Clustering with negative couplings

In this subsection, we study how clustering may appear as a pure effect of structured diffusive couplings. We assume that the agents’ dynamics are completely determined by their couplings:

x9iptq “ ´

N

ÿ

j“1

lijxjptq, (5.24)

or in a compact form

xptq “ ´Lxptq,9 (5.25)

Comparing to (5.1), we have taken Γ to be an identity matrix, c “ 1, and m “ 1.

The results derived below can be easily extended to the general case when c ą 0 and m ě 1. From Theorem 2.4.1 we know that, if the weights of the edges in G are all positive, i.e., aij “ ´lij ě 0, i ‰ j, and G contains a directed spanning tree, then

the system achieves consensus. In this subsection, we assume that there might be negative couplings in the graphG, and as a result G is a signed graph. The adjacency matrix A becomes a signed adjacency matrix, but the Laplacian matrix L is still defined as in (2.1) here. The approach that system (5.24) models negative couplings is different from that in (4.19), which consequently gives rise to different dynamical behavior. Since the signed Laplacian matrix considered in (4.20) is a diagonally dominant matrix, its eigenvalues all have nonnegative real parts which guarantees the convergence of system (4.20). Furthermore, the system states polarize if zero is an eigenvalue of the signed Laplacian matrix and converge to zero otherwise. However, the convergence of system (5.25) is not guaranteed since the Laplacian matrix in (5.25) may have eigenvalues with negative real parts. Zero is always an eigenvalue.

In addition, the geometric multiplicity of the eigenvalue zero and thus the dimension of the null space of the Laplacian matrix, can be larger than one. This allows the possibility of the occurrence of clustering in the system as we will see later.

Let η1 “ r1^T_l1, 0^T_{N ´l1}s^T, η2 “ r0^T_l1, 1^T_l2, 0^T_{N ´l1´l2}s^T, . . . , ηn “ r0^T_{N ´ln}, 1^T_lns^T, and let α1, . . . , αn be n independent vectors satisfying η^T_i αj “ 1, if i “ j and η^T_i αj “ 0, if i ‰ j. Since the solution to (5.25) is xptq “ e^´Ltxp0q, it is obvious that if

tÑ8lim e^´Lt“

n

ÿ

i“1

ηiα^T_i, (5.26)

then n-cluster synchronization might be achieved. We provide the following necessary and sufficient condition under which (5.26) holds.

5.3.2. Lemma. Equation (5.26) holds if and only if

Lη_i“ 0, α^T_iL “ 0, i “ 1, . . . , n, (5.27) where ´L has exactly n zero eigenvalues and all the other eigenvalues have negative real parts.

Proof : We give the proof for the case when n “ 2. The proof for the general case n ě 2 can be proved following similar steps.

(Sufficiency) This has been proved as Lemma 6 in [112].

(Necessity) Let J “ diagtJ1, . . . , Jsu be the Jordan form of ´L, i.e., there exists a nonsingular matrix P such that ´L “ P J P^´1. Then

tÑ8lim e^´Lt“ P lim

tÑ8diagte^J1t, . . . , e^JstuP^´1.

lim_tÑ8e^´Lt exists if and only if Ji are zero matrices or the eigenvalues of Ji have negative real parts. Let u1, . . . , uN be the columns of P and let v₁^T, . . . , v^T_Nbe the rows

5.3. Clustering with identical self-dynamics 73

of P^´1. Then the fact that (5.26) holds implies that J has the form J “ diagtOk, Zu, where the eigenvalues of Z have negative real parts. We have

tÑ8lime^´Lt“ P

„ I_k 0 0 0

ȷ P^´1“

k

ÿ

i“1

uiv^T_i .

Since rankpuiv_i^Tq “ 1 and rankpřN

i“1uiv_i^T “ Iq “ N , řk

i“1uiv^T_i must have rank k.

Combined with (5.26), one has k “ 2 and u1v₁^T` u2v<sub>2</sub><sup>T</sup> “ η1α<sup>T</sup><sub>1</sub>` η2α^T₂, which implies that ´L has exactly two zero eigenvalues and all the other eigenvalues have negative real parts. In addition, one has

u11v₁^T ` u21v<sub>2</sub><sup>T</sup> “ α<sup>T</sup>1, ¨ ¨ ¨ , u1l1v<sup>T</sup><sub>1</sub> ` u2l1v^T₂ “ α^T1,

which implies that pu1i´ u1jqv₁^T ` pu2i´ u2jqv^T₂ “ 0. Then u1i“ u1j and u2i“ u2j

for 1 ď i, j ď l1. Using similar arguments we have

u11“ ¨ ¨ ¨ “ u1l₁, u1l₁`1“ ¨ ¨ ¨ “ u1N, u21“ ¨ ¨ ¨ “ u2l<sub>1</sub>, u2l<sub>1</sub>`1“ ¨ ¨ ¨ “ u2N. If u11“ 0, then r0^T_l1, 1^T_l

2s^T is a right eigenvector associated with 0, and so is r1^T_l1, 0^T_l

2s^T. If u11 ‰ 0, r0^T_l1,^u2Nu11_u^´u1Nu21

11 1^T_l2s^T is a right eigenvector associated with 0. So r0^T_l1, 1^T_l2s^T and r1^T_l1, 0^T_l2s^T are right eigenvectors associated with 0.

Without loss of generality, choose u1“ η1“ r1^T_l1, 0^T_l2s^T and u2“ η2“ r0^T_l1, 1^T_l2s^T; then η1pv1´ α1q^T ` η2pv2´ α2q^T “ 0, which implies that v1 “ α1 and v2 “ α2.

Hence, one has α^T₁L “ α^T₂L “ 0. l

From Lemma 5.3.2, it is clear that in order to realize n-cluster synchronization, L_ij have to have zero row sums. In the following discussion, assume that L satisfies the condition that the row sums of L_ij, 1 ď i, j ď n, are zero, then L has zero as an eigenvalue whose geometric multiplicity is at least n. Let η1“ r1^T_l1, 0^T_{N ´l1}s^T, . . . , ηn“ r0^T_{N ´ln}, 1^T_lns^T, be n right eigenvectors associated with 0, and let α1, . . . , αn be the corresponding left eigenvectors satisfying η_i^Tα_j“ 1, if i “ j, and η^T_iα_j “ 0, if i ‰ j.

The following result is a slightly modified version of the main result of [112].

5.3.5. Theorem. Suppose that the initial values of system (5.25) satisfy that α^Tixp0q with 1 ď i ď n are not equal to each other; then n-cluster synchronization can be achieved if and only if ´L has exactly n zero eigenvalues and all the other eigenvalues have negative real parts.

The conditions stipulated in Theorem 5.3.5 for achieving n-cluster synchronization is an algebraic condition, which is difficult to check in application. Now we develop algorithms to construct appropriate coupling topologies which satisfy the conditions in Theorem 5.3.5.

5.3.3. Lemma. [44] Let A and B be N ˆ N Hermitian matrices and let the eigen-values λ_ipAq, λipBq, and λipA ` Bq be arranged in decreasing order as λNp¨q ď λ_{N ´1}p¨q ď ¨ ¨ ¨ ď λ1p¨q. For each k “ 1, 2, ¨ ¨ ¨ , N , we have

λ_kpAq ` λNpBq ď λkpA ` Bq ď λkpAq ` λ1pBq.

Intuitively, if the inner couplings within the clusters are strong enough, system (5.25) can achieve cluster synchronization. This is verified by the following results.

5.3.2. Proposition. Let

L “ diagtc1L11, . . . , cnLnnu `

»

—

—

— –

0 L₁₂ ¨ ¨ ¨ L_1n L21 0 ¨ ¨ ¨ L2n

... ... . .. ... Ln1 Ln2 ¨ ¨ ¨ 0

fi ffi ffi ffi fl

be a symmetric Laplacian matrix, L₁ “ diagtc1L₁₁, . . . , c_nL_nnu, and L2 “ L ´ L1. Suppose that Lij have zero row sums, matrices Liiare irreducible and the off-diagonal elements of ´Lii are nonnegative. If cią _{´ max1ďiďn}^ρpL2_λ^q

2p´Liiq, then ´L has exactly n zero eigenvalues and all the other eigenvalues are negative.

Proof : Since Lij have zero row sums, L has at least n zero eigenvalues. Using Lemma 5.3.3, one has

λNp´L2q ď λip´Lq ´ λip´L1q ď λ1p´L2q,

which leads to |λip´Lq ´ λip´L1q| ď ρp´L2q “ ρpL2q. It follows from cią ρpL2q{

p´ max1ďiďnλ2p´Liiqq that max1ďiďnciλ2p´Liiq ` ρpL2q ă 0. Since Lii are ir-reducible and the off-diagonal elements of ´Lii are nonnegative, it follows that λ1p´L1q “ ¨ ¨ ¨ “ λnp´L1q “ 0, and λn`1p´L1q “ max1ďiďnciλ2p´Liiq. Thus one concludes that λ_n`1p´Lq ď max

1ďiďnciλ2p´Liiq ` ρpL2q ă 0. l 5.3.3. Proposition. Suppose that the graphs G1, . . . ,Gn are balanced and strongly connected and the weights of the edges in these graphs are positive. Assume that L1, . . . , Ln are the corresponding Laplacian matrices. For any positive definite matrix S with proper dimension, zero is an eigenvalue of ´SdiagtL1, . . . , Lnu of algebraic and geometric multiplicity n, and all the other eigenvalues of ´SdiagtL1, . . . , L_nu have negative real parts.

Proposition 5.3.3 can be proved using a similar argument to that in the proof of Theorem 4.5 in [55].