy of initial states from disclosure, the edge-based mutually
independent standard white noise is added to each communication channel.
Further, to attenuate the effect of noise on consensus control, we propose a
stochastic approximation type protocol for each agent. By using the tools of
stochastic analysis and graph theory, the asymptotic property and convergence
accuracy of consensus error is analyzed. Finally, a numerical simulation is
given to illustrate the effectiveness of the proposed scheme.

###### Index Terms:

Event-triggered mechanism, Privacy protection, Consensus, Multi-agent systems.

##  I Introduction

In recent years, the study of distributed consensus control problem of multi-
agent systems (MASs) have received increasing attention, due to its broad
applications in multi-robot coordination [1], autonomous underwater vehicles
[2], smart grid [3], unmanned aerial vehicle formation [4] and other fields
[5, 6, 7]. In general, traditional consensus algorithms require each agent to
share its local state, or even its local input, with neighboring agents. This
means that an agent’s exact information could be monitored by some malicious
adversaries, and such privacy disclosure is very dangerous. Therefore, from
the perspective of network security, the designed consensus algorithm should
be secure and able to prevent the privacy of each agent from being leaked,
which brings a new challenge to the consensus control research of MASs. To
achieve the consensus while protecting the privacy of agents from disclosure,
the researchers have conducted extensive research on the privacy preserving
consensus control problem. Consequently, considerable progress has been made
in this direction.

Generally speaking, the related research on privacy protection can be roughly
divided into four categories, namely, cryptology-based approach, state
decomposition-based approach, output mask-based approach and random noise or
deterministic perturbation signal-based approach. To mention a few, in [8],
the authors proposed a privacy preserving method based on homomorphic
encryption for undirected networks, which can guarantee the security and
privacy of an agent as long as it has at least one legitimate neighbor.
Further, in [9], a ratio consensus algorithm combined with homomorphic
encryption is proposed. In [20], the author proposed a state decomposition
based privacy-preserving average consensus method. Furthermore, in [11], this
method is extended to the privacy preserving dynamic average consensus
problem. In [12], the authors proposed a privacy protection scheme based on
the output masking. In [13], a random noise-based privacy preserving average
consensus scheme is proposed. In [14], a deterministic perturbation signals-
based privacy preserving average consensus scheme is proposed.

In all the above privacy-preserving consensus algorithms, the communication
between agents occurs frequently in each iterative step. Considering this
time-triggered communication mode will lead to the improvement of
communication network utilization and a large loss of computing resources,
relevant scholars have introduced the event-triggered communication mechanism
into the design of privacy protection scheme. To list a few, in [15], an
event-triggered differential privacy scheme is proposed to protect the initial
states of MASs from disclosure. In [16], a kind of event-triggered privacy-
preserving consensus algorithm based on output mask is proposed for MASs. In
[17], an event-triggered privacy-preserving bipartite consensus algorithm
based on encryption is proposed for discrete-time nonlinear MASs.

Since the privacy protection scheme based on adding random noise is simple to
design and easy to implement, most of the existing privacy preserving schemes
make privacy protection by adding random noise to the transmitted information.
They do this using a tool called differential privacy. Relevant results can be
referred to [18, 19, 20, 21, 22]. In these schemes, each agent adds a random
noise to its original state, and then transmits this obfuscated signal to all
its neighbors. Since each agent usually has a different number of neighbors,
the above scheme does not fully exploit the heterogeneity of MASs, which will
damage the privacy protection performance. In addition, these works only focus
on discrete-time MASs, while there are few related research results on
continuous-time ones.

Inspired by the above works, in this paper, we aim to propose an event-
triggered privacy preserving consensus control scheme with edge-based additive
noise for continuous-time MASs. Specifically, we first design an event-
triggered communication mechanism to determine whether the current state is
transmitted to the corresponding neighbor agents, which avoids the frequent
real-time communication. Then, we develop a novel privacy preserving scheme,
which ensures the states of agents are not disclosed by adding edge-based
mutually independent standard white noises to the communication channels with
neighbors. Furthermore, we propose a stochastic approximation type protocol
for each agent to attenuate the effect of noise on consensus as much as
possible.

The main contribution of this paper is reflected in the following three
aspects.

1) An appropriate event triggering mechanism is proposed, which avoids
frequent communication and prevents Zeno behavior.

2) A privacy preserving method of adding edge-based random noise is presented,
which improves the privacy preserving performance and brings more degrees of
design freedom.

3) The proposed privacy preserving consensus control scheme is truly
distributed, which is convenient to implement for MASs.

The remainder of this article is structured as follows. Section II introduces
the preliminaries of graph theory and makes the problem statement. Section III
gives the main results of on consensus control and analyzes the convergence
accuracy and Zeno behavior. Section IV presents a numerical simulation to
verify the effectiveness and section V concludes this paper.

Notations: $\mathbb{R}$, $\mathbb{R}_{\geq 0}$ and $\mathbb{C}$ denote the set
of real numbers, non-negative real numbers and complex numbers, respectively.
$\mathbb{C}^{n}$ denotes the $n$-dimensional complex vectors.
$\mathbb{R}^{m\times n}$ and $\mathbb{C}^{m\times n}$ denote the set of
$m\times n$ real and complex matrices, respectively. The superscript $T$
denotes the transpose of a matrix or vector, and the superscript $-1$ denotes
the inverse of a matrix. $\text{diag}(\cdot)$ denotes a diagonal matrix.
$|\cdot|$ denotes the absolute value of a scalar. Let $\mathbf{1}_{N}$ and
$\mathbf{0}_{N}$ be $n$-dimensional column vectors with all elements
respectively being $1$ and $0$. For a vector $x$, $\|x\|$ denotes the
Euclidean norm. For a random variable $X$, $\mathbb{E}(X)$ and $\mathbb{V}(X)$
indicate the expectation and variance, respectively. The notion $f(t)=O(g(t))$
denotes $\text{lim sup}_{t\to\infty}|f(t)/g(t)|<\infty$ and $f(t)=o(g(t))$
denotes $\text{lim sup}_{t\to\infty}|f(t)/g(t)|=0$.

##  II Problem Statement

In this paper, we consider the privacy preserving consensus control problem
for a MAS consisting of $N$ interconnected agents. The communication topology
among agents can be represented by a digraph $\mathcal{G}$, which consists of
a node set $\mathcal{V}=\\{1,2,\cdots,N\\}$ and an edge set
$\mathcal{E}=\\{(j,i):i,j\in\mathcal{V}\\}$, where $(j,i)$ means that node $i$
can receive information from node $j$, but not necessarily vice versa. For
this digraph $\mathcal{G}$, the adjacency matrix
$\mathcal{A}=[a_{ij}]\in\mathbb{R}^{N\times N}$ is defined as $a_{ii}=0$,
$a_{ij}>0$ if $(j,i)\in\mathcal{E}$ and $a_{ij}=0$ otherwise, and the
Laplacian matrix $\mathcal{L}=[l_{ij}]\in\mathbb{R}^{N\times N}$ is defined as
$l_{ii}=\sum_{j\neq i}a_{ij}$, $l_{ij}=-a_{ij}$ for $i\neq j$. For each
$i\in\cal{V}$, we define the set of in-neighbors of node $i$ as
$\mathcal{N}_{i}=\\{j\in\mathcal{V}:(j,i)\in\mathcal{E},i\neq j\\}$.

The dynamics of the $i$-th agent is described by

$\dot{x}_{i}(t)=u_{i}(t),$ (1)

where $x_{i}(t)\in\mathbb{R}$ is the internal state of agent $i$ and
$u_{i}(t)\in\mathbb{R}$ is the control input to be designed.

The objective of this paper is to design a distributed event-triggered control
law for each agent to protect the agents’ states from disclosure while the
individual agents are able to collaborate with each other to achieve
consensus. In order to derive our main results, the following assumption and
lemma are necessary.

###### Assumption 1

The digraph $\mathcal{G}$ contains a spanning tree.

###### Lemma 1

[23] If digraph $\mathcal{G}$ contains a spanning tree, $0$ is a simple
eigenvalue of $\mathcal{L}$, with $\mathbf{1}_{N}$ is the right eigenvector
and $r^{T}=[r_{1},\cdots,r_{N}]$ is the left eigenvector, satisfying
$r^{T}\mathbf{1}_{N}=1$. Further, all the other eigenvalues of $\mathcal{L}$
have positive real parts.

##  III Main Results

In this section, we introduce a privacy preserving event-triggered control
scheme for each agent to solve the consensus problem and prove that the Zeno
behavior can be excluded.

###  III-A Controller design

It is assumed that each agent can monitor its own state continuously and
decide when to transmit its current state over the network based on the event-
triggered mechanism.

For each agent $i$, define $\tilde{x}_{i}(t)=x_{i}(t_{k,i}),\forall
t\in[t_{k,i},t_{k+1,i}]$ to denote the latest transmitted state of agent $i$,
where $t_{k,i}$ denotes the $k$-th event-triggering instant of agent $i$,
which will be determined later by the designed event-triggered mechanism.

To protect the privacy of initial states from disclosure, the edge-based
mutually independent standard white noise is added to each communication
channel. The information of the $j$-th agent received by the $i$-th agent is

$\tilde{y}_{ji}(t)=\tilde{x}_{j}(t)+\sigma_{ji}\eta_{ji}(t),$

where $\eta_{ji}(t),i\in\mathcal{V},j\in\mathcal{N}_{i}$ are mutually
independent standard white noises, and $\sigma_{ji}>0$ is the noise intensity.
Here, the noises are independent of the initial states.

Based on the information from the neighbors, the control law of agent $i$ is
given as

$u_{i}(t)=a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}(\tilde{y}_{ji}(t)-\tilde{x}_{i}(t)),$
(2)

where the consensus gain function $a(t):[0,\infty)\to[0,\infty)$ is a
decreasing and bounded piecewise continuous function, which satisfies the
following assumption.

###### Assumption 2

$a(t)$ satisfies that $\lim_{t\to\infty}t^{\gamma}a(t)$ exists and is
positive, where $\gamma\in(0.5,1)$.

###### Remark 1

Assumption 2 guarantees that $\int_{0}^{\infty}a(s)ds=\infty$ and
$\int_{0}^{\infty}a^{2}(s)ds<\infty$, which not only avoids the failure of
controller caused by too fast attenuation rate, but also can suppress the
noise to make the system converge eventually.

###### Remark 2

A direct consequence of Assumption 2 is that $\forall t>0$,
$\exists~{}\underline{C},~{}\overline{C}>0$ makes

$\frac{\underline{C}}{(1+t)^{\gamma}}\leq
a(t)\leq\frac{\overline{C}}{(1+t)^{\gamma}}.$

We present the design of the event-triggered mechanism here. For each agent
$i$, denote the state error as

$e_{i}(t)=\tilde{x}_{i}(t)-x_{i}(t).$ (3)

Then, the event-triggered mechanism for agent $i$ is designed as

$t_{k+1,i}=\inf\\{t>t_{k,i}:e^{T}_{i}(t)e_{i}(t)\geq a(t)\\}.$ (4)

Substituting (2) into (1), the dynamics of agent $i$ in the form of $\rm
It\hat{o}$ stochastic differential equation is given by

$\displaystyle dx_{i}(t)$
$\displaystyle=a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}(\tilde{y}_{ji}(t)-\tilde{x}_{i}(t))dt$
(5)
$\displaystyle=a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}(\tilde{x}_{j}(t)-\tilde{x}_{i}(t))dt$
$\displaystyle\quad+a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}\eta_{ji}(t)dt$
$\displaystyle=-a(t)\sum_{j=1}^{N}l_{ij}(x_{j}(t)+e_{j}(t))dt$
$\displaystyle\quad+a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}dw_{ji}(t),$

where $w_{ji}(t)$ is a one dimensional standard Brownian motion.

Let $x(t)=[x_{1}(t),\cdots,x_{N}(t)]^{T}$ and
$e(t)=[e_{1}(t),\cdots,e_{N}(t)]^{T}$, the dynamics of MAS can be described as

$dx(t)=-a(t)\mathcal{L}(x(t)+e(t))dt+a(t)\varSigma dw(t).$ (6)

Here,
$\varSigma=$diag$(\alpha_{1}\varSigma_{1},\cdots,\alpha_{N}\varSigma_{N})$,
where $\alpha_{i}$ is the $i$-th row of the weighted adjacency matrix
$\mathcal{A}$ and $\varSigma_{i}=$diag$(\sigma_{1i},\cdots,\sigma_{Ni})$ with
$\sigma_{ji}=0$ for $j\notin\mathcal{N}_{i}$.
$w(t)=[w_{11}(t),\cdots,w_{N1}(t),\cdots,w_{NN}(t)]^{T}$ is an $N^{2}$
dimensional standard Brownian motion. Let
$\mathcal{F}_{t}=\sigma\left\\{x(s):s\leq t\right\\}$ be the filtration
generated by $x(t)$.

Next, we will focus on the consensus problem. To this end, denote
$J=\mathbf{1}_{N}r^{T}$ and define the disagreement vector as

$\delta(t)=(I_{N}-J)x(t).$ (7)

Since $(I_{N}-J)\mathcal{L}=\mathcal{L}=\mathcal{L}(I_{N}-J)$, the dynamics of
$\delta(t)$ is

$d\delta(t)=-a(t)\mathcal{L}(\delta(t)+e(t))dt+a(t)(I_{N}-J)\varSigma dw(t).$

According to Lemma 1, there are matrices $T=[\mathbf{1}_{N}\ Y]$ and
$T^{-1}=[r\ W^{T}]^{T}$, where $Y\in\mathbb{C}^{N\times(N-1)}$ and
$W\in\mathbb{C}^{(N-1)\times N}$, such that

$T^{-1}\mathcal{L}T=J_{\mathcal{L}}=\begin{bmatrix}0&\mathbf{0}_{N-1}^{T}\\\
\mathbf{0}_{N-1}&\tilde{\mathcal{L}}\end{bmatrix},$ (8)

where $J_{\mathcal{L}}$ is the Jordan canonical form of the matrix
$\mathcal{L}$ and $\tilde{\mathcal{L}}$ is the block diagonal matrix in which
the diagonal entries are the nonzero eigenvalues of $\mathcal{L}$.

Using the state transformation, we introduce a new variable

$\varepsilon(t)=T^{-1}\delta(t)=[\varepsilon_{1}(t),\tilde{\varepsilon}^{T}(t)]^{T},$

where $\varepsilon_{1}(t)\in\mathbb{C}$ and
$\tilde{\varepsilon}(t)=[\varepsilon_{2}(t),\cdots,\varepsilon_{N}(t)]^{T}\in\mathbb{C}^{N-1}$.

Since $r^{T}\mathbf{1}_{N}=1$, it can be seen from (7) that

$\varepsilon_{1}(t)=r^{T}\delta(t)\equiv 0,$

and $\tilde{\varepsilon}(t)$ satisfies

$\displaystyle d\tilde{\varepsilon}(t)=$
$\displaystyle-a(t)\tilde{\mathcal{L}}\tilde{\varepsilon}(t)dt-a(t)\tilde{\mathcal{L}}We(t)dt$
(9) $\displaystyle+a(t)W(I_{N}-J)\varSigma dw(t).$

Then, the consensus problem can be converted to the convergence problem of
$\delta(t)$, which is equivalent to the convergence problem of
$\tilde{\varepsilon}(t)$. For further analysis, we present here the following
auxiliary lemma regarding $a(t)$, the proof of which can be found in the
Appendix.

###### Lemma 2

It is supposed that Assumption 2 holds. Then, for any given $\mu>0,p>2$, we
have

$e^{-\mu\int_{0}^{t}a(s)ds}=o(t^{-\frac{p\gamma}{2}}).$ (10)

and

$\lim_{t\to\infty}t^{\frac{p\gamma}{2}}\int_{0}^{t}a^{\frac{p+2}{2}}(s)e^{-\mu\int_{s}^{t}a(r)dr}ds=\frac{(\lim_{t\to\infty}t^{\gamma}a(t))^{\frac{p}{2}}}{\mu}.$
(11)

###  III-B Convergence analysis

Next, we will conduct the convergence analysis of $\tilde{\varepsilon}(t)$.
The relevant result is given by the following theorem.

###### Theorem 1

For a MAS with communication topology $\mathcal{G}$, it is supposed that
Assumption 1 holds. Apply the protocol (2) to the system (1). If Assumption 2
holds, then $\tilde{\varepsilon}(t)$ converges in the $p$-th moment, which
means that $\lim_{t\to\infty}\mathbb{E}(x_{j}(t)-x_{i}(t))^{p}=0$ for $\forall
p\geq 2$.

Proof. Since for $p>2$, the convergence of $\tilde{\varepsilon}(t)$ in the
$p$-th moment implies the convergence in mean square, we can assume $p>2$ in
the following proof.

Choose a Lyapunov function
$V(t)=(\tilde{\varepsilon}^{T}(t)\tilde{\varepsilon}(t))^{\frac{p}{2}}$. It
follows from (9) that

$\displaystyle dV(t)$
$\displaystyle=p(\tilde{\varepsilon}^{T}(t)\tilde{\varepsilon}(t))^{\frac{p-2}{2}}\tilde{\varepsilon}^{T}(t)d\tilde{\varepsilon}(t)$
$\displaystyle\quad+\frac{p}{2}a^{2}(t)(\tilde{\varepsilon}^{T}(t)\tilde{\varepsilon}(t))^{\frac{p-2}{2}}$
$\displaystyle\quad\times\text{trace}(\varSigma^{T}(I_{N}-J)^{T}W^{T}W(I_{N}-J)\varSigma)dt$
$\displaystyle\quad+\frac{p(p-2)}{2}a^{2}(t)(\tilde{\varepsilon}^{T}(t)\tilde{\varepsilon}(t))^{\frac{p-4}{2}}\text{trace}(\tilde{\varepsilon}(t)\tilde{\varepsilon}^{T}(t)$
$\displaystyle\quad W(I_{N}-J)\varSigma\varSigma^{T}(I_{N}-J)^{T}W^{T})dt$
$\displaystyle=-pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)\tilde{\mathcal{L}}\tilde{\varepsilon}(t)dt$
$\displaystyle\quad-
pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)\tilde{\mathcal{L}}We(t)dt$
$\displaystyle\quad+pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)W(I_{N}-J)\varSigma
dw(t)$
$\displaystyle\quad+\frac{p}{2}a^{2}(t)V^{\frac{p-2}{p}}(t)\left\|W(I_{N}-J)\varSigma\right\|_{F}^{2}dt$
$\displaystyle\quad+\frac{p(p-2)}{2}a^{2}(t)V^{\frac{p-4}{p}}(t)$
$\displaystyle\quad\tilde{\varepsilon}^{T}(t)W(I_{N}-J)\varSigma\varSigma^{T}(I_{N}-J)^{T}W^{T}\tilde{\varepsilon}(t)dt$
$\displaystyle\leq-p\tilde{\lambda}_{\min}a(t)V(t)dt$ $\displaystyle\quad-
pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)\tilde{\mathcal{L}}We(t)dt$
$\displaystyle\quad+\bigg{(}\frac{p}{2}\left\|W(I_{N}-J)\varSigma\right\|_{F}^{2}+\frac{p(p-2)}{2}\tilde{\mu}_{max}\bigg{)}$
$\displaystyle\quad\times a^{2}(t)V^{\frac{p-2}{p}}(t)dt$
$\displaystyle\quad+pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)W(I_{N}-J)\varSigma
dw(t),$

where $\tilde{\lambda}_{\min}$ is the minimal eigenvalue of
$\frac{\tilde{\mathcal{L}}+\tilde{\mathcal{L}}^{T}}{2}$, $\tilde{\mu}_{\max}$
is the maximal eigenvalue of
$W(I_{N}-J)\varSigma\varSigma^{T}(I_{N}-J)^{T}W^{T}$.

Choose two constants $\theta_{1},\theta_{2}$ satisfying

$\displaystyle C_{0}\overset{\Delta}{=}$ $\displaystyle
p\tilde{\lambda}_{\min}-(p-1)\theta_{1}^{\frac{p}{p-1}}-\frac{p-2}{2}\theta_{2}^{\frac{p}{p-2}}$
$\displaystyle\qquad\qquad\
\big{(}\left\|W(I_{N}-J)\varSigma\right\|_{F}^{2}+(p-2)\tilde{\mu}_{max}\big{)}>0.$

Then by Young’s inequality and the definition of operator norm, we have

$\displaystyle-
pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)\tilde{\mathcal{L}}We(t)$
$\displaystyle\leq$ $\displaystyle
pa(t)\bigg{(}\frac{p-1}{p}\theta_{1}^{\frac{p}{p-1}}V(t)+\frac{1}{p}\theta_{1}^{-p}\|\tilde{\mathcal{L}}W\|^{p}\|e(t)\|^{p}\bigg{)},$
$a^{2}(t)V^{\frac{p-2}{p}}(t)\leq\frac{p-2}{p}\theta_{2}^{\frac{p}{p-2}}a(t)V(t)+\frac{2}{p}\theta_{2}^{-\frac{p}{2}}a^{\frac{p+2}{2}}(t).$

Therefore,

$\displaystyle dV(t)\leq$ $\displaystyle-
C_{0}a(t)V(t)dt+C_{1}a^{\frac{p+2}{2}}(t)dt$ (12)
$\displaystyle+pa(t)V^{\frac{p-2}{p}}(t)\tilde{\varepsilon}^{T}(t)W(I_{N}-J)\varSigma
dw(t),$ $\mathbb{E}V(t)\leq-
C_{0}\int_{0}^{t}a(s)EV(s)ds+C_{1}\int_{0}^{t}a^{\frac{p+2}{2}}(s)ds,$ (13)

where
$C_{1}=\theta_{1}^{-p}\|\tilde{\mathcal{L}}W\|^{p}\|e(t)\|^{p}+\big{(}\left\|W(I_{N}-J)\varSigma\right\|_{F}^{2}+(p-2)\tilde{\mu}_{max}\big{)}\theta_{2}^{-\frac{p}{2}}$.

By the comparision theorem, we have

$\displaystyle\mathbb{E}V(t)$ $\displaystyle\leq
e^{-C_{0}\int_{0}^{t}a(s)ds}V(0)$ (14)
$\displaystyle\quad+C_{1}\int_{0}^{t}a^{\frac{p+2}{2}}(s)e^{-C_{0}\int_{s}^{t}a(r)dr}ds.$

Further, according to Lemma 2, we can get that

$\mathbb{E}V(t)\leq C_{2,p}t^{-\frac{p\gamma}{2}},$ (15)

where $C_{2,p}$ is a constant. Therefore,

$\lim_{t\to\infty}\mathbb{E}V(t)=0,$

which completes the proof. $\hfill\blacksquare$

###  III-C Accuracy analysis

For common weighted consensus algorithms, the state of each agent can converge
to the weighted average of the initial states with certainty. However, due to
the inherent property of the privacy protection mechanism, the proposed
control algorithm cannot reach the weighted average of the initial states
deterministically. Therefore, we will further analyze the statistical
properties of the convergence value corresponding to the proposed algorithm.

###### Definition 1

For any given initial state $x(0)$, $\rho\in(0,1)$, $\alpha\in\mathbb{R}_{\geq
0}$, a stochastic system is said to achieve $(\rho,\alpha)$ accuracy if the
state of every agent converges in the mean-square sense to a random variable
$x^{*}$ such that $\mathbb{P}\\{|x^{*}-\mathbb{E}(x^{*})|<\alpha\\}\geq
1-\rho$.

###### Corollary 1

Under the proposed privacy preserving consensus scheme, each agent can achieve

$\Bigg{(}\rho,\sqrt{\frac{r^{T}\varSigma\varSigma^{T}r}{\rho(2\gamma-1)}}\Bigg{)}$

accuracy and the mean of the convergence value $x_{\infty}$ is an unbiased
estimate of their initial states’ weighted average.

Proof. Multiply $J$ by both sides of (6). Since $J\mathcal{L}=0$, we can get
that

$Jdx(t)=a(t)J\varSigma dw(t),$

which means that

$Jx(t)=Jx(0)+\int_{0}^{t}a(s)J\varSigma dw(s).$

Thus, we can get that the covergence value
$x_{\infty}=\lim_{t\to\infty}r^{T}x(t)$.

Since $w(t)=[w_{11}(t),\cdots,w_{N1}(t),\cdots,w_{NN}(t)]^{T}$ is an $N^{2}$
dimensional standard Brownian motion, we can further obtain that

$\displaystyle\mathbb{E}(x_{\infty})$ $\displaystyle=r^{T}x(0),$
$\displaystyle\mathbb{V}(x_{\infty})$
$\displaystyle=\frac{r^{T}\varSigma\varSigma^{T}r}{2\gamma-1}.$

According to Chebyshev’s inequality,

$\mathbb{P}\\{|x_{\infty}-\mathbb{E}(x_{\infty})|<\alpha\\}\geq
1-\frac{\mathbb{V}(x_{\infty})}{\alpha^{2}}.$

Choosing $\alpha=\sqrt{\frac{r^{T}\varSigma\varSigma^{T}r}{\rho(2\gamma-1)}}$,
we can get that
$\mathbb{P}\\{|x_{\infty}-\mathbb{E}(x_{\infty})|<\alpha\\}\geq 1-\rho$. This
completes the proof. $\hfill\blacksquare$

###  III-D Zeno behavior analysis

The following result excludes the Zeno behavior a.s..

###### Theorem 2

Consider system (1) with the control law given in (2). Under the same
conditions as in Theorem 1, for any agent $i$, the following equality holds:

$\mathbb{P}\\{\inf_{k}(t_{k+1,i}-t_{k,i})=0\\}=0,$ (16)

which means the lower bound for inter-execution time of agent $i$ is strictly
positive a.s.. Therefore, the Zeno behavior can be excluded a.s..

Proof. Without loss of generality, we assume $a(t)=(t+1)^{-\gamma}$ in this
proof.

Fix a constant $c>0$, for any agent $i$, define

$\displaystyle\triangle t_{k,i}:$ $\displaystyle=t_{k+1,i}-t_{k,i},$
$\displaystyle d_{n,i}:$ $\displaystyle=\min_{0\leq k\leq n-1}\triangle
t_{k,i},$ $\displaystyle A_{n,i}:$ $\displaystyle=\left\\{t_{n,i}<\infty,\
\triangle t_{n,i}\leq d_{n,i}\leq c\right\\}.$

Then

$\displaystyle\mathbb{P}\\{\inf_{k}(t_{k+1,i}-t_{k,i})=0\\}$ (17)
$\displaystyle=$ $\displaystyle\mathbb{P}\\{\exists\ \text{a decreasing
subsequence}\ \triangle t_{k_{l},i}\ \text{converges to 0}\\}$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\\{A_{n,i}\ \text{i.o.}\\}.$

To prove $\mathbb{P}\\{\inf_{k}(t_{k+1,i}-t_{k,i})=0\\}=0$, it suffices to
show $\mathbb{P}\\{A_{n,i}\ \text{i.o.}\\}=0$. Further, according to Borel-
Cantelli lemma, to prove $\mathbb{P}\\{A_{n,i}\ \text{i.o.}\\}=0$, it suffices
to show

$\sum_{n=1}^{\infty}\mathbb{P}\\{A_{n,i}\\}<\infty.$ (18)

Therefore, the remainder of this proof aims to show (18).

According to (3) and (5), for $t\in[t_{k,i},t_{k+1,i})$,

$\displaystyle de_{i}(t)$ $\displaystyle=-dx_{i}(t)$ (19)
$\displaystyle=-a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}(\tilde{x}_{j}(t)-\tilde{x}_{i}(t))dt$
$\displaystyle\quad-a(t)\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}dw_{ji}(t).$

Then, we can get

$\displaystyle e_{i}(t)=$
$\displaystyle-\sum_{j\in\mathcal{N}_{i}}a_{ij}\int_{t_{k,i}}^{t}(\tilde{x}_{j}(t)-\tilde{x}_{i}(t))a(t)dt$
(20)
$\displaystyle-\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}\int_{t_{k,i}}^{t}a(t)dw_{ji}(t),$

and

$\displaystyle e^{T}_{i}(t)e_{i}(t)\leq$ $\displaystyle
2\Bigg{[}N\sum_{j\in\mathcal{N}_{i}}a_{ij}^{2}\bigg{(}\int_{t_{k,i}}^{t}\left\|\tilde{x}_{j}(t)-\tilde{x}_{i}(t)\right\|a(t)dt\bigg{)}^{2}$
(21)
$\displaystyle+\bigg{(}\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}\int_{t_{k,i}}^{t}a(t)dw_{ji}(t)\bigg{)}^{2}\Bigg{]}.$

And then we derive the expression for $\mathbb{P}\\{A_{n,i}\\}$. For the sake
of further derivation, we first define
$\Pi_{i_{1}}(t)=2N\sum_{j\in\mathcal{N}_{i}}a_{ij}^{2}\Big{(}\int_{t_{n,i}}^{t}\left\|\tilde{x}_{j}(t)-\tilde{x}_{i}(t)\right\|a(t)dt\Big{)}^{2}$
and
$\Pi_{i_{2}}(t)=2\Big{(}\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}\int_{t_{n,i}}^{t}a(t)dw_{ji}(t)\Big{)}^{2}$.
Then,

$\displaystyle\quad\mathbb{P}\\{A_{n,i}\\}$ (22)
$\displaystyle=\mathbb{P}\\{(t_{n,i}<\infty,\ \triangle t_{n,i}\leq
d_{n,i}\leq c)\\}$ $\displaystyle\leq\mathbb{P}\\{t_{n+1,i}\leq
t_{n,i}+d_{n,i}<\infty,\ d_{n,i}\leq c\\}$
$\displaystyle=\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{e^{T}_{i}(t)e_{i}(t)}{a(t)}\geq 1,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}}$
$\displaystyle\leq\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{1}}(t)+\Pi_{i_{2}}(t)}{a(t)}\geq 1,$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}}$
$\displaystyle\leq\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{1}}(t)}{a(t)}\geq 1,\ t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}}$
$\displaystyle\quad+\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{2}}(t)}{a(t)}\geq 1,\ t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}},$
$\displaystyle\overset{\Delta}{=}\Theta_{i_{1}}(t)+\Theta_{i_{2}}(t).$

Next, we will estimate $\Theta_{i_{1}}(t)$ and $\Theta_{i_{2}}(t)$
respectively.

To this end, we first define
$\breve{x}_{ji}(t)=\int_{t_{k,i}}^{t}\tilde{x}_{ji}(t)a(t)dt$, where
$\tilde{x}_{ji}(t)=\|\tilde{x}_{j}(t)-\tilde{x}_{i}(t)\|$. Then, it is easy to
get

$\displaystyle\Theta_{i_{1}}(t)=$
$\displaystyle\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}4N^{3}\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\breve{x}_{ji}^{4}(t)}{a^{2}(t)}\geq
1,$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\quad t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}}$

and

$\displaystyle\quad\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\breve{x}_{ji}^{4}(t)}{a^{2}(t)}$
(23)
$\displaystyle\leq\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}(t)a(t)dt\big{)}^{4}}{a^{2}(t_{n,i}+d_{n,i})}$
$\displaystyle\leq\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\frac{\big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}^{2}(t)dt\big{)}^{2}\big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}a^{2}(t)dt\big{)}^{2}}{a^{2}(t_{n,i}+d_{n,i})}$
$\displaystyle\leq\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\frac{d_{n,i}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}^{4}(t)dt\big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}a^{2}(t)dt\big{)}^{2}}{a^{2}(t_{n,i}+d_{n,i})},$

where the first inequality follows from the monotonicity of
$\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\breve{x}_{ji}^{4}(t)}{a^{2}(t)}$,
the second and third inequality take advantage of the Cauchy-Schwarz
inequality.

Note that $d_{n,i}\leq\frac{t_{n,i}}{n}<\frac{t_{n,i}+1}{n}$, then we have

$\displaystyle\quad\frac{\Big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}a^{2}(t)dt\Big{)}^{2}}{a^{2}(t_{n,i}+d_{n,i})}$
(24)
$\displaystyle\leq\frac{(1+t_{n,i})^{-2\gamma}d_{n}^{2}}{(1+t_{n,i}+d_{n,i})^{-2\gamma}}$
$\displaystyle=\Big{(}\frac{1+t_{n,i}+d_{n,i}}{1+t_{n,i}}\Big{)}^{2\gamma}\Big{(}\frac{d_{n,i}}{1+t_{n,i}}\Big{)}^{2\gamma}d_{n,i}^{2-2\gamma}$
$\displaystyle\leq 2^{2\gamma}d_{n,i}^{2-2\gamma}\frac{1}{n^{2\gamma}}.$

Substituting (24) into (LABEL:brevexji), we can get that

$\displaystyle\quad\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\breve{x}_{ji}^{4}(t)}{a^{2}(t)}$
$\displaystyle\leq
4^{\gamma}d_{n,i}^{3-2\gamma}\frac{1}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}^{4}(t)dt.$

Therefore,

$\displaystyle\Theta_{i_{1}}(t)$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\bigg{\\{}4^{\gamma+1}N^{3}d_{n,i}^{3-2\gamma}\frac{1}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}^{4}(t)dt\geq
1,$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}}$ $\displaystyle\leq$
$\displaystyle\mathbb{P}\bigg{\\{}4^{\gamma+1}N^{3}c^{3-2\gamma}\frac{1}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}^{4}(t)dt\geq
1,$ $\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\qquad t_{n,i}<\infty,\
d_{n,i}\leq c\bigg{\\}}$ $\displaystyle\leq$ $\displaystyle
4^{\gamma+1}N^{3}c^{3-2\gamma}\frac{1}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\mathbb{E}\bigg{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}\tilde{x}_{ji}^{4}(t)dt\bigg{)}$
$\displaystyle\leq$ $\displaystyle
4^{\gamma+1}N^{3}c^{3-2\gamma}\frac{1}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\mathbb{E}\bigg{(}\int_{0}^{\infty}\tilde{x}_{ji}^{4}(t)dt\bigg{)}$
$\displaystyle=$ $\displaystyle
4^{\gamma+1}N^{3}c^{3-2\gamma}\frac{1}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\int_{0}^{\infty}\mathbb{E}(\tilde{x}_{ji}^{4}(t))dt.$

From (15) we know

$\mathbb{E}V(t)\leq C_{2,p}t^{-\frac{p\gamma}{2}},$ (25)

which means there exists a constant $\tilde{C}>0$, such that for $\forall
1\leq i,j\leq N$ and $\forall t>0$, the following inequality holds:

$\mathbb{E}\left\|\tilde{x}_{j}(t)-\tilde{x}_{i}(t)\right\|^{4}\leq\tilde{C}(t+1)^{-2\gamma}.$
(26)

It is easy to figure out

$\int_{0}^{\infty}\mathbb{E}\left\|\tilde{x}_{j}(t)-\tilde{x}_{i}(t)\right\|^{4}\leq\frac{\tilde{C}}{2\gamma-1}.$
(27)

Therefore,

$\Theta_{i_{1}}(t)\leq\bigg{(}4^{\gamma+1}N^{3}c^{3-2\gamma}\frac{\tilde{C}}{2\gamma-1}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\bigg{)}\frac{1}{n^{2\gamma}}.$
(28)

Secondly, for $\Theta_{i_{2}}(t)$, we first prove that $\exists$ a constant
$M_{1}>0$, such that

$\displaystyle\mathbb{P}\Bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{2}}(t)}{a(t)}\geq
1|\mathcal{F}_{t_{n,i}}\Bigg{\\}}$ (29) $\displaystyle\leq$ $\displaystyle
4M_{1}N^{3}\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}\big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}a^{2}(t)dt\big{)}^{2}}{a^{2}(t_{n,i}+d_{n,i})}.$

By Chebyshev inequality for conditional expectation and monotonicity of
$a(t)$, we have

$\displaystyle\mathbb{P}\Bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{2}}(t)}{a(t)}\geq
1|\mathcal{F}_{t_{n,i}}\Bigg{\\}}$ (30) $\displaystyle\leq$
$\displaystyle\mathbb{E}\Bigg{(}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{2}}^{2}(t)}{a^{2}(t)}|\mathcal{F}_{t_{n,i}}\Bigg{)}$
$\displaystyle\leq$ $\displaystyle\frac{\mathbb{E}\big{(}\sup_{t_{n,i}\leq
t\leq
t_{n,i}+d_{n,i}}\Pi_{i_{2}}^{2}(t)|\mathcal{F}_{t_{n,i}}\big{)}}{a^{2}(t_{n,i}+d_{n,i})}.$

Define

$X_{n,i}(t)\overset{\Delta}{=}\sum_{j\in\mathcal{N}_{i}}a_{ij}\sigma_{ji}\int_{0}^{t}a(r)1_{r\geq
t_{n,i}}dw_{ji}(r).$ (31)

It is a continuous martingale with $X_{n,i}(0)=0$. Therefore, by conditional
Burkholder-Davis-Gundy inequality (see Theorem 2.3.8 in [24]), $\exists$ a
constant $M_{1}>0$, such that

$\displaystyle\mathbb{E}\bigg{(}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}X_{n,i}^{4}(t)|\mathcal{F}_{t_{n,i}}\bigg{)}$ (32)
$\displaystyle\leq$ $\displaystyle M_{1}\mathbb{E}(\left\langle
X_{n,i},X_{n,i}\right\rangle_{(t_{n,i}+d_{n,i})}^{2}|\mathcal{F}_{t_{n,i}})\
\text{a.s.},$

where $\left\langle X_{n,i},X_{n,i}\right\rangle$ is the quadratic variation
of $X_{n,i}$. Thus,

$\displaystyle\mathbb{E}\Bigg{(}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\Pi_{i_{2}}^{2}(t)|\mathcal{F}_{t_{n,i}}\Bigg{)}$ (33)
$\displaystyle=$ $\displaystyle 4\mathbb{E}\bigg{(}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}X_{n,i}^{4}(t)|\mathcal{F}_{t_{n,i}}\bigg{)}$
$\displaystyle\leq$ $\displaystyle 4M_{1}\mathbb{E}(\left\langle
X_{n,i},X_{n,i}\right\rangle_{(t_{n,i}+d_{n,i})}^{2}|\mathcal{F}_{t_{n,i}})$
$\displaystyle\leq$ $\displaystyle
4M_{1}N^{3}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}(\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}a^{2}(t)dt)^{2}\
\text{a.s.}.$

Substituting (33) into (30), it is easy to obtain (29).

Then, by the property of conditional expectation, we have

$\displaystyle\Theta_{i_{2}}(t)=$
$\displaystyle\mathbb{E}\Bigg{(}\mathbb{P}\Bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{2}}(t)}{a(t)}\geq 1,$ (34)
$\displaystyle\qquad\qquad\qquad\qquad\ t_{n,i}<\infty,\ d_{n,i}\leq
c|\mathcal{F}_{t_{n,i}}\Bigg{\\}}\Bigg{)}$ $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{(}\mathbb{P}\Bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+d_{n,i}}\frac{\Pi_{i_{2}}(t)}{a(t)}\geq
1|\mathcal{F}_{t_{n,i}}\Bigg{\\}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad 1_{t_{n,i}<\infty,\
d_{n,i}\leq c}\Bigg{)}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\Bigg{(}4M_{1}N^{3}\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}\big{(}\int_{t_{n,i}}^{t_{n,i}+d_{n,i}}a^{2}(t)dt\big{)}^{2}}{a^{2}(t_{n,i}+d_{n,i})}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad 1_{t_{n,i}<\infty,\
d_{n,i}\leq c}\Bigg{)}$ $\displaystyle\leq$
$\displaystyle\mathbb{E}\Bigg{(}4M_{1}N^{3}\frac{\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}\big{(}d_{n,i}(1+t_{n,i})^{-2\gamma}\big{)}^{2}}{(1+t_{n,i}+d_{n,i})^{-2\gamma}}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad\quad 1_{t_{n,i}<\infty,\
d_{n,i}\leq c}\Bigg{)}$ $\displaystyle=$
$\displaystyle\mathbb{E}\Bigg{(}4M_{1}N^{3}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}d_{n,i}^{2-2\gamma}\bigg{(}\frac{d_{n,i}}{1+t_{n,i}}\bigg{)}^{2\gamma}$
$\displaystyle\qquad\qquad\quad\bigg{(}\frac{1+t_{n,i}+d_{n,i}}{1+t_{n,i}}\bigg{)}^{2\gamma}1_{t_{n,i}<\infty,d_{n,i}\leq
c}\Bigg{)}$ $\displaystyle\leq$
$\displaystyle\bigg{(}4^{\gamma+1}M_{1}N^{3}c^{2-2\gamma}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}\bigg{)}\frac{1}{n^{2\gamma}},$

where the first inequality follows from (29), the second inequality follows
from $a(t)=(1+t)^{-\gamma}$, the third inequality makes use of
$d_{n,i}\leq\frac{t_{n,i}}{n}<\frac{t_{n,i}+1}{n}$ like before.

Therefore,

$\Theta_{i_{2}}(t)\leq\frac{4^{\gamma+1}M_{1}N^{3}c^{2-2\gamma}}{n^{2\gamma}}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}.$
(35)

Combining (28) and (35), we have

$\displaystyle\sum_{n=1}^{\infty}\mathbb{P}\\{A_{n,i}\\}$ $\displaystyle\leq$
$\displaystyle\sum_{n=1}^{\infty}\big{(}\Theta_{i_{1}}(t)+\Theta_{i_{2}}(t)\big{)}$
$\displaystyle\leq$
$\displaystyle\bigg{(}4^{\gamma+1}N^{3}c^{3-2\gamma}\frac{\tilde{C}}{2\gamma-1}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}$
$\displaystyle+4^{\gamma+1}M_{1}N^{3}c^{2-2\gamma}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}\bigg{)}\sum_{n=1}^{\infty}\frac{1}{n^{2\gamma}}$
$\displaystyle<$ $\displaystyle\infty,$

i.e.,

$\sum_{n=1}^{\infty}\mathbb{P}\\{A_{n,i}\\}<\infty.$ (36)

According to Borel-Cantelli lemma,

$\mathbb{P}\\{A_{n,i}\ \text{i.o.}\\}=0.$ (37)

So

$\displaystyle\mathbb{P}\\{\inf_{k}(t_{k+1,i}-t_{k,i})=0\\}$ (38)
$\displaystyle=$ $\displaystyle\mathbb{P}\\{\exists\ \text{a decreasing
subsequence}\ \triangle t_{k_{l},i}\ \text{converges to 0}\\}$
$\displaystyle\leq$ $\displaystyle\mathbb{P}\\{A_{n,i}\ \text{i.o.}\\}.$

Therefore, the Zeno behavior can be excluded a.s.. $\hfill\blacksquare$

The above theorem proves that there is a strict positive lower bound between
two adjacent execution times of each agent $i$, thus excluding Zeno behavior.
To better clarify this conclusion, we give the following estimates.

###### Corollary 2

For any constant $\iota>0$, the inequality

$t_{n+1,i}-t_{n,i}>\iota$ (39)

holds with probability at least
$1-\Big{(}\frac{4N^{3}\overline{C}^{4}\tilde{C}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}}{\underline{C}^{2}(2\gamma-1)}\iota+\frac{4M_{1}N^{3}\overline{C}^{4}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}}{\underline{C}^{2}}\Big{)}\iota^{2}(1+\iota)^{2\gamma}$.

Proof.

From (21) we have

$\displaystyle\quad\mathbb{P}\\{t_{n+1,i}-t_{n,i}\leq\iota\\}$
$\displaystyle=\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+\iota}\frac{e^{T}_{i}(t)e_{i}(t)}{a(t)}\geq 1,\
t_{n,i}<\infty\bigg{\\}}$
$\displaystyle\leq\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+\iota}\frac{\Pi_{i_{1}}(t)+\Pi_{i_{2}}(t)}{a(t)}\geq
1,t_{n,i}<\infty\bigg{\\}}$
$\displaystyle\leq\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+\iota}\frac{\Pi_{i_{1}}(t)}{a(t)}\geq 1,\ t_{n,i}<\infty\bigg{\\}}$
$\displaystyle\quad+\mathbb{P}\bigg{\\{}\sup_{t_{n,i}\leq t\leq
t_{n,i}+\iota}\frac{\Pi_{i_{2}}(t)}{a(t)}\geq 1,\ t_{n,i}<\infty\bigg{\\}}$
$\displaystyle\overset{\Delta}{=}\Theta_{i_{1},\iota}(t)+\Theta_{i_{2},\iota}(t).$

And note that

$\displaystyle\frac{\Big{(}\int_{t_{n,i}}^{t_{n,i}+\iota}a^{2}(t)dt\Big{)}^{2}}{a^{2}(t_{n,i}+\iota)}$
$\displaystyle\leq$
$\displaystyle\frac{\overline{C}^{4}(1+t_{n,i})^{-4\gamma}\iota^{2}}{\underline{C}^{2}(1+t_{n,i}+\iota)^{-2\gamma}}$
$\displaystyle=$
$\displaystyle\frac{\overline{C}^{4}}{\underline{C}^{2}}\Big{(}\frac{1+t_{n,i}+\iota}{1+t_{n,i}}\Big{)}^{2\gamma}\Big{(}\frac{\iota}{1+t_{n,i}}\Big{)}^{2\gamma}\iota^{2-2\gamma}$
$\displaystyle\leq$
$\displaystyle\frac{\overline{C}^{4}}{\underline{C}^{2}}\iota^{2}(1+\iota)^{2\gamma}$

Then, by proving method similar to (28) and (35), we can obtain the following
two inequalities:

$\Theta_{i_{1},\iota}(t)\leq\frac{4N^{3}\overline{C}^{4}\tilde{C}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}}{\underline{C}^{2}(2\gamma-1)}\iota^{3}(1+\iota)^{2\gamma},$
(40)
$\Theta_{i_{2},\iota}(t)\leq\frac{4M_{1}N^{3}\overline{C}^{4}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}}{\underline{C}^{2}}\iota^{2}(1+\iota)^{2\gamma}.$
(41)

Therefore,

$\displaystyle\quad\mathbb{P}\\{t_{n+1,i}-t_{n,i}\leq\iota\\}$
$\displaystyle\leq\Theta_{i_{1},\iota}(t)+\Theta_{i_{2},\iota}(t)$
$\displaystyle\leq\frac{4N^{3}\overline{C}^{4}\tilde{C}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}}{\underline{C}^{2}(2\gamma-1)}\iota^{3}(1+\iota)^{2\gamma}$
$\displaystyle\quad+\frac{4M_{1}N^{3}\overline{C}^{4}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}}{\underline{C}^{2}}\iota^{2}(1+\iota)^{2\gamma}$

which means the inequality

$t_{n+1,i}-t_{n,i}>\iota$ (42)

holds with probability at least
$1-(\frac{4N^{3}\overline{C}^{4}\tilde{C}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}}{\underline{C}^{2}(2\gamma-1)}\iota+\frac{4M_{1}N^{3}\overline{C}^{4}\sum_{j\in\mathcal{N}_{i}}a_{ij}^{4}\sigma_{ji}^{4}}{\underline{C}^{2}})\iota^{2}(1+\iota)^{2\gamma}$.
$\hfill\blacksquare$

##  IV Simulation

In this section, a numerical simulation is presented to illustrate the
effectiveness of the proposed distributed event-triggered privacy preserving
weighted consensus scheme. The communication topology of the MAS is shown in
Fig. 1. The initial conditions of the agents are $x_{1}(0)=5$, $x_{2}(0)=3$,
$x_{3}(0)=1$, $x_{4}(0)=2$, and $x_{5}(0)=4$. The parameters are choosen as
$\gamma=0.6$, $k=0.01$ and $h=0.3$.

Figure 1: The communication topology.

We first verify the effectiveness of the proposed event triggering mechanism.
Take agent 1 as an example, and its event triggering interval is shown in Fig.
2, where the abscissa and ordinate represent the time instant and the time
interval between two adjacent events, respectively. Fig. 2 shows that the time
interval between two adjacent events has a consistent lower bound, which
indicates that Zeno behavior can be ruled out.

Figure 2: Event-triggering intervals of agent 1. Figure 3: The state
trajectory of the agents. Figure 4: The consensus error of the agents. Figure
5: The mean square of the consensus error.

Fig. 3 and Fig. 4 show the state trajectory and consensus error of each agent,
respectively. As can be seen from these figures, since privacy protection is
taken into account in the consensus control algorithm, each agent cannot
converge to the weighted average of the initial states deterministically.
However, it can be seen from Fig. 5 that the mean-square convergence of
consensus error can be achieved. These results show that our proposed privacy
preserving consistency control algorithm is effective.

##  V Conclusion

In this paper, we proposed a distributed event-triggered privacy preserving
weighted consensus control scheme for MASs. In order to reduce unnecessary
communication loss, we have designed a novel event triggering mechanism for
each agent to determine the information transmission instants. Then, we have
developed a privacy preserving scheme, which protects the state information of
each agent from disclosure by adding edge-based mutually independent standard
white noises to the communication process with neighbors. Further, we proposed
a stochastic approximation type protocol for each agent to attenuates the
effect of noises on consensus control. Using the tools of stochastic analysis
and algebraic theory, the asymptotic property and convergence accuracy of
consensus error is analyzed. Theoretical analysis and numerical simulation
verify that the proposed method can successfully make privacy preserving
weighted consensus without incurring Zeno behavior.

## Appendix

### Proof of Lemma 2

According to Assumption 2, there is $t_{0}>0$ and $\epsilon>0$ such that
$\forall t\geq t_{0}$, $a(t)\geq\epsilon t^{-\gamma}$. Thus, it yields that

$e^{-\mu\int_{0}^{t}a(s)ds}\leq
e^{-\mu\int_{0}^{t_{0}}a(s)ds}e^{\frac{\mu\epsilon}{1-\gamma}t_{0}^{1-\gamma}}e^{-\frac{\mu\epsilon}{1-\gamma}t^{1-\gamma}}.$

Let $\nu=t^{1-\gamma}$. Since $\gamma\in(0.5,1)$, then as $t\to\infty$,
$\nu\to\infty$. And because of $\frac{\gamma}{1-\gamma}>0$ and
$-\frac{\mu\epsilon}{1-\gamma}<0$, we can get that

$\lim_{t\to\infty}t^{\frac{p\gamma}{2}}e^{-\frac{\mu\epsilon}{1-\gamma}t^{1-\gamma}}=\lim_{\nu\to\infty}\nu^{\frac{2\gamma}{p(1-\gamma)}}e^{-\frac{\mu\epsilon}{1-\gamma}\nu}=0.$

Thus,

$\lim_{t\to\infty}t^{\frac{p\gamma}{2}}e^{-\mu\int_{0}^{t}a(s)ds}=0,$

i.e.,

$e^{-\mu\int_{0}^{t}a(s)ds}=o(t^{-\frac{p\gamma}{2}})$

Further, according to Assumption 2 and (10), using L’Hôpital’s rule, it
follows that

$\displaystyle\lim_{t\to\infty}t^{\frac{p\gamma}{2}}\int_{0}^{t}a^{2}(s)e^{-\mu\int_{s}^{t}a(r)dr}ds$
$\displaystyle=$
$\displaystyle\lim_{t\to\infty}\frac{d(\int_{0}^{t}a^{2}(s)e^{\mu\int_{0}^{s}a(r)dr}ds)/dt}{d(t^{-\frac{p\gamma}{2}}e^{\mu\int_{0}^{t}a(r)dr})/dt}$
$\displaystyle=$
$\displaystyle\frac{(\lim_{t\to\infty}t^{\gamma}a(t))^{\frac{p}{2}}}{\mu}.$

This completes the proof. $\hfill\blacksquare$

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