CHEN et al.: CRITICAL CONNECTIVITY AND FASTEST CONVERGENCE RATES OF DISTRIBUTED CONSENSUS WITH SWITCHING TOPOLOGIES 6155
On the other hand, for any positive δ, (A1) is weaker than
the uniform joint-connectivity assumption. In fact, the uniform
joint-connectivity is a special case of (A1) with δ =0.
Remark 1: Compared to the uniform joint-connectivity, one
advantage of the extensible joint-connectivity is that it can be
used to analyze systems with random topology, even if the prob-
ability of connectivity of the topology is not stationary and de-
cays in a negative power rate. In this case [by (59) in Appendix
D] with probability 1 there exists a finite time T>0 such that
(A1) is satisfied for all t ≥ T . This property has been applied to
distributed consensus computation of mobile ad-hoc networks
in Section 5-A, where the probability of successful communi-
cations between two agents depends on their distance.
B. Sufficient Conditions for Consensus
We first give a key lemma deduced from [29]. Before the
statement of this key lemma some definitions are needed. For
protocol (4), define
Φ(t, i):=[I −a(t)L(t)] ···[I −a(i)L(i)] .
Take
t
l=i
(·):=I when t<i. For any x ∈ R
n
,letx
ave
:=
1
n
n
i=1
x
i
be the average value of x, and define
V (x):=x − x
ave
1
2
=
n
i=1
(x
i
− x
ave
)
2
.
For an integer sequence {t
k
}
k≥1
, denote
k
i
:= min{k : t
k
≥ i +1}and
k
t
:= max{k : t
k
− 1 ≤ t}.
Set
d
max
:= sup
t,i
j∈N
i
(t)
a
t
ij
≤ (n − 1)a
max
.
Also, following the common practice [ 21], [27], [30], [36], [38],
[39], we focus on balanced topologies on the system (1)–(2).
(A2) The topology G(t) is balanced for all t ≥ 1.
According to the definition of balancedness, if {G(t)}is undi-
rected and the weight matrix A(t) is symmetric for all t ≥ 1,
then {G(t)} are all balanced.
Under (A1) and (A2), we have the following lemma, whose
proof is postponed to Appendix A.
Lemma III.1: Suppose that (A1) and (A2) are satisfied. Let
z(t)=Φ(t, i +1)z(i) for t>i.Ifa(t) ∈ (0, 1/d
max
) then
V (z(t)) ≤ V (z(i))
k
t
−1
l=k
i
1 −
δ
l
(1 −δ
l
)
2
ε
l
n(n − 1)
2
where δ
l
= min
t
l
≤t<t
l +1
a(t) and ε
l
= min
t
l
≤t<t
l +1
(1 −a(t)
d
max
).
We also characterize robustness of protocol (1)–(2) with re-
spect to noise. This will be accomplished by accommodating a
large class of noises as specified below.
For any random variables X and Y ,letCorr(X, Y ):=
EXY −EXEY
√
Var X Va rY
denote the linear correlation coefficient between
X and Y . Following [50], we employ the notion of ρ-mixing
sequences of random variables. Let {X
i
}
i≥1
be a random vari-
able sequence. For any subset S, T ⊂ IN , t h e s u b σ-algebra
F
S
:= σ(X
i
,i∈ S) and
ρ (F
S
, F
T
):= sup{Corr(X, Y ): X ∈ L
2
(F
S
),Y ∈ L
2
(F
T
)}.
Define the ρ-mixing coefficients by
ρ(m):=sup
ρ (F
S
, F
T
):
finite sets S, T ⊂ IN such that min
i∈S,j∈T
|i −j|≥m
for any integer m ≥ 0. By definition, 0 ≤ ρ(m +1)≤ ρ(m) ≤
1 for all m ≥ 0, and ρ(0) = 1 except for the special case when
all X
i
are degenerate.
Definition III.1: A sequence of random variables {X
i
}
i≥1
is
said to be a ρ-mixing sequence if there exists an integer m>0
such that ρ(m) < 1.
Under this definition, we give the following assumption for
protocol (1)–(2).
(A3) For any network topology sequence {G(t)}
t≥1
, the noise
sequence {w
ji
(t)}
t≥1,i=1,...,n,j∈N
i
(t)
is a zero-mean ρ-mixing
sequence satisfying v := sup
i,j,t
Var (w
ji
(t)) < ∞.
Remark 2: It is well known that ρ-mixing noises include as
special cases φ-mixing noises [42], i.i.d. noises and martingale
difference noises, see [51].
A basic property of ρ-mixing sequences is cited here.
Lemma III.2 (see Th. 2.1 in [52]): Suppose that for an in-
teger m ≥ 1 and a real number 0 ≤ r<1, {X
i
}
i≥1
is a se-
quence of random variables with ρ(m) ≤ r, with EX
i
=0and
E|X
i
|
2
< ∞ for every i ≥ 1. Then there is a positive constant
D = D(m, r) such that for all k ≥ 1,
E
k
i=1
X
i
2
≤ D
k
i=1
E|X
i
|
2
.
Before the statement of the main result of this section, we
give the following lemma first.
Lemma III.3: Suppose that (A1) is satisfied with δ ≤ 1/2.
Then for any constant c
1
> 0 and integer t
∗
≥ 0,
k
t
−1
j=k
i
1 −
c
1
t
1−δ
j+1
+ t
∗
<
i
1−δ
+2c + t
∗
(t +1)
1−δ
+ t
∗
c
1
2 c
(5)
and
k
t
−1
j=k
i
1 −
c
1
(t
1−δ
j+1
+ t
∗
)(log t
j+1
+ t
∗
)
<
log[2c + i
1−δ
+ t
∗
]
log[(t +1)
1−δ
+ t
∗
]
c
1
(1−δ )
2 c
(6)
where c is the same constant appearing in (A1).
The proof of Lemma III.3 is in Appendix A. The following
theorem presents a sufficient condition for consensus.
Theorem III.1: Suppose that (A1) is satisfied with δ ≤ 1/2,
and (A2) and (A3) hold. Then for any initial state x(1), there
exists an open-loop control of the gain sequence {a(t)} such
that the system (1)–(2) reaches unbiased mean-square average-
consensus, with a convergence rate E[V (x(t))] = O(1/t
1−2δ
)
if δ<1/2, and E[V (x(t))] = O(1/ log t) if δ =1/2.
Proof. Case I: δ<1/2. Choose a(t)=
α
t
1 −δ
+t
∗
with
α ≥
32n(n −1)
4
c
(2n −3)
2
and t
∗
≥2α(n − 1)a
max
. (7)
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