Control technique for synchronization of selected nodes in directed networks
Bruno Ursino, Lucia Valentina Gambuzza, Vito Latora, Mattia Frasca
aa r X i v : . [ n li n . C D ] J a n Control technique for synchronization of selectednodes in directed networks
Bruno Ursino, Lucia Valentina Gambuzza, Vito Latora, Mattia Frasca ∗ Abstract —In this Letter we propose a method to control a setof arbitrary nodes in a directed network such that they followa synchronous trajectory which is, in general, not shared by theother units of the network. The problem is inspired to thosenatural or artificial networks whose proper operating conditionsare associated to the presence of clusters of synchronous nodes.Our proposed method is based on the introduction of distributedcontrollers that modify the topology of the connections in orderto generate outer symmetries in the nodes to be controlled. Anoptimization problem for the selection of the controllers, whichincludes as a special case the minimization of the number of thelinks added or removed, is also formulated and an algorithm forits solution is introduced.
Index Terms —Network analysis and control; Control of net-works.
I. I
NTRODUCTION I N the last few decades most of the works on synchro-nization control in complex networks have focused on theproblem of steering the network towards a collective stateshared by all the units. Such a synchronized state has beenobtained by means of techniques ranging from pinning control[1], [2] to adaptive strategies [3], discontinuous coupling [4],stochastic broadcasting [5] and impulsive control [6]. Otherstudies have focused on the control of a more structured statewhere the units split into clusters of synchronized nodes, andeach one of these groups follows a different trajectory [7]–[12].In the above mentioned works, the control action is suchthat all network nodes are forced to follow a given dynamicalbehavior. However, the number of nodes and links can be verylarge in real-world systems, so that the question of whetherit is possible to control the state of only a subset of thenetwork units, disregarding the behavior of the other units,becomes of great importance. Solving the problem can leadto potentially interesting applications. Consider a team ofmobile agents and the case in which a particular task canbe accomplished by a subset of the agents only. In such ascenario, one could exploit the relationship between oscillatorsynchronization and collective motion [13] and apply controltechniques for synchronizing a subset of nodes to recruit onlya group of all the mobile agents and coordinate them. In adifferent context, it is well known that synchronization of
B. Ursino, L.V. Gambuzza, and M. Frasca are with the Department of Elec-trical Electronic and Computer Science Engineering, University of Catania,Catania, Italy. V. Latora is with the School of Mathematical Sciences, QueenMary University of London, London E1 4NS, UK and with the Departmentof Physics and Astronomy, University of Catania and INFN, Catania, Italy.This work was supported by the Italian Ministry for Research and Education(MIUR) through Research Program PRIN 2017 under Grant 2017CWMF93. ∗ Email: [email protected] the whole brain network is associated to pathological states,whereas neural areas are actively synchronized to engage infunctional roles [14]. Our approach could provide controltechniques supporting neuromorphic engineering applicationsrelying on the principles of neuronal computation [15].Recently, it has been argued that a subset of network nodescan be controlled by adopting a distributed control paradigmwhose formulation relies on the notion of symmetries in agraph [16], [17]. The approach there presented is restrictedto undirected graphs, whereas here we propose a controltechnique for the more general case of directed networks.We find that, in order to form a synchronous cluster, thenodes to control must have the same set of successors, andthe common value of their out-degree has to be larger thana threshold, which decreases when the coupling strength inthe network is increased. Both conditions can be matchedby a proper design of controllers adding to or removinglinks from the original network structure. The selection ofthe controllers is addressed by formulating an optimizationproblem, minimizing an objective function which accounts forthe costs associated to adding and/or removing links. We showthat an exact solution to the problem can be found, and wepropose an algorithm to calculate it.The rest of the paper is organized as follows: Sec. II containsthe preliminaries; the problem is formulated in Sec. III; atheorem illustrating how to design the controllers is illustratedin Sec. IV; the optimization problem and its solution are dealtwith in Sec. V; an example of our approach is provided inSec. VI and the conclusions are drawn in Sec. VII.II. P
RELIMINARIES
In this section we introduce notations and definitions used inthe rest of the paper [18]. A graph G = ( V , E ) consists of set of vertices or nodes V = { v , ..., v n } and a set of edges or links E ⊆ V × V . Network nodes are equivalently indicated as v i or, shortly, as i . If ∀ ( v i , v j ) ∈ E ⇒ ( v j , v i ) ∈ E , the graph is undirected , otherwise it is directed . Only simple (i.e., contain-ing no loops and no multiple edges) directed graphs are con-sidered in what follows. The set S i = { v j ∈ V| ( v i , v j ) ∈ E} is the set of successors of v i (in undirected graphs S i coincideswith the set of neighbors).The graph G can be described through the adjacency matrix A , a N × N matrix, with N = |V| , and whose elements are a ij = 1 if v j ∈ S i and a ij = 0 , otherwise. We define the out-degree of a node i as the number of its successors, k i = |S i | = P Nj =1 a ij . The Laplacian matrix, L , is defined as L = D − A ,where D = diag { k , ..., k N } . Its elements are: L ij = k i , if i = j , L ij = 0 , if i = j and v j / ∈ S i , and L ij = − , if = j and v j ∈ S i . From the definition it immediately followsthat L N = 0 N, and, so, is an eigenvalue of the Laplacianmatrix with corresponding eigenvector N .While arguments based on network symmetries are usedfor controlling groups of nodes in undirected networks [17],directed topologies require the notion of outer symmetricalnodes , here introduced. We define two nodes v i and v j outersymmetrical if S i = S j and ( v i , v j ) / ∈ E . This notion ismore restrictive than that of input equivalence given in [19]for networks including different node dynamics and couplingfunctions. In [19] the input set of a node v i is defined as I ( v i ) = { e ∈ E : e = ( v i , v j ) for some v j ∈ V } . Two nodes v i and v j are called input equivalent if and only if there existsa bijection β : I ( v i ) → I ( v j ) such that the type of connection is preserved, that is the coupling function is the same andthe extremes of the edges have the same dynamics. Fornetworks of identical dynamical units and coupling functions,as those considered in our work, input equivalent nodes arenodes with the same out-degree. To be outer symmetrical,a further condition is required: outer symmetrical nodes areinput equivalent nodes where the bijection is the identity. Thisproperty is fundamental for the control problem dealt with inour paper. III. P ROBLEM FORMULATION
Let us consider a directed network of N identical, n -dimensional units whose dynamics is given by ˙ x i = f ( x i ) − σ N X j =1 L ij H x j + u i , ∀ i = 1 , ..., N (1)with x i = ( x i , x i , ..., x in ) T . Here, f : R n → R n is theuncoupled dynamics of each unit, H ∈ R n × n is a constantmatrix with elements taking values in { , } that representsinner coupling, i.e., it specifies the components of the statevectors through which node j is coupled to node i , and σ > is the coupling strength. u i represent the control actions onthe network. Equations (1) with u i = 0 are extensively usedto model diffusively coupled oscillators in biology, chemistry,physics and engineering [20].Equations (1) can be rewritten in compact form as ˙ x = F ( x ) − σ L ⊗ H x + u (2)where x = (cid:2) x T , x T , ..., x TN (cid:3) T , F ( x ) = (cid:2) f T ( x ) , f T ( x ) , ..., f T ( x N ) (cid:3) T and u = (cid:2) u T , u T , ..., u TN (cid:3) T .In the following we use distributed controllers of the form u i = − σ N X j =1 L ′ ij H x j (3)where L ′ is a matrix whose elements are − , or ; if L ij =0 , setting L ′ ij = − introduces a link between two nodes, i and j , not connected in the pristine network; on the contrary,setting L ′ ij = 1 in correspondence of L ij = − removes theexisting edge ( v i , v j ) in the pristine network; finally, setting L ′ ij = 0 indicates no addition or removal of links between i and j . The diagonal elements of L ′ are L ′ ii = − P Nj =1 ,j = i L ′ ij .Notice that, even if L ′ is not a Laplacian, the resulting matrix L ′′ = L + L ′ (a matrix representing the network formed bythe original topology and the links added or removed by thecontrollers) is instead a Laplacian.The system with the controllers reads ˙ x = F ( x ) − σ L ′′ ⊗ H x (4)The problem tackled in this paper is twofold: i) given anarbitrary subset V n of n < N nodes, to determine a set ofcontrollers u i with i = 1 , . . . , N such that the nodes in V n synchronizes to each other; ii) to formulate an optimizationproblem for the selection of the controllers u i .Without lack of generality, we relabel the network nodes sothat the nodes to control are indexed as i = n + 1 , . . . , N ,such that V n = { v n +1 , ..., v N } . Objective of the controllersis, therefore, to achieve a synchronous evolution of the type x ( t ) = s ( t ) ... x n ( t ) = s n ( t ) x n +1 ( t ) = x n +2 ( t ) = ... = x N ( t ) = s ( t ) , t → + ∞ (5)In compact form the synchronous state is denoted as x s ( t ) = (cid:2) s T ( t ) , ..., s Tn ( t ) , s T ( t ) , ..., s T ( t ) (cid:3) T . In the most general case,the trajectories of the first n nodes are different from eachother and from s ( t ) , that is, s i ( t ) = s j ( t ) = s ( t ) for i, j = 1 , . . . , n , but eventually some of them may coincideor converge to s ( t ) . In the next section, we demonstrate howto select the controllers such that the state x s ( t ) exists and islocally exponentially stable, while, in the second part of thepaper, we consider the optimization problem.IV. D ESIGN OF THE CONTROLLERS
To achieve a stable synchronous state x s ( t ) the controllers u i as in Eq. (3) have to satisfy the conditions expressed bythe following theorem. Theorem IV.1.
Consider the dynamical network (1) andthe controllers (3) such that the Laplacian L ′′ satisfies thefollowing conditions: L ′′ i ,j = L ′′ i ,j for i = n + 1 , . . . , N , i = n +1 , . . . , N , j = 1 , . . . , N and j = i , j = i ; L ′′ i ,i = 0 for i = n + 1 , . . . , N , i = n + 1 , . . . , N with i = i ;then, a synchronous behavior x s ( t ) = (cid:2) s T ( t ) , ..., s Tn ( t ) , s T ( t ) , ..., s T ( t ) (cid:3) T exists.In addition, define k i = P j L ′′ i,j with i = n + 1 , . . . , N ,and, since from hypothesis 1) k n +1 = . . . = k N , define k , k n +1 = . . . = k N . If there exists a diagonal matrix L > and two constants q > and τ > such that the following linear matrixinequality (LMI) is satisfied ∀ q ≥ q and t > : [ Df ( s ( t )) − qH ] T L + L [ Df ( s ( t )) − qH ] ≤ − τ I n , (6) where Df ( s ( t )) is the Jacobian of f evaluated on s ( t ) ; k is such that k > qσ ;then, the synchronous state is locally exponentially stable.roof. Existence of the synchronous solution. Hypotheses 1)and 2) induce some structural properties in the new net-work defined by the original topology and the controllerlinks. In particular, hypothesis 1) is equivalent to requirethat each node in V n has the same set of successors, thatis, S ′′ n +1 = . . . = S ′′ N , while hypothesis 2) requires thatthere are no links between any pair of nodes in V n , thatis, ∀ v i , v j ∈ V n ⇒ ( v i , v j ) / ∈ E . Consequently, selectingthe controllers such that hypotheses 1) and 2) hold makes thenodes in V n outer symmetrical.In turns this means that, with reference to the system inEq. (4), if we permute the nodes in V n , the dynamical networkdoes not change, and the n nodes have the same equationof motion. If the nodes in V n start from the same initialconditions, then they remain synchronized for t > t , andthus a synchronous solution x s ( t ) as in Eq. (5) exists. Local exponential stability of the synchronous solution.
Toprove the stability of x s ( t ) , we first prove that the synchronoussolution x s ( t ) is locally exponentially stable if ˙ ζ = ( Df − σkH ) ζ (7)is locally exponentially stable.We first consider Eq. (4) and linearize it around x s ( t ) . Wedefine η = x − x s and calculate its dynamics as ˙ η = DF η − σ ( L ′′ ⊗ H) η (8)Let us indicate as Df i the Jacobian of F evaluatedon x si ( t ) . Taking into account Eq. (5), it follows that Df n +1 = ... = Df N , Df s and, hence, DF = diag { Df , ..., Df n , Df s , ..., Df s } .From the structure of x s , it also follows that the syn-chronous behavior is preserved for all variations belongingto the linear subspace P generated by the column vectors ofthe following matrix M s = ... ... ... ... . . . ... ... ... ... √ n ... ... . . . ... ... ... √ n ⊗ I n Such variations in fact occur along the synchronizationmanifold where all the last n units have the same evolution.The column vectors of M s represent an orthonormal basis forthe considered linear subspace with dim ( P ) = n ( n + 1) . Theremaining vectors in R nN \ P represent transversal motionswith respect to the synchronization manifold.An orthonormal basis for R nN is built by consideringa linear vector space O of dim ( O ) = n ( n − that isorthogonal to P . All vectors of R nN can be thus expressed as linear combinations of vectors in P and vectors in O , thatis, η = ( M ⊗ I n ) ξ with M = ... ...
00 1 ... ... ... ... . . . ... ... ... . . . ... ... ...
00 0 ... √ n o n +1 , ... o n +1 ,n − ... √ n o n +2 , ... o n +2 ,n − ... ... . . . ... ... ... . . . ... ... √ n o N, ... o N,n − (9)For easy of compactness, matrix M is rewritten as M = (cid:20) I n R n (cid:21) .The evolution of the first n ( n + 1) elements of ξ isthe evolution of motions along the synchronization manifold,while the remaining elements of ξ are transversal to thesynchronization manifold. As a consequence of this, to provethe exponential stability of the synchronization manifold, wehave to prove that the evolution of the last n ( n − elementsof vector ξ decays exponentially to as t → + ∞ .We now apply the transformation ξ = ( M ⊗ I n ) − η toEqs. (8): ˙ ξ = (cid:0) M − ⊗ I n (cid:1) DF ( M ⊗ I n ) ξ − σ (cid:0) M − L ′′ M (cid:1) ⊗ Hξ (10)Straightforward calculations yield that (cid:0) M − ⊗ I n (cid:1) DF ( M ⊗ I n ) = DF . Let us now focuson M − L ′′ M . To calculate this term, we partition L ′′ in (4)as follows: L ′′ = (cid:20) A n × n B n × n C n × n D n × n (cid:21) (11)From hypothesis 2), it follows that D = kI n . Consider nowthe block C n × n . From hypothesis 1) we have that L ′′ i ,j = L ′′ i ,j , ∀ j ≤ n , ∀ i , i > n . Denoting with C i the i -th rowof C we obtain that C i = C i , ∀ i , i = 1 , ..., n .Given that M − L ′′ M = (cid:20) A BR n R Tn C R Tn DR n (cid:21) (12)since D = kI n and all the rows in C are equal, we canrewrite L ′′ as: L ′′ = A Bc c ... c ... c n c n ... c n k ... k ... ... ... . . . ... ... k (13)where c i ∈ { , } if node i is connected or not with the nodesof V n .Notice that the first row of R Tn is a vector parallel to [1 , , ..., while the remaining ones are all orthogonal to it,so: R Tn C = a √ n ... a n √ n ... ... . . . ... ... (14)oreover R Tn DR n = R Tn kI n R n = kI n = D .It follows that Eq. (10) becomes: ˙ ξ = Df · · · · · · ... ... ... ... ... · · · Df n · · · · · · Df s · · · · · · Df s · · · ... ... ... ... ... · · · · · · Df s ξ − σ · l . . . l n l n +1 l n +2 . . . l N ... ... ... ... ... l n . . . l n n l n n +1 l n n +2 . . . l n N c √ n . . . c n √ n k . . . . . . k . . . ... ... ... ... ... . . . . . . k ⊗ Hξ (15)where the lines in the matrices suggest a partition highlightingthe last n ( n − elements of ξ .The system describing the evolution of variations transversalto the synchronization manifold is uncoupled from the rest ofthe equations and composed of identical blocks, taking thefollowing form: ˙ ζ = ( Df s − σkH ) ζ, with ζ ∈ R n (16)It only remains to prove the exponential stability of (16).By hypothesis 4) we have that k > qσ , thus kσ > q . Fromthe LMI (6) we can use the Lyapunov function V = ζ T Lζ toprove exponential stability:1) V (0) = 0 ;2) V ( ζ ) = ζ T Lζ > , ∀ ζ = 0 because L > ;3) ˙ V < , ∀ ζ = 0 in fact: ddt (cid:2) ζ T Lζ (cid:3) = ˙ ζ T Lζ + ζ T L ˙ ζ = ζ T h ( Df s − σkH ) T L + L ( Df s − σkH ) i ζ ≤ − τ ζ T ζ < . (17)We note that, in the application of Theorem IV.1, we canfirst consider the set formed by the union of the successors ofthe nodes to control. If the cardinality of this set is greater than qσ , then we can add links such that the successors of each nodeof V n are all the elements of this set. Otherwise, one needsto expand this set by including other nodes of the network.Interestingly, the choice of such nodes is totally arbitrary andany node, not yet included in the set of successors, fits for thepurpose.The upper bound of k is the cardinality of V \ V n and, sinceTheorem IV.1 requires that k > qσ , a necessary condition forthe application of the proposed technique is that σ > qn : ifthis condition is not met, then, there are not enough nodes in V \ V n to which the nodes of V n can be connected by thecontrollers. V. O PTIMIZATION
In this section we address the problem of optimizing thecontrollers with respect to the cost of the links added orremoved. Let w − ij ( w + ij ) be the cost associated to the removal ofan existing link (addition of a new link) between i and j . Theseparameters account for a general scenario where different linkshave different costs to change.Formally, the following minimization problem is consid-ered: min u ∈U X L ′ i,j =1 L ′ ij w − ij + X L ′ i,j = − |L ′ ij | w + ij (18)where U is the set of controllers that satisfies Theorem IV.1and, thus, ensures the existence and stability of x s ( t ) . In thespecial case, when the costs are equal and unitary, i.e., w − ij = w + ij = 1 , the optimization problem reduces to min u ∈U X i,j |L ′ ij | (19)i.e., minimization of the number of links added or removedby the controllers.Let ¯ k , ⌈ ¯ qσ ⌉ . Theorem IV.1 requires that the nodes in V n have a number of successors greater than or equal to ¯ k , i.e.,since |S ′′ | = k , k ≥ ¯ k . The optimization problem is thusequivalent to determine the nodes in S ′′ which minimize theobjective function (18). Consider the set ¯ S = S v i ∈ V n S i \ V n ,containing the successors of at least one node of the pristinenetwork that are not in V n . Depending on the cardinality ofthis set we can have two different scenarios: 1) if | ¯ S| < k ,then, S ′′ needs to contain all the nodes in ¯ S and some othernodes of the set V n \ ¯ S ; 2) if (cid:12)(cid:12) ¯ S (cid:12)(cid:12) ≥ ¯ k , then, one has toselect S ′′ ⊂ ¯ S . In both cases, the choice of the nodes in S ′′ isaccomplished taking into account the costs associated to thenetwork links.First, note that, given V n , to fulfill condition 2) of The-orem IV.1 the links between nodes in this set need to beremoved. This yields a fixed cost ¯ c = P i,j ∈ V n a ij w − ij such that min u ∈U P L ′ i,j =1 L ′ ij w − ij + P L ′ i,j = − |L ′ ij | w + ij ≥ ¯ c .Let c + i be the cost to have node i in S ′′ and c − i the costof not including it in S ′′ . It follows that c − i = P j ∈ V n a ij w − ij and c + i = P j ∈ V n (1 − a ij ) w + ij . Once calculated c + i and c − i , we reformulate the optimization problem in terms ofminimization of the overall cost of the control: Cost = P v i ∈ ¯ S (cid:2) c + i x i + c − i (1 − x i ) (cid:3) + ¯ c , where x i ( i = 1 , . . . , N ) aredecisional variables, such that x i = 1 if v i ∈ S ′′ and x i = 0 otherwise. The optimization problem now reads: min P v i ∈ ¯ S (cid:2) c + i x i + c − i (1 − x i ) (cid:3) + ¯ c P v i ∈ ¯ S x i ≥ ¯ k (20)where the constraint P v i ∈ ¯ S x i ≥ ¯ k guarantees that condition k ≥ ¯ k holds. Since the overall cost can be rewritten as Cost = v i ∈ ¯ S (cid:0) c + i − c − i (cid:1) x i + P v i ∈ ¯ S c − i + ¯ c and the terms P v i ∈ ¯ S c − i and ¯ c do not depend on the variables x i , the optimization problembecomes min P v i ∈ ¯ S c i x i P v i ∈ ¯ S x i ≥ ¯ k (21)where c i , c + i − c − i .This formulation prompts the following solution for theoptimization problem. Defining k ′ , (cid:12)(cid:12)(cid:8) v i ∈ ¯ S | c i ≤ (cid:9)(cid:12)(cid:12) andsorting the nodes in ¯ S in ascending order with respect to theircost c i , we take k max = max (cid:8) ¯ k, k ′ (cid:9) and assign x i = 1 tothe first k max nodes and x i = 0 to the remaining ones. Theoverall cost to achieve synchronization of the nodes in the set V n is given by Cost = X v i ∈ ¯ S c − i + X v i ∈S ′′ c i + ¯ c (22)Algorithm 1 is based on the above observations and returnsthe nodes belonging to S ′′ . The inputs are V n , ¯ k and A (theadjacency matrix of the network) and the outputs are the set S ′′ and the overall cost. Algorithm 1
Algorithm to select the nodes in S ′′ Input: V n , ¯ k and A Output: S ′′ , overall cost Cost
Initialization : Create ¯ S = S v i ∈ V n S i \ V n and determine its cardinality | ¯ S| Procedure : Calculate c − i , c + i , ¯ c and c i = c + i − c − i Sort the nodes in ¯ S in ascending order and set k ′ = |{ v i ∈ S | c i ≤ }| Calculate k max = max (cid:8) ¯ k, k ′ (cid:9) if | ¯ S| ≥ ¯ k then return Cost = ¯ c + P v i ∈ ¯ S c − i + P v i ∈ ¯ S ′′ c i and build S ′′ taking the first k max nodes in ¯ S else Build S ′′ taking all the elements of ¯ S and add nodesfrom V n \ ¯ S , in ascending order of c i , until |S ′′ | = ¯ k end if return Cost and S ′′ VI. E
XAMPLES
We now discuss an example of how the proposed controlworks in the directed network with N = 20 nodes shownin Fig. 1. We refer to several cases, corresponding to twodistinct sets of nodes to control, V n , two values of ¯ k , anddifferent costs associated to the links. For each of these cases,the controllers that satisfy Theorem IV.1 and are the result ofthe optimization procedure of Sec. V are discussed; we willshow that they depend on the control goal, on the link costsand, through ¯ k , on the coupling coefficient.More specifically, we first consider unitary costs for thelinks and synchronization of two different triplets of nodes, -2 -1 0 1 2 3 x-axis -3-2-10123 y - a x i s
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
Fig. 1. Case study: a directed network of N = 20 nodes. The spatial positionof the network nodes is used to define costs for link addition proportional tothe Euclidean distance of the nodes to be connected. i.e., either V n = { , , } or V n = { , , } , with twovalues of ¯ k , i.e., ¯ k = 1 and ¯ k = 3 . This leads to cases 1-4 inTable I. Case 5, instead, refers to a scenario where the costsare not unitary. Case 1: V n = { , , } , ¯ k = 1 . Here, we have that ¯ S = { , , , , , } and | ¯ S| > ¯ k . Following Algorithm 1, wefind k ′ = 2 , so that k ′ > ¯ k and S ′′ = { , } . Synchronizationof the nodes in V n is achieved if two links are added to theoriginal network, and four links are removed. Case 2: V n = { , , } , ¯ k = 3 . Here again | ¯ S| > ¯ k , but k ′ < ¯ k . We get S ′′ = { , , } , four links to add and three toremove. Case 3: V n = { , , } , ¯ k = 1 . We have ¯ S = { , } and | ¯ S| > ¯ k . In this case, we obtain S ′′ = { } , a single link toadd and three links to remove. Case 4: V n = { , , } , ¯ k = 3 . We have (cid:12)(cid:12) ¯ S (cid:12)(cid:12) < ¯ k thus,following step 8 of the algorithm, we need to add a node from V n \ ¯ S , i.e. a node which is not a successor of any of the nodesto be synchronized. As the choice is completely arbitrary, weselect node . So, S ′′ = { , , } . Control is attained byadding seven links and removing three links. Case 5: V n = { , , } , ¯ k = 3 , non-unitary costs. For thepurpose of illustration, here we assume that the cost to adda link is proportional to the distance between the two nodes,while removing links always has a unitary cost. We considerthe synchronization problem as in case 2. Here, the differentcosts yield a different result for S ′′ , i.e., S ′′ = { , , } . Inthis scenario, optimization requires to include in S ′′ node 10rather than node 6.Finally, for case 2 we report the waveforms obtained bysimulating the network with control (Fig. 2). Chua’s circuitsstarting from random initial conditions are considered (equa-tions and parameters have been fixed as in [17]). Fig. 2 showsthat the nodes in V n = { , , } follow the same trajectory,while the remaining units are not synchronized with them.Similar results are obtained for the other scenarios. The value of ¯ k depends on the node dynamics considered and strengthof the coupling, e.g., with reference to Chua’s circuit as node dynamics andcoupling of the type H = diag { , , } , we have ¯ q = 4 . [17], and ¯ k = 1 for σ = 5 while ¯ k = 3 for σ = 2 .ABLE IA DDED AND REMOVED LINKS OBTAINED FOR DIFFERENT V n AND ¯ k FORTHE NETWORK IN F IG . 1. C OSTS ASSOCIATED TO LINKS ARE CONSIDEREDUNITARY IN ALL CASES , EXCEPT FOR CASE V n ¯ k Added links Removed links1 { } { } { } { } { } x , , t x i Fig. 2. Evolution of the first state variable for nodes in V n = { , , } (upper plot) and for all the network nodes (bottom plot). VII. C
ONCLUSIONS
In this work we have focused on the problem of controllingsynchronization of a group of nodes in directed networks.The nodes are all assumed to have the same dynamics and,similarly, coupling is assumed to be fixed to the same valuealong all the links of the network. The technique we propose isbased on the use of distributed controllers which add furtherlinks to the network or remove some of the existing ones,creating a new network structure which has to satisfy twotopological conditions. The first condition refers to the factthat, in the new network, merging the existing links andthose of the controllers, the nodes to control must be outersymmetrical, while the second condition requires that the out-degree of these nodes has to be higher than a threshold. Quiteinterestingly, the threshold depends on the dynamics of theunits and on the coupling strength, in such a way that a highercoupling strength favors control as it requires a smaller out-degree. It is also worth noticing that, when the out-degreeneeds to be increased to exceed the threshold, this can beobtained by connecting to any of the remaining nodes of thenetwork.The selection of the nodes forming the set of successors ofthe units to control is carried out by considering an optimiza-tion problem and finding the exact solution that minimizesthe cost of the changes (i.e. link additions or removals). Inthe case of unitary costs, the problem reduces to minimizationof the number of added or removed links, thereby defining astrategy for the control of synchronization of a group of nodes in a directed network with minimal topological changes.R
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