Optimizing synchronizability in networks of coupled systems

Automatica 112 (2020) 108711

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Optimizing synchronizability in networks of coupled systems✩ ∗

Saber Jafarizadeh a,b , , Farzad Tofigh b , Justin Lipman b , Mehran Abolhasan b a b

Rakuten Institute of Technology, Rakuten, Tokyo, 158-0094, Japan School of Electrical & Data Engineering, University of Technology Sydney, NSW, 2007, Australia

article

info

Article history: Received 18 December 2018 Received in revised form 23 August 2019 Accepted 23 October 2019 Available online xxxx Keywords: Network of coupled dynamical systems Synchronizability Semidefinite programming Pareto frontier

a b s t r a c t Of collective behaviors in networks of coupled systems, synchronization is of central importance and an extensively studied area. This is due to the fact that it is essential for the proper functioning of a wide variety of natural and engineered systems. Traditionally, uniform coupling strength has been the default choice and the synchronizability measure has been employed for analysis and enhancement of synchronizability. The main drawback of optimizing the synchronizability measure is that it can reach the Pareto frontier but not necessarily a unique point on the Pareto frontier. Additionally, the shortcoming of uniform coupling strength is that it can reach Pareto frontier in specific topologies including edge-transitive graphs. To achieve a unique optimal answer on the Pareto frontier, this paper takes a different approach and addresses the synchronizability in networks of coupled dynamical systems with nonuniform coupling strength and optimizing the synchronizability via maximizing the minimum distance between the nonzero eigenvalues of the Laplacian and the acceptable boundaries for the stability of the system. Furthermore, two solution methods, namely the concave–convex fractional programming and the Semidefinite Programming (SDP) formulations of the problem have been provided. The proposed solution methods have been compared over different topologies and branches of an arbitrary network, where the SDP based approach has shown to be less restricted and more suitable for a wider range of topologies. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction Understanding the collective behavior of interdependent components of a complex network and controlling it via the local coordination of the components have been studied in various disciplines (Albert & lászló Barabási, 2002; Strogatz, 2001). Since many real-world complex systems can be modeled as such complex networks (Albert & lászló Barabási, 2002; Barabási & Albert, 1999; Guo, Wang, & Perc, 2012; Strogatz, 2001). Examples of such systems include neural networks, ecosystems, social systems, world wide web, electrical power grids and large-scale sensor networks can be modeled as such complex networks. Within this context, synchronization in networks of coupled systems has been explored significantly, due to its essential role in many real-world systems. Synchronization has been studied from different aspects. Other than couplings among systems, the ✩ The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Florian Dorfler under the direction of Editor Christos G. Cassandras. ∗ Corresponding author at: Rakuten Institute of Technology, Rakuten, Tokyo, 158-0094, Japan. E-mail addresses: [email protected] (S. Jafarizadeh), [email protected] (F. Tofigh), [email protected] (J. Lipman), [email protected] (M. Abolhasan). https://doi.org/10.1016/j.automatica.2019.108711 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

self-dynamics governing the evolution of each isolated system also impact the global synchronization in the network. This is due to the fact that the global synchronization is achievable if the couplings among systems in the network can dominate the self-dynamics of each isolated system (which creates instability in terms of synchronization) (Pecora & Carroll, 1998). The structure of the network has the foremost influence on the synchronizability of the network (Lu & Chen, 2006; Pecora & Carroll, 1998). In Pecora and Carroll (1998), the authors have studied synchronization from the spectral approach. Via geometrical analysis of the synchronization manifold, authors in Lu and Chen (2006) find out that even though the synchronization manifold can be stable, the individual state may be unstable. In the literature, for improving the synchronizability, there are three general approaches: (i) adding or removing nodes/edges, (ii) rewiring the edges, (iii) adjusting the weights assigned to the edges (see Jalili (2013), for an overview). In a broader view, the synchronization of state-dependent network of oscillators has been addressed in Siljak (1978) and Siljak (2008). In the following literature, different strategies have been adopted, including edge-snapping approach where the optimal network topology is obtained by activating or deactivating links based on an adaptive coupling strength (DeLellis, diBernardo, Garofalo, & Porfiri, 2010). Analysis of synchronization on weighted topologies is of

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S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

interest in different fields including the computational neuroscience (Lonardoni, Amin, Di Marco, Maccione, Berdondini, & Nieus, 2017; Siri, Berry, Cessac, Delord, & Quoy, 2008), studying the synaptic plasticity (i.e., learning) in neural networks which is influenced by the synaptic weights network and the synaptic graph structure of the neural network. In the literature, generally, uniform coupling strength (or uniform weights) has been chosen for coupling strength. But for enhancing the synchronizability, a few weighting methods have been proposed in the literature. These methods are based on the load of the edges (Chavez, Hwang, Amann, Hentschel, & Boccaletti, 2005), node betweenness centralities (Jalili, Rad, & Hasler, 2008), gradient descent method (Kempton, Herrmann, & di Bernardo, 2017), and superposition of an undirected network and a directed network which is regarded as a gradient network and the couplings are governed by a gradient field on the network (Wang, Lai, & Lai, 2007). The majority of previous studies (Donetti, Hurtado, & Munoz, 2005; Pecora & Carroll, 1998; Tang, Qian, Gao, & Kurths, 2014) use the synchronizability measure for analysis and enhancement of synchronizability, which can reach the Pareto frontier but not necessarily a unique point on the Pareto frontier. In this paper, we have taken a different approach that can reach a unique optimal answer on the Pareto frontier. The following are the main contributions of this paper. 1.1. Main results

• We have studied the synchronizability (i.e., stability of synchronization) in networks of coupled dynamical systems, in its general form by considering nonuniform coupling strength for each individual link among neighboring systems. This has been done in order to reach the optimal synchronizability, since the uniform coupling strength can reach the Pareto frontier in specific topologies including edge-transitive graphs. • The usual practice in the literature (Donetti et al., 2005; Pecora & Carroll, 1998; Tang et al., 2014) is to treat the synchronizability problem by optimizing the synchronizability measure (i.e., the ratio between the second smallest and largest eigenvalues of the Laplacian). Optimizing the synchronizability measure results in a Pareto frontier (i.e., the set of Pareto optimal points). Therefore, instead of optimizing the synchronizability measure, we have proposed a new formulation (with unique optimal answer on the Pareto frontier) for optimizing the synchronizability by maximizing the minimum distance between the nonzero eigenvalues of the Laplacian and the acceptable boundaries for the stability of the system. The resultant unique optimal point on the Pareto frontier (in terms of synchronizability measure) is also optimal in terms of robustness to changes in the weights, since the corresponding nonzero eigenvalues of the Laplacian have the largest distance to the acceptable boundaries for the stability of the system. • To achieve optimal synchronizability, we have provided two general solution methods. The first method uses the results obtained from optimization of the synchronizability measure. It also utilizes the fact that in the optimal point the distances between the second smallest and the largest eigenvalues of Laplacian and the acceptable boundaries for the stability of the system are equal. For determining optimal weights assigned to links among neighboring systems, this method utilizes the property that eigenvalues of the Laplacian are homogeneous functions of degree one of the weights.

• The second method is based on semidefinite programming (SDP) formulation of the proposed optimization problem, which does not depend on the optimization of synchronizability measure. Compared to the first proposed method, the SDP method is less restricted and more suitable for general topologies. This is due to the fact that unlike the first method, the SDP formulation does not rely on the closed form formulation of the eigenvalues in terms of weights. • For certain types of topologies, namely Cartesian product of edge transitive graphs, complete cored symmetric star, lollipop, path and symmetric star topologies, we have provided the analytical solution based on both of the proposed solution methods. These analytical solutions provide a good comparison between proposed solution methods, in terms of their ease of use and practicality. • Using the proposed SDP based approach towards optimizing the synchronizability, we have provided the optimal weights over different types of branches of an arbitrary graph, that correspond to the optimal synchronizability. These branches include Path, Lollipop, semi-complete and extended complete ladder branches. The optimal weights over the edges of these branches are determined without having complete knowledge of the network topology. The rest of the paper is organized as follows. Preliminaries on graph theory, synchronization in networks and optimization of synchronizability measure are presented in Section 2. In Section 3, we formulate the optimal synchronizability as a convex optimization problem and propose two solution methods, based on optimization of the synchronizability measure and the semidefinite programming formulation of the problem. Analytical solution of the synchronizability problem for different topologies based on first and second proposed solution methods are provided in Sections 4 and 5, respectively. In Section 6, the optimal weights over different types of branches of an arbitrary graph, that correspond to the optimal synchronizability, have been determined. Section 7 concludes the paper. 2. Preliminaries In this section, we present fundamental concepts on graph theory, synchronization in networks of coupled dynamical systems, and optimization of synchronizability measure. 2.1. Undirected graph We consider an undirected connected graph G = (V , E ) consisting of the vertex set V = {1, . . . , N } and the edge set E ⊆ V × V , with {i, j} ∈ E and i ̸ = j. 2.1.1. Automorphism of graph An automorphism of the graph G = (V , E ) is a permutation σ of V such that {i, j} ∈ E if and only if {σ (i), σ (j)} ∈ E . The set of all such permutations, with composition as the group operation, is called the automorphism group of the graph and denoted by Aut(G ). For a vertex i ∈ V , the set of all images σ (i), as σ varies through a subgroup G ⊆ Aut(G ), is called the orbit of i under the action of G. The vertex set V can be written as disjoint union of distinct vertex orbits. Similarly the edge set E can be written as disjoint union of distinct edge orbits. An edge (vertex) transitive graph is a graph that has only one edge (vertex) orbit. Note that the edge transitivity and vertex transitivity of a graph are two different properties.

S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

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2.1.2. Weighted Laplacian The weighted Laplacian matrix L(w ) of graph G is defined as Li,j (w ) =

⎧ , ⎨−w ∑ i,j

for {i, j} ∈ E ,

j∈N (i)

⎩

wi,j , for i = j,

0

(1)

otherwise

where wi,j = wj,i ∈ R and N (i) denotes the neighborhood of node (i). L(w ) is a real symmetric matrix with zero-sum rows and columns. We assume that the weights are nonnegative which guarantees the non-negative spectrum of the Laplacian. The special case of wi,j = wc for {i, j} ∈ E , with wc a constant is referred to as ‘‘uniform coupling strength’’, which has been the focus of studies in Donetti et al. (2005), Pecora and Carroll (1998) and Tang et al. (2014). We denote the eigenvalues of the Laplacian matrix L(w ) in non-decreasing order, 0 = λ1 < λ2 ≤ · · · ≤ λN with corresponding orthonormal eigenvectors φi for i = 1, . . . , N, where φ1 = [1, . . . , 1]T . 2.2. Synchronization in networks Consider N identical coupled systems connected according to the undirected graph G = (V , E ) and governed by the following equation,

(dxi /dt ) = F (xi ) −

N ∑

( )

(2)

xi is the m-dimensional state vector of the ith system, and functions F : Rm → Rm and H : Rm → Rm represent its isolated (uncoupled) and coupling dynamics. The synchronized state xs (t) (where xi (t) = xs (t), ∀i ∈ {1, . . . , N }) is the solution s = F (xs ). Analyzing the stabilof the uncoupled equation dx dt ity of the synchronized state, in Pecora and Carroll (1998), the master stability function (MSF) ηmax (α) is introduced, which is determined based on functions F (.) and H(.) and independent of the network. Based on their MSF, network of coupled systems can be categorized into three main categories (Boccaletti, Latora, Moreno, Chavez, & Hwang, 2006). First category includes systems that can never synchronize (i.e, empty synchronized regions), and second category (with unbounded synchronized regions) contains systems that have stable synchronization if their corresponding λ2 (L (w)) is above a given threshold. Third category (with bounded synchronized region), which is the focus of this paper, includes systems that have a ‘‘V-shape’’ MSF (ηmax (α)) as shown in Barahona and Pecora (2002, Fig. 1) and Pecora and Carroll (2015, Fig. 4). It can be concluded that the synchronized state is stable if all the non-trivial eigenvalues of L(w ), {λk : k = 2, .., N }, lie within the interval (α1 , α2 ) (Pecora & Carroll, 1998). Example 2.1. Here, we provide a numerical example for determining α1 and α2 based on Rössler model (Rssler, 1976), defined as below, x˙ 2 = x1 + α x2 ,

In Donetti et al. (2005), Pecora and Carroll (1998) and Tang et al. (2014), the authors aim at either minimizing the ratio λN /λ2 or maximizing the ratio λ2 /λN , (both with uniform coupling strength) to achieve better synchronizability. The ratio λ2 /λN is known as the synchronizability measure (Donetti et al., 2005; Donetti, Hurtado, & Muñoz, 2006; Pecora & Carroll, 1998; Tang et al., 2014), i.e., Q = λ2 /λN ,

Li,j (w )H xj .

j=1

x˙ 1 = −x2 − x3 ,

Fig. 1. Largest Lyapunov exponent in terms of λ (the nominal eigenvalue of the weighted Laplacian matrix L(w )) for Rössler model.

x˙ 3 = β + (x1 − γ )x3 ,

where the parameters are α = 0.2, β = 0.2 and γ = 9 (Rssler, 1976). The Jacobian matrix of the isolated dynamics is a 3 × 3 matrix with the following rows, [0, −1, −1], [1, α, 0] and [x3 , 0, x1 − γ ]. Considering linear coupling through the x1 component, Jacobian of the coupling dynamics (i.e., DH) is obtained as a 3 × 3 matrix with 1 as the element in first row and column and zero for the rest. Let λ be the nominal eigenvalue of the weighted Laplacian matrix L(w ). In Fig. 1, the largest Lyapunov exponent is depicted in terms of λ. From Fig. 1, it is evident that as long as the non-trivial eigenvalues of L(w ) are between α1 = 0.2025 and α2 = 5.0025, the largest Lyapunov exponent and thus all Lyapunov exponents are negative. And the synchronization in network of coupled dynamical systems with Rössler model is stable.

(3)

Considering λ2 > α1 and λN < α2 , it is obvious that Q >

α1 . α2

2.3. Optimization of synchronizability measure Optimization of the synchronizability measure can be written as the following fractional programming problem max λ2 (L(w ))/ λN (L(w )) wi,j

s.t . Li,j (w ) = 0 for {i, j} ∈ / E. The ratio

λ2 (L(w )) λN (L(w ))

(4)

is a quasi concave function (Boyd & Vanden-

berghe, 2004). Also, since λ2 (L(w )) and λN (L(w )) are concave and convex functions of w , respectively, then problem (4) is referred to as a concave–convex fractional program and it can be transformed to an equivalent concave optimization problem (Boyd & Vandenberghe, 2004). The set of points w ∗ which is the optimal answer to the optimization problem (4) is known as the Pareto frontier (Boyd & Vandenberghe, 2004). It can be shown that problem (4) is equivalent to the following optimization problem, max t wi,j ,t

s.t .

λ2 (L(w)) − t ≥ 0, Li,j (w) = 0 for {i, j} ∈ / E. λN (L(w))

(5)

Lemma 2.1. Considering the concavity of λ2 and convexity of λN (Boyd, 2006; Boyd & Vandenberghe, 2004), it can be concluded that all optimal weights over edges within an edge orbit are equal. Proof of this lemma is provided in Appendix A. 3. Optimization of synchronizability In this section, we formulate the optimal synchronizability with reference to problem (4) as a convex optimization problem and propose two solution methods. First method is based on optimization of the synchronizability measure and the second method is based the semidefinite programming formulation of the problem.

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S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

3.1. Problem formulation The constraint on boundedness of the eigenvalues of the Laplacian matrix within the interval (α1 , α2 ) can be written as below,

α1 < λi (L(w)) < α2

(6)

for i = 2, . . . , N, where λi (L(w )) is the ith eigenvalue of the Laplacian L(w ). The aim of optimal synchronizability is to maximize the difference between the eigenvalues of the Laplacian and the boundaries of the interval (α1 , α2 ), i.e., λ2 (L(w )) − α1 and α2 − λN (L(w)). This is equivalent to maximizing the following function min (λ2 (L(w )) − α1 , α2 − λN (L(w )))

(7)

which can be written as the following optimization problem min max (α1 − λ2 (L(w )) , λN (L(w ) − α2 )) , wi,j

s.t . α1 − λ2 (L(w )) < 0, λN (L(w )) − α2 < 0,

(8)

Li,j (w ) = 0 for {i, j} ∈ / E. Since λ2 (L(w )) and λN (L(w )) are concave and convex functions of wij , respectively, then it can be concluded that the objective function in (8) and the functions α1 −λ2 (L(w )), and λN (L(w ))−α2 , are convex functions (Boyd & Vandenberghe, 2004). Thus the optimization problem (8) is a convex optimization problem. The epigraph form of the problem (8) is as below, min − s, wi,j ,s

s.t . λN (L(w )) − α2 ≤ −s, s > 0,

α1 − λ2 (L(w)) ≤ −s, Li,j (w ) = 0 for {i, j} ∈ / E.

(9)

Theorem 1. Optimal synchronizability (i.e., maximizing s) results s+α in optimal synchronizability measure Q = α −1s (i.e., maximum 2 λ2 (L(w)) /λN (L(w))). Proof. From the constraints of (9), we can deduce that α1 + s ≤ α2 −α1 , where the equality holds only 2 for the case of complete graph topology. On the other hand, we have λ2 ≥ s + α1 and λ1 ≥ α 1−s , and therefore

λ2 ≤ λN ≤ α2 − s. Thus, s ≤ N

λ2 /λN ≥ (s + α1 ) /(α2 − s) .

2

(10)

From (10), we can conclude that maximizing s results in maxiλ mum λ 2 , i.e., Q = (s + α1 ) /(α2 − s). N

3.1.1. Pareto Frontier The weights obtained from the optimization problem (4), which correspond to optimal synchronizability measure are the Pareto frontier of the synchronizability problem. The synchronizability problem seeks to maximize both of the distances α2 − λN (L(w)) and λ2 (L(w)) − α1 . This is a multi-objective optimization problem (Boyd & Vandenberghe, 2004) and as a typical feature of multi-objective optimization problems, there is no feasible solution that maximizes both of the objective functions (i.e., distances), simultaneously. Thus, the optimal point should be selected from the set of Pareto optimal points, i.e., the Pareto frontier. For set of points on the Pareto frontier, λ2 (L(w )) /λN (L(w)) is equal to the optimal synchronizability measure Q . Defining x = α2 − λN (L(w )) and y = λ2 (L(w )) − α1 , the Pareto frontier can be represented as the following line, y = Q (α2 − x) − α1 .

(11)

Fig. 2. Pareto frontier and optimal synchronizability with reference to the optimization problem (9). The optimal synchronizability obtained for Q = 1 corresponds to the case of complete graph topology. The rest of the optimal points are between this point and the origin, along the line x = y, as highlighted.

Lemma 3.1. The optimal synchronizability with reference to the optimization problem (9) is the intersection of the Pareto frontier (11) with line x = y. Proof. From Theorem 1, it is obvious that the optimal synchronizability (obtained from (9)) is on the Pareto frontier (11). Without loss of generality, we assume that α2 − λN (L(w )) ≥ λ2 (L(w)) − α1 . Then, the objective function (7) reduces to λ2 (L(w)) − α1 . The synchronizability problem aims at maximizing the objective function λ2 (L(w )) − α1 . The function λ2 (L(w )) − α1 is a concave and monotonically increasing function of the weights (w ), while α2 −λN (L(w )) is a concave and monotonically decreasing function of the weights (w ). Thus, it can be concluded that λ2 (L(w)) − α1 reaches its maximum when λ2 (L(w)) − α1 reaches its upper limit according to the constraint α2 − λN (L(w )) ≥ λ2 (L(w)) − α1 i.e.,

α2 − λN (L(w)) = λ2 (L(w)) − α1 .

(12)

Remark 1. The optimal synchronizability with reference to the optimization problem (9) for complete graph topology is the point in the intersection of the Pareto frontier for Q = 1 with line x = y. The rest of the optimal points are between this point and the origin, along the line x = y, (as highlighted in Fig. 2), where as Q decreases, the resultant optimal synchronizability with reference to the optimization problem (9) moves toward the origin. A major difference between this paper’s approach and the previous ones in the literature is evident from the relation between the optimal answer of optimization problems (4) and (9). Optimization problem (9) results in the unique point on the Pareto frontier that optimize s which in turn results in the optimal Q as well. On the other hand, optimization problem (4) results in set of points (on Pareto frontier) that optimize Q while each one of these points corresponds to different values of s. Another major distinction between this paper’s approach and the previous ones in the literature is the choice of nonuniform coupling strength. This is due to the fact that using the uniform coupling strength Pareto frontier can be reached only in specific topologies including edge-transitive graphs. 3.2. Solution method based on optimization of synchronizability measure Here, we describe a solution method for finding the optimal synchronizability with reference to the optimization problem (9) using the Pareto frontier. In the first step the optimal synchronizability measure along with the ratio between the optimal weights is obtained from solution of either optimization problems (4) or (5). Note that

S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

the definite value of optimal weights cannot be obtained from solution of either (4) or (5). In Lemma B.1 (Appendix B), it is shown that λ2 (L(w )) and λN (L(w )) are homogeneous functions of degree one of weights, i.e.,

ˆi ) , ˆi ) , λN (L(w)) = wi κi,N (w λ2 (L(w)) = wi κi,2 (w i

wi+1 , . . .. wi

i

1, The weights w1 , w2 ,. . . ,are the homogeneous variables w w while the ratios w1 , w2 , . . . are the inhomogeneous variables. For i i the weights corresponding to points on the Pareto frontier, we have Q = κi,2 /κi,N , which is independent of i. From the optimal value of Q , the optimal value of s can be obtained by setting x = y Q α −α in (11), which results in s = Q2+1 1 . Optimal value of wi (and thus other weights) can be determined based on either one of the relations λ2 (L(w )) = α1 + s or λN (L(w )) = α2 − s.

ˆi ) are indeˆi ) and κi,N (w Remark 2. In the optimal case, κi,2 (w pendent of the values of α1 and α2 , since they are functions of w w the inhomogeneous variables { w1 , w2 , . . .}. i

i

3.3. Semidefinite Programming (SDP) Formulation The problem of reducing the gap between λ2 (L(w )) and λN (L(w )) which has been derived as the optimization problem (9), can be written as the following SDP problem, wi,j ,s

s.t . (α1 + s) (I − J /N ) ⪯ L(w ), L(w ) ⪯ (α2 − s) (I − J /N )

(14)

where J = 1 × 1T is a N × N matrix with all elements equal to one. In the following, we transform the optimization problem (14) in the standard form of the semidefinite programming (Boyd & Vandenberghe, 2004). Considering ei as a N × 1 column vector with one in the ith position and )T L (w) can be written ( ( zero) elsewhere, ∑ and the optimization e − e e − e as L (w) = w i j i j i , j {i,j}∈E problem (14) can be written as below, wi,j ,s

s.t .

]

[

Z

s.t . Tr Fs · Z · Z T = −c|x| = 1,

[

]

[

] T

Tr Fi · Z · Z

(19)

= −ci = 0, for i = 1, . . . , |x| − 1.

The dual variable Z can be written as Z = Z1T , Z2T

[

Z1 =

∑

]T

,

(20)

ai,j ei − ej ,

(

)

Z2 =

{i,j}∈E

∑

(

bi,j ei − ej

)

(21)

{i,j}∈E

[

Using (20) and (21), the dual constraints in (19) (i.e. Tr Fi · Z · Z T = 0) reduce to the following Z1T ei − ej

(

(

))2

))2 ( ( = Z2T ei − ej

]

(22)

The complementary slackness condition (Boyd & Vandenberghe, 2004), reduces to

((α1 + s) − L (w)) Z1 = 0, ((−α2 + s) + L (w)) Z2 = 0

(23)

From (23), it is obvious that α2 − s and α1 + s are both eigenvalues of L (w), for optimal value of s. Considering the constraints of the optimization problem (9) stating that all eigenvalues of L (w) are bounded in between α1 + s and α2 − s, it can be concluded that for optimal value of s, we have λ2 (L (w)) = α1 + s and λN (L (w)) = α2 − s, which in turn confirms (12). Hence, the α +s optimal synchronizability ratio can be written as Q = α1 −s = t. In the case of tree topologies (and those ) that can be reduced to tree topologies where vectors ei − ej for i ̸ = j are independent of each other, (23) can be interpreted as, N ∑

( )T (α1 + s) ai,j − wi,j ei − ej Z1 = 0,

(24a)

)T ( (−α2 + s) bi,j + wi,j ei − ej Z2 = 0,

(24b)

j=1 N ∑ j=1

Using relation (22), from (24) the following can be concluded

−s

( )2 ( )2 (α1 + s) ai,j = (−α2 + s) bi,j

(α1 + s) (I − J /N ) −

∑

wi,j ei − ej

(

)(

ei − ej

)T

⪯ 0,

{i,j}∈E

∑

− (α2 − s) (I − J /N ) +

( )( )T wi,j ei − ej ei − ej ⪯ 0.

{i,j}∈E

(15)

[ ] Introducing x = ..., wi,j , . . . , s , c = [0, . . . , 0, −1], Fi as below, [ ] α1 0 F0 = ⊗ (I − J /N ) , Fs = I2 ⊗ (IN − J /N ) , (16) 0 −α2 (( )( )T ) Fi,j = −σz ⊗ ei − ej ei − ej , (17) [ ] 1 0 for {i, j} ∈ E , with σz = and ⊗ as the Kronecker 0 −1 product, problem (15) can be written in the standard form of the semidefinite programming (Boyd & Vandenberghe, 2004) as below, T

min c · x x

s.t . F (x) =

Tr F0 · Z · Z T ,

max

2

min − s

min

The dual problem is as following,

(13)

where wi can be the weight over any one of edges in the graph, w and w ˆi refers to the weights normalized by wi , i.e., ww1 , . . . , wi−1 ,

5

|x| ∑ i=1

(18) xi F i + F 0 ⪯ 0

(25)

for {i, j} ∈ E . One of the main differences between the solution methods provided in Sections 3.2 and 3.3 is that the solution method based on the optimization of Synchronizability measure (Section 3.2) requires either one of the eigenvalues λ2 (L(w )) or λN (L(w)) to be derived in terms of weights. This is a very strict requirement, which cannot be satisfied all the time. 4. Analytical solution: Based on optimization of synchronizability measure In this section, we solve the synchronizability problem for Path, Lollipop topologies and the Cartesian product of edge transitive graphs, according to the solution method described in Section 3.2. 4.1. Path topology (P4 ) In this subsection, we consider a path topology with four vertices as shown in Fig. 3. From Fig. 3, it is obvious that permuting vertices (0) with (3) and (1) with (2) does not changes the topology. Thus, edges (0, 1) and (2, 3) are in the same edge orbit and based on Lemma 2.1, the optimal weights on these edges are equal. In Fig. 3, these weights are denoted by w1 .

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S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

Fig. 3. Path topology.

Fig. 4. Lollipop topology.

The } matrix for this topology are { eigenvalues of the Laplacian

√ 0, 2w1 , w1 + w0 ± w12 + w02 . Obviously, λ2 (L(w )) = w1 + √ √ w0 − w12 + w02 and λ4 (L(w)) = w1 + w0 + w12 + w02 . The synchronizability as Q = (λ2 /λ4 ) = √ measure (Q ) can be written √ (w1 + w0 −

w12 + w02 )/(w1 + w0 +

w12 + w02 ). Defining the √ w ratio of x = w0 , we have Q = (1 − (1 + x2 )/(1 + x)2 )/(1 + 1 √ (1 + x2 )/(1 + x)2 ). Function (1 + x2 )/(1 + x)2 = 1 − (2x/(1 + x)2 ) reaches its minimum in x = 1, √ which is equivalent to w√0 = w1 . This results in √ λ2 = w0 (2 − 2) (i.e.,√κ0,2 = 2 − 2) √and Qmax = 3 − 2 2, and thus smax = ((3 − 2 2)α2 − α1 )/(4 − 2 2), α +α and w0 = w1 = 1 4 2 . 4.2. Lollipop topology In this subsection, we consider a Lollipop topology with a complete graph of size three connected to a path graph of size one, as shown in Fig. 4. From Fig. 4, it is obvious that permuting vertices (−1) and (−2) does not changes the topology. Thus, edges (0, −1) and (0, −2) are in the same edge orbit and considering Lemma 2.1, the weights can be assigned as depicted in Fig. 4. The eigenvalues of the Laplacian matrix for this topology are as

{0, w0 + 2w−1 , x± =

w0 + 2w1 ±

1 (3 2

√

9w02 + 4w12 − 4w0 w1 )}.

Choosing w−1 such that w0 + 2w−1 is between x− and x+ , i.e., x− ≤ w0 + 2w−1 ≤ x+ , then λ2 = x− and λ4 = x+ . Then, the synchronizability √ measure (Q ) can be written as Q = x− /x+ = (3w0 + 2w1 − (3w0 − 2w1 )2 + 8w0 w1 )/(3w0 + 2w1 +

√

(3w0 − 2w1 )2 + 8w0 w1 ). √ Defining the ratio x = w0 /w1 , we have Q = (1 − ((3x − 2)2 + 8x)/((3x + 2)2 ))/(1 +

((3x − 2)2 + 8x)/((3x + 2)2 )) The function ((3x−2)2 +8x)/((3x+ 2)2 ) reaches its minimum in x = 23 , which is equivalent to √ √ 3w0 = 2w1 . This results in λ2 = w0 (3 ( − ) 3) (i.e., κ0,2 = 3 − 3)

√

√

and Qmax = 2− 3, and thus smax =

w0 =

α1 +α2

√ 2− 3 α2 −α1 √ , 3− 3

and w1 =

α1 +α2 4

,

the constraint (. The√valid) range for w−(1 to satisfy √ ) above is w0 1 − 3/2 ≤ w−1 ≤ w0 1 + 3/2 , where w−1 = w0 is an acceptable answer. 6

4.3. Cartesian product of edge transitive graphs This topology is obtained from Cartesian product of m edgetransitive graphs. Based on Lemma 2.1, it can be concluded that the optimal weights over all edges of an edge transitive graph are equal. In the case of edge transitive graphs, uniform coupling strength can achieve the optimal answer. Thus, the weighted Laplacian matrix (Li (w )) of ith graph can be written as (Li (w )) = wi · Li in terms of its unweighted Laplacian matrix (Li ). Using this

Fig. 5. (a) Symmetric star topology with m = 5, n = 3, (b) CCS star topology with m = 5, n = 2.

relation the weighted Laplacian ∑m matrix for the whole graph can be derived as below, L(w ) = i=1 wi · IN1 × IN2 × · · · × INi−1 × Li × INi+1 ×· · ·× INm . INj is the identity matrix with size Nj and Nj is the number of vertices in the jth graph. We denote the eigenvalues of the ith unweighted Laplacian matrix Li in their sorted order by λi,βi where βi varies from 1 to Ni . Using this notation the eigenvalues of the weighted Laplacian of the whole graph can be written as

λβ1 ,β1 ,...,βm = w1 · λ1,β1 + w2 · λ2,β2 + · · · + wm · λm,βm

(26)

where βi for i = 1, . . . , m varies from 1 to Ni . Based on the derivation in (26) and considering the fact that first eigenvalue of each unweighted Laplacian matrix Li is zero (i.e. λi,1 = 0 for i = 1, . . . , m), the second smallest eigenvalue of the weighted Laplacian of the whole graph can be written as λ2 (L(w )) = min{w1 λ1,2 , w2 λ2,2 , . . . , wm λm,2 }, and the largest eigenvalue of the weighted Laplacian of the whole graph can be written as λN (L(w)) = w1 λ1,N1 + w2 λ2,N2 + · · · + wm λm,Nm . Assuming that wi λi,2 ≤ wj λj,2 for all(∑ i ̸ = j, then/ we )have λ2 (L(w )) = wi λi,2 , and m λN (L(w)) ≥ wi λi,2 j=1 λj,Nj λj,2 . Considering the fact that λ2 (L(w )) =/wi λi,)2 , we can conclude that ( λN (L(w ))/ λ2 (L(w ))) ≥ ( ∑ m j=1 λj,Nj λj,2 . Equality holds when wj λj,2 = wi λi,2 for all j ̸ = i. Thus the optimal synchronizability measure can be written ∑m as t ∗ = Q = λ2 (L(w )) /λN (L(w )) = 1/ j=1 (λj,Nj /λj,2 ), where t ∗ is the optimal answer of the optimization problem (5), and ∗ 1 , κi,2 = λi,2 . The optimal value of s is obtained from s∗ = t tα∗2+−α 1 where using the relation wi κi,2 = λ2 (L(w )) = α1∑ + s∗ , the optimal m weights are obtained as wi = (α1 +α2 )/(λi,2 (1 + j=1 (λj,Nj /λj,2 ))) for i = 1, . . . , m. 5. Analytical solution: Using SDP formulation In this section, we provide the analytical solution to the optimization problem (14) for Complete Cored Symmetric (CCS) star, lollipop topology, path topology, and symmetric star topologies. 5.1. Complete Cored Symmetric (CCS) star CCS star topology of order (m, n) consists of m path branches of length n, connected to each other at one end forming a complete graph, called core. Each one of path branches contains n edges. A CCS star graph of order m = 5, n = 2 is depicted in Fig. 5(b). A CCS star graph has N = m(n + 1) nodes and |E | = nm + m(m + 1)/2 edges. We denote the nodes of the graph by V = {i(n + 1) + j − n|i = 1, . . . , m, j = 0, . . . , n}. The automorphism group of CCS Star topology is Sm (permutation group of branches). In other words, nodes with the same distance

S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

7

from the central core are in the same vertex orbit. Similarly, edges with the same distance from the central core are in the same edge orbit, and from Lemma 2.1, it is obvious that the optimal weight over edges with the same distance from the central core is the same. Thus, for the edge between nodes i(n + 1) + j − n − 1 and i(n + 1) + j − n (for i = 1, . . . , m, j = 1, . . . , n), we use wj for j = 1, . . . , n and for the edges in the central core, we use w0 . For the optimal weights we have,

w0 = (α1 + α2 ) /(2m) .

(27)

wi = (α1 + α2 ) /4. for i = 1, . . . , n.

(28)

s = ((α2 − α1 ) /2) − ((α2 + α1 ) /2) cos (π/(2(n + 1)))

(29)

From (29), it can be concluded that the optimal synchronizability measure (λ2 /λN ) (i.e., the optimization variable t in (5)) for the CCS star topology is equal to ((α1 + s) /(α2 − s)) = tan2 (π /(4 (n + 1))). The detailed solution for CCS Star Topology is provided in Appendix C. Interestingly, the number of branches (i.e. m) has no impact on the optimal s. Imposing the constraint s > 0, it can be concluded that synchronization is not achievable π 1 for CCS star topology with n ≥ 2(arccos − 1 (with µ = αα2 −α ), +α (µ)) 2

1

which is independent of number of branches. For n = 0 (i.e. a α −α complete graph), the optimal value of s is equal to 2 2 1 and the synchronizability measure is equal to one, which could have been obtained from uniform coupling strength as well, since complete graph is an edge transitive graph. For n =√ 1, the optimal synchronizability measure is equal to 3 + 2 2. The synchronizability measure using uniform coupling strength, can be obtained from the eigenvalues of (C.2) (with uniform coupling √ strength), which in the case of n = 1, it is equal to √

2+m+

2 +4

2+m−

m2 +4

√m

Fig. 6. Lollipop topology with n = 3, m = 5.

and from Lemma 2.1, it is obvious that the optimal weights over these two groups of edges are the same. Thus, for the weight on the edge between nodes (i − 1) and (i) for i = 1, . . . , n, we use wi for i = 1, . . . , n; and for the edges between nodes (0) to (−l) for l = 1, . . . , m − 1, we use w0 ; and for the edges connecting node (−l) to node (−k) we use w−1 for k > l = 1, . . . , m − 1. For the α +α α +α optimal weights, we have w−1 = w0 = 12m 2 , wi = 1 4 2 , i = α −α

α +α1

1, . . . , n. The optimal value of s is equal to 2 2 1 − 2 2 where cos (θ ∗ ) is the smallest root of the following m sin ((n + 2) θ) = (m − 2) sin (nθ)

cos (θ ∗ ),

(31)

Using the optimal value of s, the optimal synchronizability meaα +s sure (λ2 (L(w )) /λN (L(w ))) can be obtained based on Q = α1 −s . 2

These results are in agreement with those obtained in Section 4.2 for the Lollipop topology with n = 1 and m = 3. The detailed solution for the Lollipop topology is provided in Appendix D. 5.3. Path topology

, For a path topology with N vertices, the Laplacian matrix L (w) can be written as below,

which is greater than 2+√2 for all m ≥ 2. For m = 2, the CCS star 2− 2 topology reduces to a path topology with 2 (n + 1) vertices. Even though, the resultant topology is not edge transitive, but uniform coupling strength can reach the optimal synchronizability. Also, these results are in agreement with those obtained in Section 4.1 for path topology as a special case of CCS Star topology with m = 2, n = 1. Using uniform coupling strength (which has been studied in Donetti et al., 2005; Pecora & Carroll, 1998; Tang et al., 2014), the optimal value of θ is equal to the smallest root of the following equation in the range [0, π ],

A path topology with N vertices is a special case of Lollipop topology with parameters n = N − 2 and m = 2. Thus, based on the results provided in Section 5.2 for the Lollipop topology, α +α it can be concluded that the optimal value of wi is equal to 1 4 2 and the optimal value of s is as below,

sin ((n + 2) θ) = (m − 1) sin (nθ) − (m − 2) sin ((n + 1) θ) ,

s = ((α2 − α1 ) /2) − ((α2 + α1 ) /2) cos (π/N ) (30)

where the optimal value of s is equal to cos (θ) and the optimal value of synchronizability measure can be obtained from (λ2 /λN ) = (α1 + s) /(α2 − s). 5.2. Lollipop topology The (m, n)-lollipop topology L(m,n) is the topology obtained by joining a complete graph Km to a path graph Pn , as shown in Fig. 6 for n = 3, m = 5. We denote the set of nodes of path graph Pn by {(1), (2), . . . , (n)} and the nodes of complete graph Km by {(0)} ∪ {(−l)|l = 1, . . . , m − 1}, where (0) is the node connected to path graph Pn , (see Fig. 6 for n = 3, m = 5). The automorphism group of Lollipop topology is Sm−1 (permutation group of m − 1 nodes {(−l)|l = 1, . . . , m − 1}). In other words, m − 1 nodes in the complete graph Km of the Lollipop topology which are not connected to the path graph of Lollipop topology are in the same vertex orbit. Similarly, edges between nodes (0) and (−l) for l = 1, . . . , m − 1 are in the same edge orbit, as well as the edges between nodes (−l) and (−k) for l < k = 1, . . . , m − 1,

L (w) =

N −1 ∑

wi (ei − ei+1 ) (ei − ei+1 )T .

(32)

i=1

(33)

Even though, the path topology is not an edge transitive graph, but the results obtained in this subsection are same as those obtained with uniform coupling strength, i.e., the optimal weights (i.e. wi ) obtained for path topology are all equal. From (33) and the constraint on positivity of s, it can be concluded that synchroπ nization is not achievable for path topology with N ≥ arccos( , µ) where µ =

α2 −α1 . α2 +α1

5.4. Symmetric star Symmetric star topology of order (m, n) consists of m path branches of length n, connected to one central node. Each one of path branches contains n edges. A symmetric star graph of order m = 5, n = 3 is depicted in Fig. 5(a). A symmetric star graph has N = 1 + nm vertices and |E | = nm edges. We denote the nodes of the graph by V = {(i − 1)n + j|i = 1, . . . , m, j = 1, . . . , n} ∪ {(0)}. The automorphism group of symmetric star topology is Sm (permutation group of branches). In other words, nodes with the same distance from the central vertex are in the same vertex orbit. Similarly, edges with the same distance from the central

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S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

vertex are in the same edge orbit, and from Lemma 2.1 it is obvious that the optimal weight over edges with the same distance from the central vertex is the same. Thus, for the weight on the edge between nodes (i−1)n+j and (i−1)n+j+1 (for i = 1, . . . , m, j = 1, . . . , n − 1), we use wj for j = 1, . . . , n − 1 and for the weight on the edges connected to the central node, we use w0 . For a symmetric star topology of order (m(, n), Laplacian ) ( matrix )LT(w) ∑m w e − e e0,0 − ei,1 + can be written as below, 0 0 ,0 i,1 i=1

)( )T ∑ ( ei,j − ei,j+1 , where ei,j for {(i, j)|i = wj m i=1 ei,j − ei,j+1 1, . . . , m, j = 1, . . . , n}∪{(0, 0)} is a column vector of size 1+mn with 1 in the position corresponding to the ((i − 1)n + j)th vertex

∑n−1 j=1

and zero, elsewhere. Following the solution procedure similar to that of the CCS star or Lollipop topologies (without the change of basis), it can be concluded that the optimal value of w0 is equal to (α1 + α2 ) /(m + 2); the optimal value of wj is equal to (α1 + α2 ) /2 for j = 1, . . . , n − 1 and the optimal value of s is α −α α +α equal to 2 2 1 − 2 2 1 cos (θ), where θ is the root of the following

equation with smallest cos (θ),

(m + 2) cos ((n + (1/2)) θ) = (m − 2) cos ((n − (1/2)) θ)

(34)

For m = 2, the symmetric star topology reduces to a path topology with 2n + 1 vertices. The results obtained in this subsection for symmetric star topology (with m = 2) are in agreement with those obtained in Section 5.3 for path topology. For n = 1, m = N − 1 ≥ 2, the symmetric star topology is an edge transitive topology. Thus, uniform coupling strength (Donetti et al., 2005), can achieve the optimal answer, i.e., s = ((α2 − α1 ) /2) − (m(α2 + α1 )) /(2(m + 2)), and Q = 1/N. 6. Optimal transition probabilities on branches of an arbitrary graph

[

Z2 TB , −bN ,N +1 +

∑

j∈NN′ +1

bN +1,j , Z2′

T

]

where NN′+1 is the set of

′ ′ neighbors ∑ of (N + 1)th node within E ′ and ∑ Z1 = {i,j}∈E ′ aij ( e i − e′ j ) − j∈N ′ aN +1,j e′ j , and Z2′ = {i,j}∈E ′ bij ( e′ i − e′ j ) −

∑

∑

N +1

j∈NN′ +1

Z1B =

bN +1,j e′ j .

∑

)

(

aij eBi − eBj + aN ,N +1 eBN

(36)

bij + bN ,N +1 eBN

(37)

{i,j}∈EB

Z2B =

∑ {i,j}∈EB

with VB = {1, . . . , N } and V ′ as the set of nodes in the unknown part of the topology. Based on the above derivation of the dual variable Z , and vectors ei , Eqs. (22) and (23) for the arbitrary branch part of topology can be written as Z1 TB eBi − eBj

(

(

))2

))2 ( ( =0 − Z2 TB eBi − eBj

(38)

with i, j ∈ VB and

((s + α1 ) IB − LB (w)) Z1B ⎛ −wN ,N +1 ⎝−aN ,N +1 +

⎞ ∑

aN +1,j ⎠ = 0,

(39a)

j∈NN′ +1

((s − α2 ) IB + LB (w)) Z2B ⎛ ⎞ ∑ +wN ,N +1 ⎝bN ,N +1 − bN +1,j ⎠ = 0,

(39b)

j∈NN′ +1

In general, the optimal value of wi,j for edges of the branch is obtained from solving Eqs. (38) and (39) (excluding the equations corresponding to Nth node).

In this section, we determine the optimal weights over different types of branches of an arbitrary graph, that correspond to the optimal synchronizability. The optimal weights over the edges of these branches are determined independent of the rest of topology. The branches considered in this section are connected to one node in the rest of topology. Thus, L (w) for a branch with N nodes can be written as below,

6.1. Path branch

LB (w) L(w ) = ⎣AT (w) 0

The optimal weights for path branch are as below

⎡

A (w)

0

⎤

BT (w)⎦ , L′ (w)

LN +1 (w) B (w)

(35)

where LB (w) is a N × N submatrix defined based on the topology of the branch [and A (w) is a N] × 1 column vector in the form of A (w)T = 0, . . . , 0, wN ,N +1 which denotes the connection between the branch and the (N + 1)th node in the whole topology. LN +1 (w) is a 1 × 1 block of the matrix corresponding to the (N + 1)th node. The square submatrix L′ (w) and the column vector B (w) correspond to the rest of graph indicating the unknown part of the topology. We define vectors ei for i ∈ V as column vectors with ith element equal to 1 and zero elsewhere. The size of vectors ei for i ∈ V is same as the number of vertices in the whole graph. These vectors can be divided into three parts i.e. eTi =

[

]

eB Ti , ei (N + 1), e′ i , where eB Ti is the first N vertices T

This branch is formed by a path graph (with N nodes). For this branch LB (w) can be written as below, LB (w) =

parts as Z1T =

Z1 TB , −aN ,N +1 +

∑

j∈NN′ +1

aN +1,j , Z1′

T

Z2T =

)T )( ( wi eBi − eBi+1 eBi − eBi+1

(40)

i=1

wi = (α1 + α2 ) /4,

for i = 1, . . . , N − 1

(41)

The detailed solution for path branch is provided in Appendix F. 6.2. Lollipop branch This branch is formed by a Lollipop graph (with parameters m and n), as shown in Fig. 7. For this branch LB (w) can be written as below, LB (w) = w−1

m−1 ∑ (

eB −k − eB −l

)(

eB −k − eB −l

)T

l>k=1

+ w0

m−1 ∑ (

eB −k − eB0

)(

eB −k − eB0

)T

(42)

k=1

T

(corresponding to vertices in the branch) and e′ i corresponds to the other vertices which are in the unknown part of the topology. For i = 1, . . . , N, the ith element of eBi is one and the rest is zero, ei (N + 1) is zero and e′ i is a vector of all zeros. For i = N + 1, eBi and e′ i is a vector of all zeros and ei (N + 1) is one. For i = N + 1, . . . , |V |, the (i − N)th element of e′ i is one and the rest is zero, ei (N + 1) is zero and eBi is a vector of all zeros. Similarly, we divide the dual variable into [ ] three

N −1 ∑

+

n ∑

( )( )T wk eBk−1 − eBk eBk−1 − eBk

k=1

where eBi are column vectors of size n + m, each representing vertex (i) for i = −(m−1), . . . , n. The optimal weights for Lollipop branch are as below

w0 = (α1 + α2 ) /(2m) .

(43)

wi = (α1 + α2 ) /4,

(44)

for i = 1, . . . , N − 1

S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

9

∑n

topology, where N = i=1 mi is the number of vertices in the branch. We use wi,i+1 for denoting the weights on the edge {(i, αi ), (i + 1, αi+1 )} for i = 1, . . . , n − 1, αi = 1, . . . , mi and αi+1 = 1, . . . , mi+1 , and wi for denoting the weights on the edge {(i, αi ), (i, βi )} for i = 1, . . . , n, αi ̸ = βi = 1, . . . , mi . The α +α optimal value of w1 is equal to 2(m1 +m2 ) for α1 ̸ = β1 = 1, . . . , m1 . 1

The optimal value of wi is equal to Fig. 7. Lollipop branch with parameters n = 3 and m = 5.

2 ( ) (α1 +α2 ) m2i −mi−1 mi+1

(

2mi mi +mi−1

)(mi +mi+1 )

for i =

2, . . . , n − 1 and αi ̸ = βi = 1, . . . , mi . The optimal value of wi,i+1 α +α is equal to 2 m 1+m 2 for i = 1, . . . , n − 1, αi = 1, . . . , mi and

( i i +1 ) αi+1 = 1, . . . , mi+1 .

6.5. Applicability to generic graphs

Fig. 8. Semi-complete branch with n1 = 2, n2 = 3 and m = 3.

6.3. Semi-complete branch A semi-complete branch consists of a complete graph connected to two path graphs, such that the nodes at the end of path graphs are connected to all of the nodes in the complete graph part. Semi-complete branch is identified by three parameters, namely m, n1 and n2 . n1 and n2 are the number of nodes in path graphs and m is the number of nodes in the complete graph part. A semi-complete branch with parameters n1 = 2, n2 = 3 and m = 3 is depicted in Fig. 8. In semi-complete branch, the nodes in the first path graph (with length n1 ) are denoted by {(−n1 ), . . . , (−1)} where node (−1) is connected to all nodes in the complete graph part. The nodes in the complete graph part are denoted by {(0, 1), . . . , (0, m)} and nodes in the second path graph (with length n2 ) are denoted by {(1), . . . , (n2 )} where node (1) is connected to all nodes in the complete graph part and node (n2 ) is connected to node (N + 1) in the unknown part of the topology. The automorphism group of the semi-complete branch is Sm (permutation group of nodes in the complete graph part of the semi-complete branch), i.e., nodes {(0, 1), . . . , (0, m)} are in the same vertex orbit. Thus, the set of edges connecting {(0, 1), . . . , (0, m)} to nodes (1) and (−1) are in the two separate edge orbits and we use w1 and w−1 to denote their optimal weights, respectively. And for the edges connecting nodes {(0, 1), . . . , (0, m)} to each other are in the same edge orbit and we use w0 to denote their optimal weights. For the weights on edges between nodes (i) and (i + 1), we use wi for i = −2, . . . , −n1 , and for the weights on edges between nodes (i − 1) and (i), we use wi for i = 2, . . . , n2 . The optimal value of wi α +α is equal to 1 4 2 for i = −2, . . . , −n1 and i = 2, . . . , n2 . The α +α optimal values of w−1 and w1 are equal to 2(1m+12) . The optimal value of w0 is

(α1 +α2 )(m−1) . 2m(m+1)

6.4. Extended complete ladder branch The extended complete ladder branch is a complete ladder graph with parameters (m1 , m2 , . . . , mn ), such that all nodes in the nth layer are connected to the rest of graph. An extended complete ladder graph ECLn is n complete graphs with m1 , m2 , . . . , mn nodes each connected to each other in the form of a n-partite graph. In extended complete ladder branch, each vertex is denoted by (i, αi ) where i varies from 1 to n and αi = 1, . . . , mi for i = 1, . . . , n. Vertices (n, α ) for α = 1, . . . , mn are connected to node (N + 1) in the unknown part of the

Utilizing these results in generic graphs of arbitrary size requires the graph branches to be identified in the first place, which can be a challenging task in large random graphs. This can be done by first identifying all maximal cliques in the graph and then checking their connections with their neighboring cliques and path subgraphs. Finally recognizing the parameters of branches (i.e., their size) and assigning optimal weights using the closed form formulas provided in this section. For identifying all maximal cliques in a graph of size N, using Bron–Kerbosch ( )algorithm, the worst-case running time is shown to be O 3N /3 (Tomita, Tanaka, & Takahashi, 2006). Developing of an algorithm for performing all steps mentioned above, and its complexity analysis is beyond the scope of this paper. But the results provided in this section can reduce the number of variables in the final optimization problem. 7. Conclusions Considering networks of coupled systems, we have addressed synchronizability in such networks. One of the trends in the literature addresses optimization of the synchronizability measure using uniform coupling strength. This results in non-unique answers which can reach the Pareto frontier in specific topologies including edge-transitive graphs. In this paper, we have studied the synchronizability problem in a more general form by considering nonuniform coupling strength for each individual link among neighboring systems. Also, instead of synchronizability measure, we have optimized the synchronizability by maximizing the minimum distance between the nonzero eigenvalues of the Laplacian and the acceptable boundaries for the stability of the system. The newly proposed approach has the advantage of resulting in a unique optimal point on the Pareto frontier rather than Pareto frontier. We have provided two solution methods for the proposed formulation, where their solutions over several topologies have shown that SDP based solution is more suitable in general. Furthermore, based on the SDP approach, we have provided the optimal weights over different types of branches of an arbitrary graph, that correspond to the optimal synchronizability. These weights have been determined without having complete knowledge of the network topology. Further investigation is required to study the effect of different weighting methods, along with other (mainly structural) properties of the networks, such as clustering coefficient and small world properties of the network, on synchronizability. Appendix A. Proof of Lemma 2.1 In Boyd and Vandenberghe (2004), it is shown that given a symmetric matrix X , its largest eigenvalue λmax (X ) is a convex function of X , while its smallest eigenvalue λmin (X ) is a concave

10

S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

function of X . Let w ∗ be the weights that optimize the synchro∑ ∗ nizability measure (3). Defining L = |Aut1(G )| σ ∈Aut (G ) σ (L(w )), it is obvious that σ L = L, and

( )

(Appendix E), following is concluded for eigenvalues of L0 (w ) and Li (w ),

λn (Li (w)) ≥ λn (L0 (w)) ≥ · · · ≥ λ1 (Li (w)) ≥ λ1 (L0 (w)) = 0

⎛⎛

⎞ ⎞ ∑ ( ) ( ∗) (a) λN L = λN ⎝⎝ σ L(w ) ⎠ /|Aut (G )|⎠ ≤

(C.3) From (C.3), it can be concluded that the second largest and smallest eigenvalues of L(w ) are largest and smallest eigenvalues of Li (w ). Now the problem has reduced to the synchronization problem over a Path topology with a self loop on one end with weight mw0 . Since all matrices Li (w) for i = 1, . . . , m − 1 (and their spectrum) are identical then in the rest of the solution provided for the CCS star topology, we only consider L1 (w). Thus L1 (w) can be written as below,

σ ∈Aut (G )

∑

(1/|Aut (G )|)

( ( )) ( ) λN σ L(w∗ ) ≤ λN L(w∗ ) ,

σ ∈Aut (G )

where (a) is due to convexity of λN (·). For λ2 L , we have

( )

⎛ ∑

λ2 L = λ2 ⎝(1/|Aut (G )|)

( )

⎞ ) (b) σ L(w∗ ) ⎠ ≥ (

σ ∈Aut (G )

∑

(1/|Aut (G )|)

L1 (w) = mw0 φ1,0 φ1T,0 +

( ( )) ( ) λ2 σ L(w∗ ) ≥ λ2 L(w∗ ) ,

where (b) is due to concavity of λ2 (·). It is obvious that equality holds only for L = L(w ∗ ), i.e. the symmetric case.

(C.4) Introducing x = [w0 , w1 , . . . , wn , s], c = [0, . . . , 0, −1], Fi as below,

Appendix B. Homogeneity of eigenvalues of Laplacian

[ α1

F0 = Lemma B.1. For a given connected graph, the eigenvalues of its weighted Laplacian matrix λi (L(w )) are homogeneous functions of degree one of the weights (wi ). Proof. The eigenvalue equation for L(w ) can be written as (L(w) − λ (L(w)) I ) ν = 0. Comparing the above eigenvalue equation of L(w ) with that of L(ηw ), i.e., (ηL(w ) − λ (L(ηw )) I ) ν = 0, where η is a constant, and considering the fact that L(ηw ) = ηL(w), the following relation between the eigenvalues of L(w) and L(ηw ) can be established

λi (L(ηw)) = ηλi (L(w)) . Hence, it can be concluded that λi (L(w )) is a homogeneous functions of degree one of wi . Appendix C. CCS star topology: Solution using SDP For a CCS star topology of order (m, n), the Laplacian matrix L (w) can be written as below, m−1 ∑

w0 E0 +

i>j=1

n ∑

ν=1

wν

m ∑

Ei,ν

(C.1)

i=1

where E0 = (ei,0 − ej,0 )(ei,0 − ej,0 )T , Ei,ν = (ei,ν−1 − ei,ν )(ei,ν−1 − ei,ν )T and ei,µ are orthonormal basis defined as ei,ν = ei ⊗ eν for i = 1, . . . , m; ν = 0, . . . , n, where ei and eν are m × 1 and n × 1 column vectors with one in the ith and the ν th position respectively and √ zero elsewhere. Defining the new basis φi,ν = ∑m−1 ik ω e / m − 1, i = 0, . . . , m − 1, ν = 0, . . . , n with ω = i ,ν k=1 2π

e−j m , the operator L(w ) is transformed into the block diagonal form as L(w ) = diag(L0 (w ), L1 (w ), . . . , Lm−1 (w )) where Li (w ) = mw0 e1 eT1 +

n ∑

wi (ei − ei+1 )(ei − ei+1 )T

( )( )T wν φ1,ν−1 − φ1,ν φ1,ν−1 − φ1,ν .

ν=1

σ ∈Aut (G )

L (w) =

n ∑

(C.2)

i=1

for i = 1, . . . , m − 1, where ei for i = 1, . . . , n + 1 are column vectors with 1 in the ith position and zero elsewhere, and L0 (w ) = Li (w ) − mw0 e1 eT1 , where e1 is n × 1 column vector with one in first position and zero elsewhere. Considering the relation between Li (w ) and L0 (w ), and using Courant–Weyl inequalities

0

0

−α2

]

( ) ⊗ I , Fs = I2N , Fw0 = −σz ⊗ mφ1,0 φ1T,0 , (C.5)

and Fwν = −σz ⊗ ((φ1,ν−1 − φ1,ν )(φ1,ν−1 − φ1,ν ) ), for {i, j} ∈ E , synchronization problem over a CCS star topology can be written in the standard form of the semidefinite programming (Boyd & Vandenberghe, 2004). Substituting L1 (w) from (C.4) in (23), Eqs. (21)(for the CCS topology can be written ) ) ∑ ∑n as (Z1 = a0 φ1,0 + n ν=1 aν φ1,ν−1 − φ1,ν , Z2 = b0 φ1,0 + ν=1 bν φ1,ν−1 − φ1,ν . Substituting them in the complementary slackness condition (23), we have T

(s + α1 ) a0 − mw0 (a0 − a1 ) = 0

(C.6a)

(s + α1 ) ai − wi (2ai − ai−1 − ai+1 ) = 0,

(C.6b)

(s + α1 ) an − wn (2an − an−1 ) = 0

(C.6c)

(s − α2 ) b0 + mw0 (b0 − b1 ) = 0

(C.6d)

(s − α2 ) bi + wi (2bi − bi−1 − bi+1 ) = 0,

(C.6e)

(s − α2 ) bn + wn (2bn − bn−1 ) = 0

(C.6f)

where (C.6b) and (C.6e) hold for i = 1, . . . , n − 1. From (25), we have ((s + α1 )ai )2 = ((s − α2 )bi )2 for i = 1, . . . , n. These relations can also be obtained from (22). From these relations we can conclude the following

(ai /ai+1 )2 = (bi /bi+1 )2 , for i = 0, . . . , n − 1.

(C.7)

From (C.6a) and (C.6d), we have (s + α1 − mw0 )2 a20 = (mw0 a1 )2 , (s − α2 + mw0 )2 b20 = (mw0 b1 )2 . From these equations and considering (C.7) for i = 1, we have (s + α1 − mw0 )2 = (s − α2 + α −α mw0 )2 . For this equality to be satisfied, either s should be 2 2 1 α −α

(which is only possible for complete graph topology, since 2 2 1 is the maximum possible value for s) or (27) should hold. Substituting (27) in (C.6a) and (C.6d), a1 (b1 ) can be written in terms of a0 (b0 ). Using these results in (C.6b) and (C.6e) for i = 2, and α1 +α2 considering (C.7) for i = 2, we have (s + α −α + α1 − 2w1 )2 = −2s (s + α

α1 +α2

2 −α1 −2s

2

1

− α2 + 2w1 )2 where (28) for i = 1 can be concluded.

Continuing this procedure recursively, (28) is obtained for the optimal value of wi . Substituting (28) in (C.6), we have

(s − ((α2 − α1 ) /2)) a0 + ((α2 + α1 ) /2) a1 = 0,

(C.8a)

(s − ((α2 − α1 ) /2)) ai + ((α2 + α1 ) /2) (ai−1 + ai+1 ) = 0, (C.8b)

S. Jafarizadeh, F. Tofigh, J. Lipman et al. / Automatica 112 (2020) 108711

(s − ((α2 − α1 ) /2)) an + ((α2 + α1 ) /2) an−1 = 0,

(C.8c)

(s − ((α2 − α1 ) /2)) b0 − ((α2 + α1 ) /2) b1 = 0,

(C.8d)

(s − ((α2 − α1 ) /2)) bi − ((α2 + α1 ) /2) (bi−1 + bi+1 ) = 0, (C.8e) (s − ((α2 − α1 ) /2)) bn − ((α2 + α1 ) /2) bn−1 = 0, (C.8b) and (C.8e) hold for i = 2, . . . , n − 2. Assuming s = α2 +α1 cos (θ) and substituting (28) in (C.8), we have 2 ai = cos (iθ) a0 ,

bi = cos (i(π − θ )) b0

(C.8f) α2 −α1 2

−

(C.9)

for i = 0, . . . , n − 1. In the (n)th stage, from (C.8c) and (C.8f) and Eqs. (C.9), we get that sin ((n + 1)θ) = 0, which in turn results in θk = ((2k + 1)π )/(2(n + 1)) for k = 0, . . . , n. Thus the optimal value of s is equal to the largest root of θk = ((2k + 1)π )/(2(n + 1)), which is (29). Appendix D. Lollipop topology: Solution using SDP For Laplacian L (w) of Lollipop topology, have L (w) = ∑m−we ∑the 1 −1 T w−1 m k=1 (e−k − e0 ) (e−k l>k=1 (e−k − e−l ) (e−k − e−l ) + w0

∑n

wk (ek−1 − ek ) (ek−1 − ek )T , where ei are column vectors of size n + m, each representing vertex √ i = ∑ −1 (i) for −(m − 1), . . . , n. We define new basis α−1 = m e−k / m − 1, k=1 ∑ m−1 lk α0 = e0 , αk = ek , for k = 1, . . . , n, and βl = k=1 e−k ω / √ 2iπ m − 1, for l = 1, . . . , m − 2, where ω = e m−1 . On the new basis the operator L (w) becomes block diagonal with following −e0 )T +

k=1

Li (w ) = (m − 1)w−1 + w0 , for i = 1, . . . , m − 2.

Appendix F. Path branch: Solution using SDP (36) and (37) Substituting LB (w) (given in (40)) in (39), Eqs. ∑ N −1 for the path branch can be written as Z1B = i=1 ai (eBi − ) ) ∑N −1 ( b e − e , and substituting the results eBi+1 , Z2B = i Bi Bi + 1 i=1 in (39), we have

(s + α1 ) a1 − w1 (2a1 − a2 ) = 0

(F.1a)

(s + α1 ) ai − wi (2ai − ai−1 − ai+1 ) = 0, ( ) (s + α1 ) aN −1 − wN −1 2aN −1 − aN −2 − aN ,N +1 = 0

(F.1b)

(s − α2 ) b1 + w1 (2b1 − b2 ) = 0

(F.1d)

(s − α2 ) bi + wi (2bi − bi−1 − bi+1 ) = 0, ( ) (s − α2 ) bN −1 + wN −1 2bN −1 − bN −2 − bN ,N +1 = 0

(F.1e)

∑

w i Ei √

((s + α1 )ai )2 = ((s − α2 )bi )2 ,

(F.2)

for i = 1, . . . , N − 1. Relations in (F.2) can also be obtained from (38). From (F.2), we can conclude the following for i = 1, . . . , N − 2

(D.2)

(s + α1 − 2w1 )2 a21 = (w1 a2 )2 , (s − α2 + 2w1 )2 b21 = (w1 b2 )2 (F.4)

T

(s + α1 )ai − wi (2ai − ai−1 − ai+1 ) = 0,

(D.3c)

(s + α1 )an − wn (2an − an−1 ) = 0

(D.3d)

(s − α2 )b0 + w0 (mb0 −

m − 1b1 ) = 0

(D.3e)

m − 1b0 − b2 ) = 0

(D.3f)

(s − α2 )b1 + w1 (2b1 −

√

(s − α2 )bi + wi (2bi − bi−1 − bi+1 ) = 0,

(D.3g)

(s − α2 )bn + wn (2bn − bn−1 ) = 0

(D.3h)

where (D.3c) and (D.3g) hold for i = 2, . . . , n − 1. Following the same procedure as in the case of CCS topology (Section 5.1), for α +α α +α optimal weights, we have w0 = 12m 2 , wi = 1 4 2 , i = 1, . . . , n, and Eq. (31) is obtained. Appendix E. The Courant–Weyl Inequalities (Cvetkovic, Doob, & Sachs, 1980) Let A and B be Hermitian matrices of size n, and let 1 ≤ i, j ≤ n.

(F.3)

On the other hand from (F.1a) and (F.1d), we have

where E0 = (ˆ e0 − m − 1ˆ e1 )(ˆ e0 − m − 1ˆ e1 ) , Ei = (ˆ ei −ˆ ei+1 )(ˆ ei − ˆ ei+1 )T , and ˆ ej for j = 0, . . . , n + 1 are column vectors with 1 in the jth position and zero elsewhere. Setting w−1 such that λ2 (L(w)) < (m − 1) w−1 + w0 < λN (L(w)), ensures that both of eigenvalues λ2 (L(w )) and λN (L(w )) are among the eigenvalues of L0 (w ) in (D.2). Substituting (D.2) in (23),( Eqs. (21) for )the √ e0 − m − 1ˆ Lollipop topology can be written as, Z1 = a0 ˆ e1 + ( ) ∑n √ ∑n ˆ ˆ ˆ ˆ e − e − e , and Z = b ei − a + b m − 1 e ) ( (ˆ 0 i i + 1 2 0 i i 1 i=1 i=1 ˆ ei+1 ). Substituting Z1 and Z2 in (24), we have √ (s + α1 )a0 − w0 (ma0 − m − 1a1 ) = 0, (D.3a) √ (s + α1 )a1 − w1 (2a1 − m − 1a0 − a2 ) = 0, (D.3b)

√

(F.1f)

(F.1b) and (F.1e) hold for i = 2, . . . , N − 2. Since edges in path branch are not part of any loop, then from (25) we have,

i=1

√

(F.1c)

(D.1)

n

L0 (w) = w0 E0 +

(i) If i + j − 1 ≤ n, then λi+j−1 (A + B) ≤ λi (A) + λj (B), (ii) If i + j − n ≥ 1, then λi+j−n (A + B) ≥ λi (A) + λj (B), (iii) If B is positive definite, then λi (A + B) ≥ λi (A).

(ai /ai+1 )2 = (bi /bi+1 )2 ,

blocks

11

Considering (F.3) for i = 1, from (F.4) we have (s + α1 − 2w1 )2 = (s − α2 + 2w1 )2 . For this equality to be satisfied, either s should α −α be 2 2 1 (which is only possible for complete graph topology)

or following should hold w1 = (α1 + α2 ) /4. Substituting w1 in (F.1a) and (F.1d), a1 (b1 ) can be written in terms of a2 (b2 ). Using these results in (F.1b) and (F.1e) for i = 2, and considering α1 +α2 (F.3) for i = 2, we have (s + α −α + α1 − 2w2 )2 = (s + −2s α1 +α2 α2 −α1 −2s

2

1

− α2 + 2w2 )2 where the following can be concluded

that w2 = (α1 + α2 ) /4. Continuing this procedure recursively, the optimal value of wi is obtained as in (41). References Albert, Réka, & lászló Barabási, Albert (2002). Statistical mechanics of complex networks. Reviews of Modern Physics. Barabási, Albert-lászló, & Albert, Réka (1999). Emergence of scaling in random networks. Science, 286(5439), 509–512. Barahona, Mauricio, & Pecora, Louis M. (2002). Synchronization in small-world systems. Physical Review Letters, 89, 054101. Boccaletti, S., Latora, V., Moreno, Y., Chavez, M., & Hwang, D.-U. (2006). Complex networks: Structure and dynamics. Physics Reports, 424(4), 175–308. Boyd, S. (2006). Convex optimization of graph laplacian eigenvalues. In Proceedings of the international congress of mathematicians. Boyd, S., & Vandenberghe, L. (2004). Convex optimization. New York, NY, USA: Cambridge University Press. Chavez, M., Hwang, D-U, Amann, Arno, Hentschel, H. G. E., & Boccaletti, Stefano (2005). Synchronization is enhanced in weighted complex networks.. Physical Review Letters, 94 21, 218701. Cvetkovic, D. M., Doob, M., & Sachs, H. (1980). Spectra of graphs: Theory and application. New York: Academic Press. DeLellis, P., diBernardo, M., Garofalo, F., & Porfiri, M. (2010). Evolution of complex networks via edge snapping. IEEE Transactions on Circuits and Systems. I. Regular Papers, 57(8), 2132–2143.

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Saber Jafarizadeh received his B.S. and M.S. in Electrical Engineering from University of Tabriz in 2007 and Sharif University of Technology in 2009, respectively. He received his Ph.D. degree from the Department of Electrical & Information Engineering at the University of Sydney in 2015. He is currently a researcher at Rakuten Institute of Technology (the R&D organization of Rakuten Inc.) and a visiting scholar in University of Technology Sydney. His research interests lie in the fields of wireless ad-hoc networks and distributed control with emphasis on distributed consensus algorithms and distributed optimization. Farzad Tofigh received his B.S. and M.S. in Electrical Engineering from Urmia University, Iran, in 2011 and 2013, respectively. He is currently working on developing an infrastructure to estimate density distribution using distributed sensor networks in pursuit of his Ph.D. in Engineering at University of Technology Sydney. His research interests include wireless sensor networks, graph theory and distributed estimation methods.

Dr Justin Lipman is an Industry Associate Professor at the University of Technology Sydney and Director of the RF Communications Technologies (RFCT) Lab, where he leads industry engagement in RF technologies, Internet of Things, Tactile Internet and Software Defined Communication. He received his Ph.D. Telecommunications and B.E. in Computer Engineering from the University of Wollongong, Australia in 2003 and 1999 respectively. From 2004 to 2017, Dr. Lipman was based in Shanghai, China and held a number of leadership roles at Intel and Alcatel. He is an IEEE Senior Member. His research interests are in all "things" adaptive, connected, distributed and ubiquitous. A/Prof. Mehran Abolhasan completed his B.E. in Computer Engineering and Ph.D. in Telecommunications on 1999 and 2003 respectively at the University of Wollongong. He is currently an Associate Professor and Deputy Head of School at School of Electrical and Data Engineering at UTS. A/Prof. Abolhasan has authored over 130 international publications and has won over 3 million dollars in research funding. His current research Interests are in Software Defined Networking, IoT, Wireless Mesh, Wireless Body Area Networks, 5G and next generation Wireless networks. He is a Senior Member of IEEE.