Novel decentralized adaptive strategies for the
synchronization of complex networks
Pietro De Lellis a,∗ Mario di Bernardo a Francesco Garofalo a
a
Department of Systems and Computer Science,
University of Naples Federico II, Naples 80125, Italy
Abstract
This paper is concerned with the analysis of the synchronization of networks of nonlinear oscillators through an innovative
local adaptive approach. In particular, time-varying feedback coupling gains are considered, whose gradient is a function of
the local synchronization error over each edge in the network. It is shown that, under appropriate conditions, the strategy is
indeed successful in guaranteeing the achievement of a common synchronous evolution for all oscillators in the network. The
theoretical derivation is complemented by its validation on a set of representative examples.
Key words: synchronization; oscillators; Lyapunov function; stability analysis; adaptive control.
1
Introduction
Complex networked control systems have been the subject of much ongoing research (see for example [1] and
references therein). They usually consist of many interacting agents communicating over a web of interconnections characterized by a complex topology. To model this
class of systems, three essential elements are needed: (i)
a model of the dynamics of each agent in the system;
(ii) a communication protocol (or interaction model) describing how agents communicate with each other and
(iii) a graph (network) describing the topology of the
interconnections between agents. In many applications,
the dynamics of each agent is assumed to be modeled
by a set of nonlinear ordinary differential equations. The
communication protocol is often taken to be dependent
on some output function of the state of each node so that
nearby nodes exchange information on the mismatch between their output functions. The resulting model is a
complex network of dynamical systems.
Recently, the problem of making such a network evolve
onto a common synchronous evolution has been the subject of much ongoing research. Specifically, a network is
considered consisting of N identical nonlinear dynamical systems [3], [21]. Each system is described by a set
∗ Corresponding author.
Email addresses: [email protected] (Pietro De
Lellis), [email protected] (Mario di Bernardo),
[email protected] (Francesco Garofalo).
Preprint submitted to Automatica
of nonlinear ordinary differential equations (ODEs) of
the form ẋ = f (x), where x ∈ Rn is the state vector
and f : Rn → Rn is a nonlinear vector field describing
the system dynamics. The coupling between neighboring nodes is assumed to be proportional to the difference
between their output functions. Hence, the equations of
motion for the generic i-th system in the network become:
N
dxi
= f (xi (t)) − σ
Lij h(xj ),
dt
j=1
i = 1, 2, . . . , N,
(1)
where xi represents the state vector of the i-th oscillator,
σ the overall strength of the coupling, h : Rn → Rn the
output function through which the systems in the network are coupled and Lij the elements of the Laplacian
matrix L describing the network topology. In particular,
L is such that its entries, Lij , are zero if node i is not
connected to node j = i, while are negative otherwise.
Moreover, | Lij | gives a measure of the strength of the
interaction. The synchronization problem is to find the
range of values of σ so that all systems in the network
synchronize on the same unknown evolution, say xs (t),
i.e. finding the values of σ such that
lim (xi (t) − xj (t)) = 0,
t→∞
(2)
for all pairs (i, j) of nodes such that i = j.
Different approaches have been used to evaluate the
range of the coupling gain σ that ensures synchroniza-
8 January 2009
tion. One method, popular with the Physics community,
is the so-called Master Stability Function (MSF), which
uses the largest Lyapunov exponent of the coupled network as an indicator of the transverse stability of the
synchronization manifold [3]. Alternatively, a Lyapunov
approach has been used to prove asymptotic stability of
the network of coupled oscillators onto the synchronous
solution [22], [25], [27]. In general, the coupling gain σ is
assumed to be identical for all edges in the network and
constant in time. Many real-world networks are characterized instead by evolving, adapting coupling gains,
which vary in time according to different environmental
conditions. A typical example are wireless sensor networks that gather and communicate data to a central
base station [19]. Adaptation is also necessary to control networks of robots when the operating conditions
change unexpectedly (i.e. a robot loses a sensor) [23].
Moreover, examples of adaptive networks can be found
in biology and ecology, as, for instance, the social insect
colonies, studied in [9].
[14] indicated that the proposed strategies were a viable
approach to synchronization but no theoretical proof
was given of their effectiveness. A first theoretical proof
of asymptotic stability was given in [13], under the assumption that neighboring were select their gains as a
function of the distance between their outputs. While
proving stability for this specific adaptation strategy,
the proof did not encompass other cases proposed in the
literature such as the one presented in [29].
Adaptive gains have been seldom considered in the existing literature on synchronization of complex networks.
A more common approach found in the literature is to
guarantee synchronization by means of an additional
adaptive control input acting on all (or a fraction of)
nodes in the network. Examples include the results presented in [4], [16] to guarantee synchronization in the
presence of uncertainties, the work in [28] on the case
of hybrid coupled networks with time-varying delays or
the case of time varying network topologies shown in
[17]. More often, it is more appropriate to consider adaptive strategies involving the adaptation of the coupling
gains between neighboring nodes in the network without
any additional control input on the nodes. Some global
strategies in this direction were recently presented in [7]
and [15] where a unique adaptive coupling gain, equal
for all network nodes is proposed to achieve synchronization.
The rest of the paper is outlined as follows. In Sec. 2
we describe the adaptive strategies analyzed in the paper, we introduce the notation and give definitions used
throughout the paper. In Sec. 3 we introduce a new theorem and a subsequent corollary that give sufficient conditions for global asymptotic stability of a network of
N oscillators under the adaptive strategies described in
the previous section. In Sec. 4 we validate numerically
our theoretical derivation on a set of representative examples. Conclusions are finally drawn in Sec. 5.
The aim of this paper is to present a novel proof of
asymptotic stability for a wider class of edge-based adaptation strategies expanding and integrating the results
presented in [13]. In particular, two classes of adaptation laws are introduced and proven to guarantee asymptotic convergence of the network towards a common synchronous evolution. The theoretical findings are validated by numerical simulation of networks of 1000 nodes
with different topologies and node dynamics confirming
the effectiveness of the presented strategy.
2
Model description and mathematical preliminaries
Let us denote with E the set of edges of the network,
containing pairs of indices associated to nodes connected
by an existing link. For example (i, j) ∈ E will indicate
there exists an edge connecting node i to node j. Moreover, let us indicate with Ei ⊆ E the subset of all edges
connected to node i. We consider the following network
model:
Despite the effectiveness of the strategies described
above, in many networks it is important for the adaptation strategy to be decentralized and local so that the
coupling gain can be different for each node or edge in
the network. For example, in [5] it is proposed for each
node to update the coupling gain with its neighbours at
some discrete time-instants using the scheme of gradient networks [24]. A key implementation issue is for the
adaptive strategy to be simple while relying on a minimal amount of local information on the network nodes
and topology. To address this, a new set of decentralized local strategies, named vertex-based strategy and
edge-based strategy, were introduced in [14]. These were
shown to be effective in a number of numerical experiments. An alternative strategy was also independently
described in [29]. In both cases, the idea is for mutually
coupled nodes to negotiate the strength of their mutual
coupling on the basis of local information on their output mismatch. Numerical simulations reported in [29],
dxi
= f (xi ) −
σij (t)(h(xj ) − h(xi )), i = 1, 2, . . . , N,
dt
j∈Ei
(3)
where xi , f and h are defined as in equation (1) but now
the coupling gains are the adaptive, time-varying functions σij (t) associated to each edge (i, j). The evolution
of the adaptive gains is described by the differential law:
σ̇ij (t) = φ(eij ),
(4)
where eij = xj − xi is the local error between node i and
node j and φ : Rn → R is a generic sufficiently smooth
function of this error. In the following, we will consider
two possible choices of the function φ. Namely, we will
assume φ to belong to one of the following classes:
2
where Λ = Λ ⊗ In , X = (xT1 , . . . , xTN )T , F (X) =
0
≥ 0, ∀(i, j) ∈ E and ⊗
(f (x1 )T , . . . , f (xN )T )T , σij
denotes the Kronecker product. Furthermore, say
Δ = In ⊗ Δ and U = U ⊗ In .
Note that, obviously, if we set e(t) = X T U X, then
limt→∞ e(t) = 0 iff (2) is verified.
Within this framework, we can state the following theorems and corollaries.
• Class 1:
φ(eij ) = α||eij ||p , with 0 < p ≤ 2
(5)
• Class 2: φ is a monotonously increasing function of
the error norm such that
(1) φ(0) = 0;
(2) for some finite constant m < +∞,
0 ≤ φ(eij ) ≤ m.
Theorem 2 If f is QUAD, the matrix [UΔ + UΛ] is
negative semidefinite for all t ≥ 0, and the function φ
belongs to classes 1 or 2, then the network globally synchronizes onto a common evolution with
(6)
Numerical experiments and preliminary analytical studies have shown that if φ belongs to class 1, with p = 1,
a common synchronous evolution can be achieved (see
[14] and [13] for further details). A pressing open problem is to prove analytically the asymptotic stability of
the synchronous trajectory for the more general choice
of φ described above.
lim e(t) = 0,
t→∞
∀e(0) = e0 ,
and
lim σij (t) = cij < +∞,
t→∞
We start by introducing some notation and giving some
definitions that will be used throughout the rest of the
paper.
PROOF. To prove the theorem, let us consider the following candidate Lyapunov function:
Definition 1 Similarly to what stated in [7], we say that
a vector field f : Rn × R+ → Rn is QUAD iff, for any
x, y ∈ Rn :
(x − y) T [f (x, t) − f (y, t)] − (x − y)T Δ ·
· (x − y) ≤ −ω̄(x − y)T (x − y),
1
1 ηX T (t)UX(t) +
(cij − σij (t))2 ,
2
2α
E
(10)
where η is a positive scalar and cij is an arbitrary scalar
associated to each edge. We have:
V (X, σij ) =
(7)
where Δ is an arbitrary diagonal matrix of order n and
ω̄ is a non-negative scalar.
Let us introduce the N dimensional row matrix Λ = [lij ]:
lij =
⎪
⎪
⎩
V̇ ≤ −η ω̄X T UX + ηX T [UΔ + UΛ]X +
1
(cij − σij )σ̇ij .
−
α
otherwise.
X(0) = X0
0
σij (0) = σij
, ∀(i, j) ∈ E
Adding and subtracting X T UΔX, yields:
(12)
Since f is QUAD, we can then state that X T U[F (X)−
ΔX] ≤ −ω̄X T UX. Consequently, we can write:
Choosing the edge-based strategy described in equations
(3), (4), the governing equations of the network can be
recast as follows:
Ẋ = F (X) + ΛX,
σ˙ij = φ(eij ),
(11)
E
σij ,
if (i, j) ∈ E,
− k∈Ei σik if i = j,
0
E
V̇ = ηX T U[F (X) − ΔX] + ηX T UΔX +
1
(cij − σij )σ̇ij .
+ ηX T UΛX −
α
Proof of asymptotic stability
⎧
⎪
⎪
⎨
1
(cij − σij )σ̇ij =
α
E
1
= ηX T U[F (X) + ΛX] −
(cij − σij )σ̇ij .
α
V̇ = ηX T UẊ −
In what follows, we define the matrices Ξ = diag{ξ1 , . . . , ξn }
and U = Ξ − ξξ T , where ξ = (ξ1 , . . . , ξn )T is the normalized left eigenvector corresponding to the unique
zero eigenvalue of the Laplacian matrix L. Clearly,
ξi = 1/N, ∀i = 1, . . . , N .
3
∀(i, j) ∈ E.
(13)
E
Let us denote as W (X, σij ) the right-hand side of
inequality (13) and label as W1 (X, σij ), W2 (X, σij )
and W3 (X, σij ) the first, second and third addend
of W (X, σij ) respectively. Clearly, W1 (X, σij ) and
(8)
(9)
3
W2 (X, σij ) are negative semidefinite from the assumptions.
It suffices, therefore, to show that the term
− α1 E (cij − σij (t))σ̇ij is non positive. Obviously, if
each σij is upper bounded, given that by assumption
σ˙ij ≥ 0 then there exists a value of each arbitrary constant cij that guarantees asymptotic stability of system
(8)-(9). Otherwise, if σij were unbounded, as explained
below, we would get a contradiction. For the sake of
clarity, we will split the rest of the proof into two parts.
Definition 4 Let M be a q-dimensional square matrix.
According to [10], the matrix M is said to be diagonally
dominant if
q
|aij |
(14)
|aii | ≥
j=1,j=0
Lemma 5 Let M = [mij ] be a Hermitian diagonally
dominant q-dimensional square matrix with nonnegative
diagonal entries. Then, M is positive semidefinite.
Case 1: φ belongs to class 1.
In this case, W3 (X, σ) = − E (cij − σij )||eij ||p . Both
W2 (X, σ) and W3 (X, σ) are linear functions of the various σij . Moreover, in the worst case when p = 2, they are
quadratic function of the local errors eij . Hence, if σij
diverged, both terms would also diverge linearly. Thus,
it is possible to find a suitable value of the constant η so
that, for all X and σij , |W2 (X, σij )| ≥ |W3 (X, σij )|. As
W2 (X, σij ) is negative semi-definite from the hypothesis,
we would then get that for all X and σij , V̇ ≤ 0 against
the assumption that σij diverged. Hence, the various
σij are upper bounded and, being σij monotone increasing (indeed σ̇ij ≥ 0), then limt→∞ σij = γij < +∞.
Choosing cij = γij , it then follows that V̇ ≤ 0 for all
X ∈ RnN , σij ∈ R, ∀(i, j) ∈ E and the synchronization
error e = X T UX is bounded. Moreover, notice that
V̇ = 0 ⇐⇒ e = 0 and σij = cij . Thus, from the global
Krasovskii-LaSalle principle [11], the synchronous trajectory is globally asymptotically stable and so e(t) → 0.
PROOF. By Gershgorin’s circle theorem [10], for each
eigenvalue λ of M an index i exists such that:
⎡
λ ∈ ⎣aii −
q
q
aij , aii +
j=1,j=0
⎤
aij ⎦ ,
j=1,j=0
which implies, from Definition 2, that λj ≥ 0, ∀j =
1, . . . , q.
We can now prove Corollary 3. Specifically, to prove the
thesis, as Δ = 0, it suffices to show that UΛ is negative
semidefinite so that Theorem 2 then holds. Now, UΛ is
a real but asymmetric matrix, so to assess if it is negative semidefinite, we can equivalently check the negative
semidefiniteness of matrix UΛ+ΛU. Thus, from Definition 4, −(ΛU + UΛ) is diagonally dominant (for further
details, see appendix A) and, having nonnegative diagonal entries, from lemma 5, we can say that ΛU + UΛ
is negative semidefinite. As a consequence, also UΛ is
negative semidefinite and the Corollary remains proved.
Case 2: φ belongs to class 2.
We look now at the case when φ is chosen to belong
to class 2. In this case, we know that as σij diverges,
W
3 (X, σij ) → +∞. Moreover, from (6), W3 (X, σij ) ≤
1
E (cij −σij )m. Choosing α = m, we get W3 (X, σij ) ≤
α
E (cij −σij ). So W3 (X, σij ) is upper bounded by a function depending linearly on σij . We know that W2 (X, σij )
diverges linearly to −∞ as σij → +∞. Thus, using the
same arguments used in case 1, we can conclude that
the synchronization manifold is globally asymptotically
stable.
4
Representative examples
We now validate the theoretical results on a set of representative examples. We start with the simplest case
of two identical Kuramoto oscillators, move then to the
consensus of N coupled integrators and the more complicated example of N coupled chaotic oscillators. In so
doing, we choose two adaptation laws φ belonging to
classes 1 and 2 as follows.
The hypotheses of Theorem 2 can be difficult to verify
in practice. In what follows, we show that the theorem
hypotheses become particularly easy to verify for the
large class of nonlinear systems which are QUAD with
Δ = 0 (see [6,7]). In particular, it is possible to state the
following corollary of Theorem 2.
S1) φ belongs to class 1 with p = 1:
φ(eij ) = α||eij ||.
(15)
This strategy was previously introduced in [14]; in
this case the growth of the adaptive gains is simply
proportional to the norm of the local error.
S2) φ belongs to class 2 with m = 1, i.e.
Corollary 3 If f is QUAD with Δ = 0, then the origin
of the synchronization error is globally asymptotically
stable with σij → σ̄ij < +∞.
To prove this corollary, we need to start with some definitions and preliminary lemmas.
φ(eij ) =
4
||eij ||
.
||1 + eij ||
(16)
xi(t)
0.5
0
σ(t)
In this case we have f = 0, so we can obviously state
that, according to Definition 1, f is QUAD with Δ = 0.
Thus, the hypothesis of Corollary 3 are verified and so
the synchronization manifold x1 (t) = x2 (t) is GAS. We
have tested numerically both strategies S1 and S2, setting α = 0.1 for strategy S1. In both simulation, w.l.o.g.,
we have chosen σ(0) = 0, so that, at t = 0, the two systems were initially uncoupled. Moreover, we have chosen
the initial conditions of the two oscillators randomly (in
our case x1 (0) = 1.08 and x2 (0) = 0.50). As expected,
the numerical simulations confirm that in both cases the
synchronous state is asymptotically reached while σ settles onto a constant value as depicted in Figs. 1 and 2.
1
1
2
1
2
t
3
4
5
3
4
5
0.5
0
0
t
xi(t)
Fig. 1. Two Kuramoto oscillators. Strategy S1. Evolution of
phases (top), σ (bottom).
1
2
t
3
4
5
We can recast the governing equations of the network as
follows:
0.5
0
0
1
2
t
3
4
Ẋ = ΛX,
σ˙ij = φ(eij ),
5
In this case it is trivial to observe that 0 ≤ φ(eij ) ≤
1, φ(eij ) = 0 ⇐⇒ eij = 0 and moreover that φ
is an increasing function of ||eij ||. Note that this
strategy is the edge-based version of the adaptation
law proposed in [29].
Network of 2 identical Kuramoto oscillators
4.3
(17)
(18)
(19)
−1
1
0.5
⇒ λ1 = −1, λ2 = 0; ξ =
⇒
1 −1
0.5
0.5 0
0.25 −0.25
T
⇒Ξ=
⇒ U = Ξ − ξξ =
.
0 0.5
−0.25 0.25
L=
Network of N Chua’s chaotic circuits
To further validate our approach, we consider a network
of N Chua’s chaotic circuits [18] coupled through all
state variables. The equations that describe the network
are (8) and (9), where this time x = (p, q, r)T and the
equation governing each node is:
and according to our notation, we have:
(20)
(21)
As shown in Figs. 3 and 4, we observe that as expected synchronization is achieved in both cases with
all adaptive gains σij settling as predicted onto some
final steady-state values. Note that since strategy S2
constraints the maximum rate of change of the adaptive
gains to be unitary, we observe a slower transient to
synchronization in Fig. 4.
Let’s consider the simplest network of two linearly coupled Kuramoto oscillators ([8], [12]). Each Kuramoto is
assumed to evolve along the flow generated by ẋi = ωi .
Notice that in this case, being E = {(1, 2)} the only edge
of the network, we have just one coupling gain in the
network σ12 = σ21 = σ.
Choosing ω1 = ω2 = 0, the equations of the network are:
ẋ1 (t) = σ(t)(x2 (t) − x1 (t)),
ẋ2 (t) = σ(t)(x1 (t) − x2 (t)),
σ̇(t) = φ(eij ),
(i, j) ∈ E.
Clearly, the integrators are QUAD with Δ = 0. From
Corollary 3, the synchronization manifold is GAS. In
our simulations, we consider a network of N = 1000
nodes connected through a Barabàsi-Albert scale-free
network, constructed from N0 = 5 starting nodes (for
further details on this topology, see [3], [21], [26]). We
choose σij (0) = 0 for all (i, j) ∈ E and, for strategy S1,
α = 0.1. Moreover, the initial condition Xi (0) of the
integrators are chosen randomly from a standard normal
distribution.
Fig. 2. Two Kuramoto oscillators. Strategy S2. Evolution of
phases (top), σ (bottom).
4.1
Adaptive consensus of N integrators
We move now to the problem of reaching consensus for
a network of N integrators studied in [20] where f (xi ) =
0, ∀i = 1, . . . , N . Again, we validate both the strategies
S1 and S2 presented above.
1
0.5
0
σ(t)
4.2
f (p, q, r) = (β(−q − γ(p)), p − q + r, ξq)T ,
(22)
where γ(p) = m0 p + 12 (m1 − m0 )(|p + 1| − |p − 1|). To
ensure the chaotic behaviour of the circuit the parameters are chosen as in [2]: β = 0.59/0.12, ξ = 0.59/0.162,
5
m0 = −0.07, m1 = 1.5. Adapting the algebraic manipulation presented in [15], it is possible to verify that the
Chua’s circuit is QUAD with Δ = 0. Firstly, notice that
we can recast the equations of the Chua’s Circuit as follows:
ẋ = Ax + g(x),
(23)
xi(t)
2
0
−2
0
2
4
2
4
t
6
8
10
6
8
10
where
⎡
0.4
0 −β 0
⎢
⎥
⎥
A=⎢
⎣ 1 −1 1 ⎦ ,
0 ξ 0
σij(t)
⎤
(24)
0.2
0
0
t
Fig. 3. Adaptive consensus of 1000 integrators. Strategy S1.
Evolution of x (top), σ (bottom).
and g(p) = [−βγ(p), 0, 0]T . Denoting with λmax
the maximum eigenvalue of A + AT and being
y = [p , q , r ] ∈ R3 , we can write:
xi(t)
2
T
(x − y) [f (x, t) − f (y, t)] =
= (x − y)T A(x − y) + (x − y)T (g(x) − g(y)) ≤
1
≤ λmax (x − y)T (x − y) + (x − y)T (g(x) − g(y)) ≤
2
1
≤ λmax (x − y)T (x − y) − m1 β(p − p ) ≤
2
1
≤ λmax (x − y)T (x − y) − m1 β(x − y)T (x − y).
2
0
−2
0
2
4
2
4
t
6
8
10
6
8
10
σij(t)
0.4
0.2
0
0
Thus, if we choose ω̄ = m1 β − 12 λmax = 4.79, we can
write:
t
Fig. 4. Adaptive consensus of 1000 integrators. Strategy S2.
Evolution of x (top), σ (bottom).
(x − y)T [f (x, t) − f (y, t)] ≤ −ω̄(x − y)T (x − y), (25)
and so f is QUAD with Δ = 0. Thus, the hypothesis of
Corollary 3 is verified and the synchronous evolution is
GAS.
To verify our theoretical conclusions, we simulate a
Erdös and Renji random network with N = 700 and
average degree k̂ = 4.5 connected with strategy S1 (5)
and S2 (6). Moreover, we select the initial conditions of
the state variables of the chaotic oscillators randomly
from a normal distribution with mean equal to 0 and
standard deviation equal to 40 and, for strategy S1, we
choose α = 0.1. Figs. 5 and 6 show the resulting synchronization dynamics. Again, as predicted by Corollary 3,
we observe that under both strategies limt→∞ e(t) = 0
and limt→∞ σij (t) = σ̄ij < +∞ for all i.
Fig. 5. 700 Chua’s chaotic circuits. Strategy S1. Evolution
of p (top), σ (bottom).
5
Conclusions
We have proven that a novel decentralized adaptive control approach can be successfully used to globally synchronize a network of nonlinear systems. The theoretical approach presented was then validated on the case
of two linearly coupled Kuramoto oscillators, N linear
integrators and a network of Chua’s chaotic circuits. All
the examples confirm the theoretical derivation showing the effectiveness of the edge-based adaptive strat-
It is worth mentioning here that again in the case of
strategy S2, the rate of change of the adaptive gains is
upper bounded and therefore we observe a slower transient to synchronization. Also, we wish to emphasize that
the specific topology of the network (scale-free, random
etc) does not affect the proof of asymptotic stability but
can have an effect on the transient dynamics which is
currently under investigation.
6
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Fig. 6. 700 Chua’s chaotic circuits. Strategy S2. Evolution
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be useful in all those application where the coupling
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7
Clearly the two addends of (A.5) are either positive or
null, then Ωij ≥ 0. From (A.2), (A.3), (A.4) (A.5) we
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A
Appendix
Firstly, notice that if −(UΛ + ΛU) is diagonally dominant, so is UΛ + ΛU. From the elementary properties
of the Kronecker product, we can write that:
UΛ + ΛU = (U ⊗ In )(Λ ⊗ In ) + (Λ ⊗ In )(U ⊗ Λ) =
= (U Λ) ⊗ In + (ΛU ) ⊗ In =
(A.1)
= (U Λ + ΛU ) ⊗ In .
As a obvious consequence of A.1, if (U Λ + ΛU ) is diagonally dominant, −(UΛ + ΛU) also has this property. In what follows, we prove that (U Λ + ΛU ) is diagonally dominant. Firstly let us summarize the properties of the symmetric matrices U and Λ. U is such that
Uij = −Uii /N = −Ujj /N for all i, j = 1, . . . , N, i = j
and Uii > 0 for all i = 1, . . . , N . Λ has nonnegative non
N
diagonal elements and is such that lii = − j=1,j=i lij .
Therefore:
(1) U and Λ have zero row and column sums;
(2) They are both diagonally dominant.
Moreover, the diagonal entries Λii ≤ 0. Also, the matrix
U Λ has zero row-sum and column-sum. In facts, we have:
N
(U Λ)ij = Ui 1
j=1
N
i=1
N
N
l1k + . . . + UiN
k=1
(U Λ)ij = l1j
N
lnk = 0, (A.2)
k=1
Uk1 + . . . + lN j
k=1
N
Ukn = 0. (A.3)
k=1
Obviously, the diagonal entries of U Λ are non positive,
i.e.:
N
Uji lij≤0
(A.4)
(U Λ)jj = Ujj lj j +
i=1,i=j
From (A.2), (A.3) and (A.4), it follows that U Λ + ΛU :=
Ω has zero row-sums and zero column-sums, and moreover it has non positive diagonal entries. The final step
is to show that all the non diagonal elements are nonnegative, so that Ω (and therefore −(UΛ + ΛU)) is diagonally dominant. Indeed, if we consider an arbitrary
non diagonal element Ωij of Ω, we have:
Ωij = (U Λ)ij + (ΛU )ij =
⎛
= (Uii + Ujj )lij + Uij ⎝
N
k=1,k=i
lkj +
N
⎞
lki ⎠ .
k=1,k=j
(A.5)
8