Efficient identification of link importance in dynamic networks

Available online at www.sciencedirect.com

Journal of the Franklin Institute 352 (2015) 3716–3729 www.elsevier.com/locate/jfranklin

Efficient identification of link importance in dynamic networks$ Yoonsoo Kim Department of Aerospace and System Engineering, Research Center for Aircraft Parts Technology, Gyeongsang National University, Jinju 660-701, Republic of Korea Received 22 June 2014; received in revised form 13 October 2014; accepted 15 November 2014 Available online 3 December 2014

Abstract For a given network topology of linking dynamical systems, determining the least or the most important link(s) or edge(s) in dynamic networks is a complex combinatorial optimization problem. The purpose of solving this problem is to modify the given network topology, in the hope of using a less number of costly communication links while keeping or improving the network's performance. In this paper, this identification of link importance is approached via finding the least or the most sensitive edge(s) by analytically obtaining the sensitivity of each edge or numerically solving LMIs (linear matrix inequalities). The proposed schemes are applied to a consensus network, a vehicle network and random networks to support their merit. & 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Problem statement and Introduction Consider a network (graph) G of n (possibly, non-homogeneous) dynamical systems (nodes) connected via communication links (edges),1 and assume that the collective network dynamics is ☆ A preliminary version of this paper [15] was presented at The 19th World Congress of the International Federation of Automatic Control in Cape Town, South Africa. The author is greatly indebted to the Associate Editor and the anonymous reviewers' invaluable comments that have greatly helped to improve the quality of this paper. This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF), funded by the Ministry of Education, Science and Technology (Grant no. NRF-2012R1A1A1040598). E-mail address: [email protected] 1 Network and graph, dynamical system and node, and communication link and edge shall be used interchangeably in this paper.

http://dx.doi.org/10.1016/j.jfranklin.2014.11.011 0016-0032/& 2014 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

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described by a typical linear state model: x_ ¼ Ax þ Bu;

y ¼ Cx;

ð1:1Þ

where x and u are vectors with entries of xi and ui ði ¼ 1; 2; …; nÞ representing the state and control input of individual node, respectively; and A ¼ ½aij , B ¼ ½bij  and C ¼ ½cij  are constant matrices with appropriate dimension and have a structure associated with the network topology which defines who-talks-to-whom between the nodes. Suppose a signal v is injected through one of the nodes. According to the signal's characteristics, v can be a reference input r that must be tracked by all the nodes on the graph; or the signal can be an unwanted disturbance w that must be rejected by the nodes.2 For the sake of reducing the inter-communication cost, one is now interested in modifying the graph, so that a smaller number of edges is used to track or reject the signal. In this regard, when v ¼ r, it is desired to remove an edge(s) so that the network dynamics (transfer function from r to y; Gry) is affected as little as possible. In contrast, when v¼ w, it is desired to remove an edge(s) so that the network dynamics keeps or gains robustness as much as possible, i.e. the H 1 -norm of Gwy is minimized. The aforementioned link identification problem is basically a combinatorial optimization problem. For a certain measure which correctly reflects the network dynamics, this identification problem boils down to determining which edge(s) removal affects the measure the most/least, which certainly has the combinatorial nature. The most relevant works to the present problem are found in [1,2]. In [1], a bisection algorithm is proposed to determine the edge(s) affecting the network's algebraic connectivity λ2 ðGÞ the most/least. This bisection algorithm quickly computes λ2 ðGÞ after each edge removal by solving the so-called secular equation. The measure of λ2 ðGÞ, however, represents how well static nodes are connected, rather than how sensitive or robust dynamic nodes are with respect to disturbances. In other words, λ2 ðGÞ does not reveal or incorporate any information on the network dynamics. Thus, this bisection algorithm is not directly applicable to the link identification problem of present interest. In [2], a network topology design problem (NTD) for relative sensing networks is discussed. Note that NTD is concerned with designing a complete network in order to meet a given performance requirement, whereas the link identification problem of present interest is concerned with selecting (and then removing) an edge(s) while minimizing the original network's performance degradation. NTD may be formulated as a computationally challenging mixed-integer semi-definite program, and requires a relaxation technique (e.g. ignoring the integer constraint on decision variables as tried in [2]) to be solved for large network cases. It is interesting to note that some NTD found in [3] may be formulated as a (convex) semi-definite program for synchronizing oscillator networks having uniform inductances. Some other relevant works are also available in the recent literature. In [4,5], the network topology is optimized in terms of formation rigidity. In [6], a probabilistic search technique is proposed to find a network topology which minimizes the expected time to detect a target. In [7], a given network topology is modified to minimize the network's diameter by adding a path. In the course of minimization, a so-called tree network system is used to approximate the network dynamics. In [8], several conditions that characterize Laplacian spectral bounds are studied and used for a network topology design. As already mentioned, all these relevant works (except [2]) try to design a network topology using graph-theoretical measures which do not fully accommodate the network dynamics. In this work, however, the modification of a given network topology (not a complex design of the entire network like [2]) is tried to explicitly incorporate the 2

See Eq. (2.1) for an example of injecting r into the system, and Eq. (3.3) for injecting w into the system.

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network dynamics into the edge removal which leads to keeping or improving the network's dynamical performance. The rest of paper is organized as follows. In Sections 2 and 3, two kinds of link identification problems are presented: one is for v¼ r; and the other is for v ¼ w. For the first kind, an edgeremoval scheme is developed in such a way that the change in the network dynamics (transfer function from r to y, Gry) is minimized, or the peak sensitivity norm JS J 1 is minimized. This developed scheme is then applied to identify the least important edge(s) in a consensus network. For the second kind, another edge-removal scheme is developed in such a way that the network dynamics keeps or gains robustness as much as possible, or the H 1 -norm of the network dynamics (transfer function from w to y), JGwy J 1 , is minimized. This developed scheme is then applied to identify the least important edge(s) in a vehicle network with the objective of formation control. These two schemes are also applied to random networks for evaluating their probabilistic performance. Concluding remarks follow in Section 4. 2. Identification of link importance for reference tracking 2.1. Problem formulation One of the main control objectives for networked systems is to let all the dynamical systems in the network track a reference signal. For example, a group of flying vehicles may need to be controlled to chase a single target together or to maintain a fixed pattern of formation. Suppose a reference signal r is injected through some of the nodes, and the following form of control input in Eq. (1.1) u ¼ K1x þ K 2r

ð2:1Þ

is applied to accomplish a control objective, where K1 and K2 are constant matrices with appropriate dimension. After combining Eqs. (1.1) and (2.1), one can obtain the closed-loop transfer functions Gry ¼ CðsI  ðA þ BK 1 ÞÞ  1 BK 2 from r to y, and Gre ¼ I  Gry from r to e ¼ r  y, where A þ BK 1 is assumed to be Hurwitz. Note that all matrices appearing in Gry may depend on the underlying network topology G, and their entries may change if an edge(s) is removed from the network. In this section, it is desired to find an edge(s) eij in such a way to minimize JGre J 1 for G  eij , in the hope of good reference tracking with a less number of communication links. Here, G  eij denotes the graph with eij removed from G, and it must be connected. It is assumed that the optimal edge eij does not render the network unstable as well.3 2.2. Link identification using sensitivity In order to see the effect of edge removal (no communication between two dynamical systems) on the r-to-y relation, the sensitivity function S ¼ ½S ij  is defined as S ij ¼ ∂Gry =∂eij for some edge weight eij.4 Suppose each entry, say aij, of the state and control matrices, is a differentiable 3

Finding the optimal edge while ensuring stability shall be a future research topic. A relevant work is found in [9] and the references therein. 4 A similar sensitivity function is defined in [10]. In this work, the authors use Mason's direct rule to derive the sensitivity function, and analyze the effect of cycles on the sensitivity function.

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function of eij. Then, one can easily arrive at the following result after some straightforward calculus. Theorem 2.1. For given i; j
 ∂C ∂B ∂K 2 ðsI  ðA þ BK 1 ÞÞ  1 BK 2 þ CðsI  ðA þ BK 1 ÞÞ  1 K2 þ B ∂eij ∂eij ∂eij
 ∂A ∂B ∂K 1 þCðsI  ðA þ BK 1 ÞÞ  1 þ K1 þ B ðsI  ðA þ BK 1 ÞÞ  1 BK 2 ; ∂eij ∂eij ∂eij

S ij ¼

ð2:2Þ

where ∂M=∂eij (M ¼ ½mij  ¼ A; B; C; K 1 or K2) is an all-zero matrix except ∂mij =∂eij at (i,j) position. If aij ¼ eij and all the other matrices are insensitive to eij, Eq. (2.2) reduces to S ij ¼ CðsI  ðA þ BK 1 ÞÞ  1 E ij ðsI  ðA þ BK 1 ÞÞ  1 BK 2 ;

ð2:3Þ

where Eij is an all-zero matrix except an ‘1’ at (i,j) position. Proof. The claim easily follows using the fact that ∂M  1 =∂eij ¼  M  1 ð∂M=∂eij ÞM  1 .

□

Since S can be given in the analytic form such as Eq. (2.3) and the matrix derivatives are almost zero matrices, it is easy and quick to check how much a small change in the edge weight (or communication strength) affects the r-to-y relation.  In fact, S caneasily  be computed due to two additional facts that (1) ðsI  ðA þ BK 1 ÞÞ  1 ¼ L eðAþBK 1 Þt ¼ QL eΛt Q  1 ; where L denotes the Laplace transform and QΛQ  1 is the Jordan decomposition of A þ BK 1 ; and (2) the computation (Laplace transform and Jordan decomposition) for the first fact is required only once, i.e. it is not repeated when a different edge is selected  for  evaluation. If Λ ¼ diagðλ1 ; …; λn Þ, i.e. a diagonal matrix with entries ðλ1 ; …; λn Þ, QL eΛt Q  1 becomes simply Q diagð1= ðs λ1 Þ; …; 1=ðs λn ÞÞQ  1 . Although S ij or JS ij J 1 (H 1 or peak norm of S ij ; maxω jS ij ðjωÞj with all frequencies ω) may not provide full information on the effect of complete edge removal, it is reported in [1] that this kind of small-change analysis (also found in [11,12]) often reveals the network's behaviour after a big change (complete edge removal).5 Therefore, the edge(s) corresponding to the smallest JS ij J 1 is highly likely to minimize the change in Gry when it is removed. Since this edge removal scheme may, however, yield a disconnected network, the edge must be chosen so that the resulting network is connected. Checking if an edge yields a disconnected network can easily be done via the Graph Scanning Algorithm [13] which runs in Oðm þ nÞ time, where m; n denote the numbers of edges and nodes, respectively.

2.3. Single edge removal from a consensus network To illustrate the use of Theorem 2.1, we consider a consensus network, e.g. in [14], with the communication topology given in Fig. 1(a), in which r is injected through node #1 and y is 5

An alternative way to perform the small-change analysis is to make use of Lagrange multipliers. In [9], this alternative way is used for design of structured optimal state feedback gains.

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r

r

y 1

10

2

9 4

3

5

8 y

7

1

6

2 3 4 5

Fig. 1. Examples of network topology.

chosen as the state value of 2 4 6 1 6 6 A ¼  LG ¼  6 6 1 6 4 1 1

node #5. Assume that6 3 1 1 1 1 3 0 1 1 7 7 7 0 1 0 0 7 7; 7 1 0 2 0 5 1 0 0 2

B ¼ ½1; 0; 0; 0; 0T , C ¼ ½0; 0; 0; 0; 1, K 1 ¼  C and K 2 ¼ 1, r ¼ 1, and the initial value of x ¼ ½1; 2; 3; 4; 5T . Note that the given data yield that all the nodes have the reference value of 1 at steady state. Then, Eq. (2.3) becomes S ij ¼  CðsI  ðA  BCÞÞ  1 F ij ðsI  ðA  BCÞÞ  1 B |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} c

b

¼  ðc i  c j Þðb i  b j Þ;

ð2:4Þ

where Fij is an all-zero matrix except an ‘1’ at (i,i) and (j,j) positions, and a ‘ 1’ at (i,j) and (j,i) positions (due to the structure of LG ), and b and c are the vectors with entries of b i and c i , respectively. Noting that A  BC is Hurwitz for all possible connected graphs obtained after removing an edge, JS ij J 1 based on Eq. (2.4) can be calculated and shown in Fig. 2(a). As seen in the figure, edge e24 yields the smallest J S ij J 1 among feasible edges (that exist in the graph and do not render the graph disconnected when one of the edges is removed; for instance, e34 is not feasible, as it does not exist in the graph.) Fig. 2(b) demonstrates that the removal of e24 does not affect the reference tracking performance much, whereas the removal of e15 yielding a large value of J S 15 J 1 does affect the transient behaviour much. 2.4. Multiple edge removal and performance test on random graphs Fig. 2(a) also shows that e25 yields the second smallest JS ij J 1 among the feasible edges, and implies that e24 and e25 may be chosen if two edges need to be removed from the network. This idea of removing multiple edges is applied to the consensus network with A ¼  LG associated with 1000 connected graphs G having 10 nodes and a randomly selected number (9–45) of edges.7 Like before, r is injected through node #1 and y is chosen as the last node(#10)'s value. First, esensi yielding the smallest sensitivity is found for each graph G and the corresponding The Laplacian matrix LG is the difference between the degree matrix and the adjacency matrix of G. The procedure of generating a random graph on 10 nodes is as follows: (1) Select a random number m between 9 and 45; (2) Randomly select m pairs (i,j) of nodes; (3) Generate a graph G with the set of edges corresponding to the m pairs; and (4) Redo the preceding steps until G is connected. 6 7

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−10

∞

log(||S|| )

0

−20 −30 −40 12

13

14

15

23

24

25

34

35

45

Edge number 5

e15 removed

y(t)

4 3

No edge removed & e removed

2

24

1 0

0

5

10

15

20

25

30

time (sec)

14

14

12

12

10

10

Relative error (%)

Relative error (%)

Fig. 2. J S J 1 and reference tracking performance after edge removal (e23 ; e34 ; e35 ; e36 are infeasible edges).

8

6

4

2

8

6

4

2

0

0 0

500

1000

Simulation number

0

500

1000

Simulation number

Fig. 3. Performance test on random graphs: (a) When one edge is removed; (b) When two edges are removed.

JGre J 1 ¼ γ sensi is computed for G  esensi (the graph with esensi removed), where Gre is the transfer function from r to e ¼ r  y. To see the quality of this edge removal scheme, an exhaustive search is also performed to find etrue that yields the smallest JGre J 1 ¼ γ true for G  etrue . Second, two edges yielding the smallest and the second smallest sensitivity are found for each graph and the corresponding H 1 norm, γ sensi , for the graph with the two edges removed.

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Also, an exhaustive search is performed to find two edges that yield the smallest H 1 norm, γ true , for the graph with the two edges removed. Fig. 3 shows the relative errors (%) in γ sensi with respect to γ true when the aforementioned edge removal scheme is applied to each of 1000 random graphs. The relative errors of less than 1% (5%) are obtained for 876 (993) graphs when one edge is removed, and the relative errors of less than 1% (5%) are obtained for 813 (975) graphs when two edges are removed. These results clearly testify the merit of the proposed edge removal scheme. 2.5. Static network versus dynamic network In the preceding section, e24 and e25 in Fig. 1(a) were identified as the edges that affect J Gry J 1 the least. It is interesting to observe that removing these two edges lets the network be a star topology whose algebraic connectivity λ2 ðGÞ is the largest among all tree graphs on five nodes. This observation may imply that identification of link importance could be done in a way to simply maximize a static measure, e.g. algebraic connectivity, of a given graph [1], not in the way to incorporate the network dynamics as was done in the preceding section. However, it is not difficult to find a case in which incorporating the network dynamics can make a significant difference. For the network topology in Fig. 1(b), it can be shown that removal of any edge yields a path graph G with λ2 ðGÞ ¼ 0:0979, and thus an edge selection via maximizing the static measure may end up removing e1;10 . As a matter of fact, removing e1;10 yields J Gre J 1 ¼ 3:0986 which is 198% larger than removing e89 does, where e89 is the edge found by the proposed edgeremoval scheme. Remark 2.2. It is important to note that the proposed edge removal scheme is to select an edge (s) in a network of fixed dynamical systems. In other words, each node shall not choose or

Fig. 4. Example of a network containing two structurally identical subsystems.

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change the edge weights (control gains) around it to improve the network's performance after an edge(s) is removed from the network. In fact, several works including [16] present decent ideas of identifying critical edges or nodes when each node is capable of changing the edge weights around it, in order to guarantee the network's performance such as controllability. In those works, critical edges are mainly determined by static or structural properties of the network rather than an individual's dynamic characteristics. For example, Fig. 4 contains two statically or structurally identical subsystems, G1 and G2 , and from the controllability point of view in [16] there is no difference between e1 and e2 in terms of their importance when achieving y-r. However, these edges' importance shall definitely change when G1 has a different dynamic characteristic than G2 , e.g. JGre J 1 for G  e1 is greater than JGre J 1 for G  e2 . Also, the injection location of r or the measurement location of y may change the link importance significantly, although the network's structure stays the same. The present work aims at detecting such importance of edges in a fixed dynamical network, in which no node shall be asked to change its dynamics (control gains) after an edge(s) is removed from the network. 3. Identification of link importance for disturbance rejection 3.1. Problem formulation When an unwanted signal w comes into the network, edge removal must be done in such a way to reject w as much as possible (again, this edge removal must not lead to a disconnected or unstable network). Therefore, it is now desired to find a feasible edge(s) eij in such a way to minimize J Gwy J 1 for G  eij , in the hope of good disturbance rejection with a less number of communication links. 3.2. Link identification using bounded real lemma Instead of checking the value of J Gwy J 1 after each edge removal, the following theorem on LMI (linear matrix inequality) can be utilized to devise a computationally tractable edge-removal rule. Theorem 3.1 (Bounded Real Lemma). Let γ40, and Gwy is a stable transfer function from w to y, where x_ ¼ Ax þ Bw and y ¼ Cx. Then, J Gwy J 1 oγ if and only if 2 3 CT XA þ AT X XB 6 BT X  γI 0 7 ð3:1Þ 4 5o0 C 0  γI with a positive definite matrix X40, an identity matrix I, and a zero matrix 0 with appropriate dimension. Proof. See the proof of Corollary 12.3 in [17].

□

Based on Theorem 3.1, one obvious solution strategy is to find an edge(s) such that the edge removal yields the smallest γ along with a feasible X. In other words, each of feasible edges is removed to obtain the corresponding γ ¼ J Gwy J 1 by solving Eq. (3.1) and the best edge yielding the smallest γ can be identified. However, as choosing such an edge(s) through the edgeby-edge check can be time-consuming (especially for multiple-edge removal), just a small

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change in the edge weight is considered instead as in Section 2. In fact, SeDuMi [20,21] requires a running time of Oðp2 q2:5 þ q3:5 Þ to solve q LMIs with p decision variables.8 Thus in the case of single-edge removal from m feasible edges, the edge-by-edge check can be done in approximately Oðkmn4 =4Þ with some constant k for symmetric X A Rnn . This running time is m times longer than the one for the LMI-based optimization to be proposed shortly. Once X is obtained after solving Eq. (3.1) for a given graph G (before edge removal), consider the effect of small changes δeij 's in eij's on A, B and C, e.g. ΔA ¼ ∑i;j ð∂A=∂eij Þδeij , and find the best edge(s) by solving the following optimization problem P which employs a modified version of Eq. (3.1) for δeij 's and ΔX with sufficiently small jδeij j and JΔX J: ðPÞ minimize β subject to X þ ΔX40 and 2 3 ðX þ ΔXÞA þ XΔA þ ðA þ ΔAÞT X þ AT ΔX ðX þ ΔXÞB þ XΔB ðC þ ΔCÞT 6 7 BT ðX þ ΔXÞ þ ΔBT X  βI 0 4 5o0: C þ ΔC

0

 βI ð3:2Þ

The above LMI is obtained via introducing small changes in A; B; C; X and then neglecting the products of the small changes. After performing this optimization to obtain δeij 's, ΔX and β, the best edge can be found by identifying the largest entry of δeij 's. Again, the Graph Scanning Algorithm can be used to check if the best edge renders the resulting graph (after the edge removal) disconnected. If this is the case, the second best edge must be chosen unless it renders the resulting graph disconnected. Remark 3.2. The aforementioned LMI-based edge-removal scheme and its performance check to be presented shortly can be summarized as follows: A graph G is given and the corresponding set E of feasible edges; Using ðA; B; CÞ associated with G, solve Eq. (3.1) to obtain X; Using the obtained X, solve Eq. (3.2) for ΔA; ΔB; ΔC; ΔX; δeij and β; Identify eij A E with the largest jδeij j; Using ðA; B; CÞ associated with G  eij , solve Eq. (3.1) to obtain γ LMI ; Perform an exhaustive search to find etrue AE that yields the smallest γ true after solving Eq. (3.1) using ðA; B; CÞ associated with G  etrue ; Step 6. Repeat the above steps for each of 1000 random graphs to obtain 100jγ true  γ LMI j=γ true (see Fig. 7(a)).

Step Step Step Step Step Step

0. 1. 2. 3. 4. 5.

Remark 3.3. Another LMI condition could be imposed together with Eq. (3.2) to ensure that the resulting graph is connected. This LMI condition can easily be formulated using the graph Laplacian, as done in [18]. Note however that the Graph Scanning Algorithm [13] can quickly perform a connectivity check after removing an edge. In fact, this algorithm runs in Oðm þ nÞ time (m: number of edges; n: number of nodes) and so needs Oðm2 þ mnÞ for checking m cases, whereas the LMI check adds the computational complexity of Oðkn4 =4Þ with some constant k. Thus, solving P and thereafter checking the connectivity using the Graph Scanning Algorithm require a total running time of Oðkðm þ n2 =2Þ2 þ m2 þ mnÞ  Oðkn4 =4Þ for m{n2 =2. 8

It is worth mentioning that existing LMI solvers need to be further improved to handle large-scale problems in a satisfactory manner, though.

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3.3. Single edge removal from a vehicle network To illustrate the use of P, consider the 2-dimensional vehicle formation control problem in [19]. Each vehicle is assumed to have the following dynamics: x_ i ¼ Aveh xi þ Bveh ui ;

i ¼ 1; 2; …; n; xi A R4 ;

where the entries of xi represent the 2-dimensional position and the velocity of vehicle i, ui represents the control inputs, and Aveh ¼ I 2  ½0; 1; a21 ; a22 ,9 Bveh ¼ I 2  ½0; 1. Here, I2 is a 2-by-2 identity matrix, and  denotes the Kronecker product. Then, using a decentralized static feedback control law u ¼ FLðx hÞ,10 where u and x are the vectors of ui's and xi's, the n-vehicle dynamics can be represented as follows: x_ ¼ Ax þ BFLðx  hÞ or x_ ¼ ðA þ BFLÞx BFLh: Here, A ¼ I n  Aveh , B ¼ I n  Bveh , L ¼ LG  I 4 and h is a constant vector associated with a desired formation (see [19] for details). In [19], it is shown that a21 ¼ 0 and F ¼ I n  F veh with F veh ¼ I n  ½ f 1 ;  f 2  with sufficiently large positive constants f1 and f2, can guarantee achieving any formation h for any connected graph. The n-vehicle dynamics is now perturbed by a disturbance w A R with a constant Bw, and the output y is defined as the vector of each vehicle's relative position with respect to the centre of n vehicles' positions, i.e. x_ ¼ ðA þ BFLÞx BFLh þ Bw w;

y ¼ Cx

ð3:3Þ

with C ¼  LK  ½1=n; 0; 0; 0; 0; 0; 1=n; 0, where LK denotes the Laplacian matrix corresponding to the complete graph K on n nodes. Note that L is the only matrix dependent on edge weights eij's, and a small change in this matrix's entry is denoted by δlij . For a properly chosen F, A þ BFL has stable eigenvalues with negative real parts and two unstable eigenvalues with nonnegative real parts. In fact, these unstable eigenvalues are those of Aveh and do not contribute to achieving a desired formation.11 Also, they may yield an unstable transfer function Gwy, and so must be removed in order to use Theorem 3.1. For this purpose, a non-square P matrix can be introduced (see [18] for how to construct such a P matrix) such that x ¼ Pz, PT P ¼ I, and eigenvalues of PT ðA þ BFLÞP are the same as the stable eigenvalues of A þ BFL. Then, Eq. (3.3) becomes z_ ¼ PT ðA þ BFLÞPz  PT BFLh þ PT Bw w ¼ Az PT BFLh þ Bw and y ¼ CPz ¼ Cz. Since A is now Hurwitz and the only matrix dependent on eij's,Eq. (3.2) can be used with A, ΔA, B and C in place of A, ΔA,B and C, respectively, and ΔB ¼ ΔC ¼ 0. Note that ΔA ¼ PT BFΔLP ¼ PT BFðΔLG  I 4 ÞP. For Aveh ¼ ½0; 1; 0; 0:1, Bw ¼ ½1; 0; …; 0T , F veh ¼ I n  ½ 1=2;  1=2 and LG corresponding to the graph in Fig. 1(a), the magnitudes of entries of ΔLG , jδlij j, are shown in Fig. 5(a) after solving P (along with move limits on ΔLG and ΔX) via SeDuMi [20] and YALMIP [22]. The figure indicates that e25 contributes to the minimization of β the most. In order to check if the selected edge indeed yields the best performance, J Gwy J 1 is The notation such as ½a; b; c; d denotes a 2-by-2 matrix with 4 entries. This control law has a static feedback gain FL and forces each vehicle to use the information only from its neighbouring vehicles with a direct communication link (no central planner). 11 Due to these unstable eigenvalues, vehicles do not normally reach a steady state but move together in a certain direction after a desired formation is achieved. The speed of convergence to a desired formation is determined by the stable eigenvalue with the smallest real part in magnitude. 9

10

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amount of variation to minimize

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wy ∞

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Fig. 6. Performance comparison between different edge removals for G in Fig. 1-(a).

calculated by minimizing γ subject to Eq. (3.1) after each edge removal and the result is shown in Fig. 5(b). The figure shows that e13 yields the smallest JGwy J 1 . However, this edge is not a feasible one as it renders the graph disconnected. Clearly, the second best edge or the best

40

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Fig. 7. Performance test on random graphs: (a) When one edge is removed; (b) When two edges are removed.

feasible edge matches to the one found by solving P. Fig. 6 shows how different edge removals affect the performance when a random signal (wðtÞA ½ 0:1; 0:1 for each t) is injected into the first vehicle's position. The figure demonstrates that the removal of the best e25 lets the vehicles converge to a desired pentagon formation (marked as ‘o’ in Fig. 6(a)) from an initial line formation (marked as ‘x’ in Fig. 6(a)), with a less deviation from the desired formation than when e12 is removed (see Fig. 6(b)). Here, the deviation is defined as dðtÞ ¼ JyðtÞ Ch J 2 ,12 where h ¼ hp  hv ¼ ½0; 0; 1; 0; 0; 1; 1; 1; 1=2; 2T  ½1; 0T defines the desired formation.13 In order to see the quantitative effect of w on d, the comparison of d12 (d(t) with no w and removal of e12), dw12 (d(t) with w and removal of e12), d25 and dw25 is performed. As expected, Jd25  dw25 J 2 ¼ 0:00984 r Jd 12  d w12 J 2 ¼ 0:0108 implies that the removal of e25 attenuates the disturbance effect better than the removal of e12 does. 3.4. Multiple edge removal and performance test on random graphs In order to verify that the proposed edge removal scheme works for other network topologies, 1000 connected random graphs G with 10 nodes and m edges are generated. Again, m is chosen as a random number between 9 and 45 to differentiate the network topologies further. For each graph G, the same data as in the preceding section except F veh ¼ I n  ½ 5;  5 are used to solve P, and obtain the edge eLMI yielding the largest jδlij j and the corresponding J Gwy J 1 ¼ γ LMI for G  eLMI (the graph with eLMI removed). Also, the best edge etrue yielding the smallest JGwy J 1 ¼ γ true for G  etrue is manually checked to see how far γ LMI is from γ true . A similar test is performed for the case of removing two edges. In this case, γ LMI denotes JGwy J 1 for the graph with two edges removed, where the two edges correspond to the largest and the second J  J 2 denotes the 2-norm or Euclidean norm of a vector. hp represents the relative position of each vehicle with respect to the first vehicle. For example, the fifth vehicle's desired relative position with respect to the first vehicle is ½1=2; 2. The ‘0’ in hv represents zero relative velocity between vehicles. 12 13

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largest jδlij j; and γ true is obtained via an exhaustive search for the graph yielding the smallest J Gwy J 1 after each of feasible pairs of edges is removed. Fig. 7(a) indicates that among these 1000 cases, the proposed edge removal scheme finds an edge yielding a relative error in γ LMI (with respect to γ true ) of less than 1% (10%) for 944 (979) cases; and it finds two edges yielding a relative error in γ LMI of less than 1% (10%) for 914 (964) cases. These results imply that the proposed edge removal scheme can indeed identify link importance with high probability using relatively low computational power. 4. Concluding remarks In this paper, efficient methods were presented to identify link importance in dynamic networks. Unlike similar problems in the literature, this link identification problem is considered using sensitivity and robustness measures to accommodate the network dynamics. Two kinds of the link identification problem were identified: one is for keeping the network dynamics the same as much as possible after removing a communication link(s); and the other is for gaining the network's robustness as much as possible after removing a communication link(s). The first kind was approached by analytically obtaining the sensitivity of the closed-loop transfer function Gry with respect to each edge, then the edge(s) which yields the least sensitivity in terms of peak norm was removed. The second kind was approached by numerically solving LMIs which help to identify the edge(s) yielding the largest decrease in the H 1 -norm of Gwy. These proposed schemes were tested on a consensus network, a vehicle network and random networks, and the obtained solutions were close (within 1% relative error) to the optimal ones with high probability. As mentioned in one of the footnotes, the proposed scheme could be improved in such a way that the resulting network after edge removal is guaranteed to be dynamically stable, not just statically connected. Also, a certain edge's importance could be identified under the assumption that each node is allowed to change the edge weights (control gains) around it after the edge is removed from the network. In this case, it is likely that the edge's importance shall be more dependent on the network's structural property than an individual's dynamic characteristics. These issues shall be discussed in the author's future work.

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