The Observability Radius of Networks
Gianluca Bianchin, Paolo Frasca, Andrea Gasparri, and Fabio Pasqualetti
Abstract—This paper studies the observability radius of network attacker gains control of some network edges, and in network science systems, which measures the robustness of a network to perturbations for the classification of edges and the design of robust topologies. of the edges. We consider linear networks, where the dynamics are described by a weighted adjacency matrix, and dedicated sensors are Related work Our study is inspired by classic works on the ob- positioned at a subset of nodes. We allow for perturbations of certain servability radius of dynamical systems [11], [12], [13], defined as edge weights, with the objective of preventing observability of some the norm of the smallest perturbation yielding unobservability or, modes of the network dynamics. To comply with the network setting, equivalently, the distance to the nearest unobservable realization. our work considers perturbations with a desired sparsity structure, thus For a linear system described by the pair (A, C), the radius of extending the classic literature on the observability radius of linear systems. The paper proposes two sets of results. First, we propose observability has been classically defined as an optimization framework to determine a perturbation with smallest ∆A Frobenius norm that renders a desired mode unobservable from the µ(A, C) = min , existing sensor nodes. Second, we study the expected observability radius ∆ ,∆ ∆ A C C 2 of networks with given structure and random edge weights. We provide fundamental robustness bounds dependent on the connectivity properties s.t. (A + ∆A,C + ∆C ) is unobservable. of the network and we analytically characterize optimal perturbations of line and star networks, showing that line networks are inherently more As a known result [12], the observability radius satisfies robust than star networks. sI − A µ(A, C) = min σn , s C I.INTRODUCTION where σn denotes the smallest singular value, and s ∈ R (s ∈ C Networks are broadly used to model engineering, social, and natu- if complex perturbations are allowed). The optimal perturbations ral systems. An important property of such systems is their robustness ∆A and ∆C are typically full matrices and, to the best of our to contingencies, including failure of components affecting the flow of knowledge, all existing results and procedures are not applicable information, external disturbances altering individual node dynamics, to the case where the perturbations must satisfy a desired sparsity and variations in the network topology and weights. It remains an constraint (e.g., see [14]). This scenario is in fact the relevant one for outstanding problem to quantify how different topological features network systems, where the nonzero entries of the network matrices enable robustness, and to engineer complex networks that remain A and C correspond to existing network edges, and it would be operable in the face of arbitrary, and perhaps malicious perturbations. undesirable or unrealistic for a perturbation to modify the interaction Observability of a network guarantees the ability to reconstruct the of disconnected nodes. An exception is the recent paper [8], where state of each node from sparse measurements. While observability is structured perturbations are considered in a controllability problem, a binary notion [2], the degree of observability, akin to the degree yet the discussion is limited to the removal of edges. of controllability, can be quantified in different ways, including the We depart from the literature by requiring the perturbation to be energy associated with the measurements [3], [4], the novelty of the real, with a desired sparsity pattern, and confined to the network output signal [5], the number of necessary sensor nodes [6], [7], and matrix (∆C = 0). Our approach builds on the theory of total least the robustness to removal of interconnection edges [8]. A quantitative squares [15]. With respect to existing results on this topic, our work notion of observability is preferable over a binary one, as it allows to proposes procedures tailored to networks, fundamental bounds, and compare different observable networks, select optimal sensor nodes, insights into the robustness of different network topologies. and identify topological features favoring observability. Contribution The contribution of this paper is threefold. First, we In this work we measure robustness of a network based on the define a metric of network robustness that captures the resilience size of the smallest perturbation needed to prevent observability. Our of a network system to structural, possibly malicious, perturbations. notion of robustness is motivated by the fact that observability is a arXiv:1612.06120v1 [cs.SY] 19 Dec 2016 Our metric evaluates the distance of a network from the set of generic property [9] and network weights are rarely known without unobservable networks with the same interconnection structure, and it uncertainty. For these reasons numerical tests to assess observability extends existing works on the observability radius of linear systems. may be unreliable and in fact fail to recognize unobservable systems: Second, we formulate a problem to determine optimal perturbations instead, our measure of observability robustness can be more reliably (with smallest Frobenius norm) preventing observability. We show evaluated [10]. Among our contributions, we highlight connections that the problem is not convex, derive optimality conditions, and between the robustness of a network and its structure, and we prove that any optimal solution solves a nonlinear generalized eigen- propose an algorithmic procedure to construct optimal perturbations. value problem. Additionally, we propose a numerical procedure based Our work finds applicability in network control problems where the on the power iteration method to determine (sub)optimal solutions. network weights can be changed, in security applications where an Third, we derive a fundamental bound on the expected observabil- ity radius for networks with random weights. In particular, we present This material is based upon work supported in part by NSF awards #BCS-1430279 and #ECCS-1405330. A preliminary and abbreviated version a class of networks for which the expected observability radius of this work will appear in the Proceedings of the 2016 American Con- decays to zero as the network cardinality increases. Furthermore, we trol Conference [1]. Gianluca Bianchin and Fabio Pasqualetti are with the characterize the robustness of line and star networks. In accordance Mechanical Engineering Department, University of California at Riverside, with recent findings on the role of symmetries for the observability [email protected], [email protected]. Paolo Frasca is with the Department of Applied Mathematics, University of Twente, and controllability of networks [16], [17], we demonstrate that line [email protected]. Andrea Gasparri is with the Department of networks are inherently more robust than star networks to pertur- Engineering, Roma Tre University, [email protected]. bations of the edge weights. This analysis shows that our measure of robustness can in fact be used to compare different network For the pair (A, CO), the network observability radius is the topologies and guide the design of robust complex systems. solution to the optimization problem (2), which quantifies the total Because the networks we consider are in fact systems with linear edge perturbation to achieve unobservability. Different cost functions dynamics, our results are generally applicable to linear dynamical may be of interest and are left as the subject of future research. systems. Yet, our setup allows for perturbations with a fixed sparsity The minimization problem (2) can be solved by two subsequent pattern, which may arise from the organization of a network system. steps. First, we fix the eigenvalue λ, and compute an optimal Paper organization The rest of the paper is organized as follows. perturbation that solves the minimization problem for that λ. This Section II contains our network model, the definition of the network computation is the topic of the next section. Second, we search observability radius, and some preliminary considerations. Section III the complex plane for the optimal λ yielding the perturbation with describes our method to compute network perturbations with smallest minimum cost. We observe that (i) the exhaustive search of the Frobenius norm, our optimization algorithm, and an illustrative optimal λ is an inherent feature of this class of problems, as also example. Our bounds on the observability radius of random networks highlighted in prior work [13]; (ii) in some cases and for certain are in Section IV. Finally, Section V concludes the paper. network topologies the optimal λ can be found analytically, as we do in Section IV for line and star networks; and (iii) in certain II.THE NETWORK OBSERVABILITY RADIUS applications the choice of λ is guided by the objective of the network Consider a directed graph G := (V, E), where V := {1, . . . , n} perturbation, such as inducing unobservability of unstable modes. and E ⊆ V × V are the vertex and edge sets, respectively. Let III.OPTIMALITY CONDITIONSAND ALGORITHMSFORTHE A = [aij ] be the weighted adjacency matrix of G, where aij ∈ R NETWORK OBSERVABILITY RADIUS denotes the weight associated with the edge (i, j) ∈ E (representing flow of information from node j to node i), and aij = 0 whenever In this section we consider problem (2) with fixed λ. Specifically, (i, j) 6∈ E. Let ei denote the i-th canonical vector of dimension n. we address the following minimization problem: given a constraint Let O = {o1, . . . , op} ⊆ V be the set of sensor nodes, and define the graph H, the network matrix A ∈ AG , an output matrix CO, and T a desired unobservable eigenvalue λ ∈ , determine a perturbation network output matrix as CO = eo1 ··· eop . Let xi(t) ∈ R C n ∆∗ ∈ n×n satisfying denote the state of node i at time t, and let x : N≥0 → R be the R ∗ 2 2 map describing the evolution over time of the network state. The k∆ kF = min k∆kF , n n×n network dynamics are described by the linear discrete-time system x∈C ,∆∈R s.t. (A + ∆)x = λx, x(t + 1) = A x(t), and y(t) = CO x(t), (1) kxk2 = 1, (3) p where y : N≥0 → R is the output of the sensor nodes O. COx = 0, In this work we characterize structured network perturbations that prevent observability from the sensor nodes. To this aim, let ∆ ∈ AH. H = (VH, EH) be the constraint graph, and define the set of matrices ∗ 2 From (3), the value k∆ kF equals the observability radius of the compatible with H as network A with sensor nodes O, constraint graph H, and fixed |V|×|V| unobservable eigenvalue λ. AH = {M : M ∈ R ,Mij = 0 if (i, j) 6∈ EH}.
Recall from the eigenvector observability test that the network (1) is A. Optimal network perturbation observable if and only if there is no right eigenvector of A that lies We now shape minimization problem (3) to facilitate its solution. in the kernel of CO, that is, COx 6= 0 whenever x 6= 0, Ax = λx, Without affecting generality, relabel the network nodes such that the and λ ∈ C [18]. In this work we consider and study the following optimization problem: sensor nodes set satisfy 2 O = {1, . . . , p}, so that CO = Ip 0 . (4) min k∆kF , s.t. (A + ∆)x = λx, (eigenvalue constraint), Accordingly, A A ∆ ∆ kxk2 = 1, (eigenvector constraint), (2) A = 11 12 , and ∆ = 11 12 , (5) A21 A22 ∆21 ∆22 COx = 0, (unobservability), p×p p×n−p n−p×p where A11 ∈ R , A12 ∈ R , A21 ∈ R , and ∆ ∈ AH, (structural constraint), n−p×n−p A22 ∈ R . Let V = [vij ] be the unweighted adjacency n where the minimization is carried out over the eigenvector x ∈ C , matrix of H, where vij = 1 if (i, j) ∈ EH, and vij = 0 otherwise. the unobservable eigenvalue λ ∈ C, and the network perturbation Following the partitioning of A in (5), let n×n n×n ∆ ∈ R . The function k · kF : R → R≥0 is the V11 V12 Frobenius norm, and AH expresses the desired sparsity pattern of V = . V V the perturbation. It should be observed that (i) the minimization 21 22 problem (2) is not convex because the variables ∆ and x are We perform the following three simplifying steps. multiplied each other in the eigenvector constraint (A + ∆)x = λx, (1–Rewriting the structural constraints) Let B = A + ∆, and notice 2 Pn Pn 2 (ii) if A ∈ AH, then the minimization problem is feasible if and that k∆kF = i=1 j=1 (bij − aij ) . Then, the minimization only if there exists a network matrix A + ∆ = A˜ ∈ AH satisfying problem (3) can equivalently be rewritten restating the constraint the eigenvalue and eigenvector constraint, and (iii) if H = G, then ∆ ∈ AH, as in the following: the perturbation modifies the weights of the existing edges only. We n n 2 2 X X 2 −1 make the following assumption: k∆kF = kB − AkF = (bij − aij ) vij . i=1 j=1 (A1) The pair (A, CO) is observable. 2 Assumption (A1) implies that the perturbation ∆ must be nonzero to Notice that k∆kF = ∞ whenever ∆ does not satisfy the structural satisfy the constraints in (2). constraint, that is, when vij = 0 and bij 6= aij . ∗ 2n ∗ (2–Minimization with real variables) Let λ = λ< + iλ=, where i for some σ > 0 and y ∈ R with ky k = 1, and where denotes the imaginary unit. Let D1 = diag(v11, . . . , v1n, v21, . . . , v2n, . . . , vn1, . . . , vnn), 1 1 x< x= x< = 2 , and x= = 2 , D2 = diag(v11, . . . , vn1, v12, . . . , vn2, . . . , v1n, . . . , vnn), x< x= ∗ T ∗ ∗ T ∗ denote the real and imaginary parts of the eigenvector x, with Sx = (I ⊗ x<) D1(I ⊗ x<),Tx = (I ⊗ x<) D1(I ⊗ x=), (10) 1 p 1 p 2 n−p 2 n−p x< ∈ , x= ∈ , x< ∈ , and x= ∈ . ∗ T ∗ T R R R R Qx = (I ⊗ x=) D1(I ⊗ x=),Sy = (I ⊗ y1) D2(I ⊗ y1), Lemma 3.1: (Minimization with real eigenvector constraint) The T T constraint (A + ∆)x = λx can equivalently be written as Ty = (I ⊗ y1) D2(I ⊗ y2),Qy = (I ⊗ y2) D2(I ⊗ y2).
(A + ∆ − λ ∗ and the optimality conditions with respect to the unobservable eigen- Proof: The expression for the perturbation ∆ comes from T 0p Lemma 3.2 and (18), and the fact that L1 = σ diag(y1), value λ (see Remark 1) simplify to `1 2 = 0. x< L2 = σ diag(y2). To show the second part notice that 2 2 ¯ ¯ 2 k∆kF = kA − BkF = kL1VX< + L2VX=kF B. A heuristic procedure to compute structural perturbations 2 X X 2 2 2 2 = σ y1ix Lemma 3.6 implies that the inverse iteration method is not directly and b22, b23, b32, and b33 satisfy: applicable to (20). In fact, the zero eigenvalue of (H,D) leads the inverse iteration to instability, while the presence of eigenvalues of b33 − b22 (b22 − a22) − (b33 − a33) + (b23 − a23) = 0, (H,D) with equal magnitude may induce non-decaying oscillations b32 in the solution vector. To overcome these issues, we employ a shifting b23 (b32 − a32) − (b23 − a23) = 0, (21) mechanism as detailed in Algorithm 1, where the eigenvector z is b32 iteratively updated by solving the equation (H − µD)zk+1 = Dzk b22 + b33 − 2λ< = 0, 2 2 until a convergence criteria is met. Notice that (i) the eigenvalues of b22b33 − b23b32 − λ< − λ= = 0. (H − µD, D) are shifted with respect to the eigenvalues of (H,D), that is, if σ ∈ spec(H,D), then σ − µ ∈ spec(H − µD, D),1 (ii) the Proof: Let Bx = λx and notice that, because λ is unobservable, T pairs (H − µD, D) and (H,D) share the same eigenvectors, and COx = [1 0 0]x = 0. Then, x = [x1 x2 x3] , x1 = 0, b11 = a11, (iii) by selecting µ = ψ · min{σ ∈ spec(H,D): σ > 0}, the pair and b21 = a21. By contradiction, let x2 = 0. Notice that Bx = λx (H −µD, D) has nonzero eigenvalues with distinct magnitude. Thus, implies b33 = λ, which contradicts the assumption that λ= 6= 0 and Algorithm 1 estimates the eigenvector z associated with the smallest b33 ∈ R. Thus, x2 6= 0. Because x2 6= 0, the relation Bx = λx and nonzero eigenvalue σ of (H,D), and converges when z and σ also x1 = 0 imply b12 = 0. Additionally, λ is an eigenvalue of satisfy equations (20). The parameter ψ determines a compromise b22 b23 between numerical stability and convergence speed; larger values of B2 = . ψ improve the convergence speed.2 b32 b33 The characteristic polynomial of B2 is Algorithm 1: Heuristic solution to (20) 2 PB (s) = s − (b22 + b33)s + b22b33 − b23b32. Input: Matrix H; max iterations maxiter; ψ ∈ (0.5, 1). 2 Output: (σ, z) satisfying (20), or fail. For λ ∈ spec(B ), we must have P (s) = (s − λ)(s − λ¯), where repeat 2 B2 λ¯ is the complex conjugate of λ. Thus, z ← (H − µD)−1Dz; φ ← kzk; 2 2 2 PB (s) = (s − λ< − iλ=)(s − λ< + iλ=) = s − 2λ When convergent, Algorithm 1 finds a solution to (20) and, 2 2 2 2 L(b22, b23, b32, b33, p1, p2) = d22 + d23 + d32 + d33 consequently, the algorithm could stop at a local minimum and return 2 2 a (sub)optimal network perturbation preventing observability of a + p1(2λ< + b22 + b33) + p2(b22b33 − b23b32 − (λ< + λ=)), desired eigenvalue. All information about the network matrix, the where p1, p2 ∈ R are Lagrange multipliers, and dij = bij − aij . By sensor nodes, the constraint graph, and the unobservable eigenvalue is equating the partial derivatives of L to zero we obtain encoded in the matrix H as in (7), (9) and (20). Although convergence of Algorithm 1 is not guaranteed, numerical studies show that it ∂L = 0 ⇒ 2d22 + p1 + p2b33 = 0, (23) performs well in practice; see Sections III-C and IV. ∂b22 ∂L = 0 ⇒ 2d33 + p1 + p2b22 = 0, (24) C. Optimal perturbations and algorithm validation ∂b33 ∂L In this section we validate Algorithm 1 on a small network. We = 0 ⇒ 2d − p b = 0, (25) ∂b 23 2 32 start with the following result. 23 ∂L Theorem 3.7: (Optimal perturbations of 3-dimensional line net- = 0 ⇒ 2d32 − p2b23 = 0, (26) ∂b32 works with fixed λ ∈ C) Consider a network with graph G = (V, E), together with (22). The statement follows by substituting the La- 1 To see this, let σ be an eigenvalue of (H,D), that is, Hx = σDx. grange multipliers p1 and p2 into (23) and (26). Then, (H − µD)x = Hx − µDx = σDx − µDx = (σ − µ)Dx. That is (H − µD)x = (σ − µ)Dx thus σ − µ is an eigenvalue of (H − µD, D). To validate Algorithm 1, in Fig. 1 we compute optimal perturba- 2In Algorithm 1 the range for ψ has been empirically determined during tions for 3-dimensional line networks based on Theorem 3.7, and our numerical studies. compare them with the perturbation obtained at with Algorithm 1. 1.4 Accordingly, the modified network matrix is reducible and reads as 1.2 A 0 A¯ = 11 . F 1 A21 A22 k ⇤ 0.8 Let x2 be an eigenvector of A22 with corresponding eigenvalue λ. ¯ T T k 0.6 Notice that λ is an eigenvalue of A with eigenvector x = [0 x2 ] . F k ) 0.4 Since O ⊆ V1, COx = 0, so that the eigenvalue λ is unobservable. i ( ¯ From the above discussion we conclude that, for each E ∈ Ωk(O), 0.2 k there exists a perturbation ∆ = [δij ] that is compatible with G 0 and ensures that one eigenvalue is unobservable. Moreover, the -0.2 0 10 20 30 40 50 60 70 80 i perturbation ∆ is defined as δij = −aij if (i, j) ∈ E¯, and δij = 0 otherwise. We thus have Fig. 1. This figure validates the effectiveness of Algorithm 1 to compute optimal perturbations for the line network in Section III-C. The plot shows s X the mean and standard deviation over 100 networks of the difference between [δ] ≤ min a2 . E E ¯ ij ∗ (i) E∈Ωk(O) ∆ , obtained via the optimality conditions (21), and ∆ , computed at the (i,j)∈E¯ i-th iteration of Algorithm 1. The unobservable eigenvalue is λ = i and the values aij are chosen independently and uniformly distributed in [0, 1]. Because any two elements of Ωk(O) have empty intersection and all edge weights are independent, we have ω IV. OBSERVABILITY RADIUSOF RANDOM NETWORKS: THE s X s X Pr min a2 ≥ x = Pr a2 ≥ x ¯ ij ij CASEOF LINEAND STAR NETWORKS E∈Ωk(O) (i,j)∈E¯ (i,j)∈E¯ In this section we study the observability radius of networks with ω ω X 2 2 X 2 2 fixed structure and random weights, when the desired unobservable = Pr aij ≥ x = 1 − Pr aij ≤ x . eigenvalue is an optimization parameter as in (2). First, we give a (i,j)∈E¯ (i,j)∈E¯ general upper bound on the size of an optimal perturbation. Next, we In order to obtain a more explicit expression for this probability, we explicitly compute optimal perturbations for line and star networks, resort to using a lower bound. Let a denote the vector of a with showing that their robustness is essentially different. ij (i, j) ∈ E¯. The condition P a2 ≤ x2 implies that a belongs to We start with some necessary definitions. Given a directed graph (i,j)∈E¯ ij the k-dimensional sphere of radius x (centered at the origin). In fact, G = (V, E), a cut is a subset of edges E¯ ⊆ E. Given two disjoint since a is sampled in [0, 1]k, it belongs to the intersection between sets of vertices S , S ⊂ V, we say that a cut E¯ disconnects S 1 2 2 the sphere and the first orthant. By computing the volume of the from S if there exists no path from any vertex in S to any vertex 1 2 k-dimensional cube inscribed in the sphere, we obtain in S1 in the subgraph (V, E\ E¯). Two cuts E1 and E2 are disjoint √ k (2x/ k) k √ if they have no edge in common, that is, if E1 ∩ E2 = ∅. Finally, √x ∞ z−1 −x X 2 2 2k = , x ≤ k, R Pr aij ≤ x ≥ k the Gamma function is defined as Γ(z) = 0 x e dx. With this notation in place, we are in the position to prove a general (i,j)∈E¯ 1, otherwise. upper bound on the (expected) norm of the smallest perturbation that Since δ takes on nonnegative values only, its expectation can be prevents observability. The proof is based on the following intuition: computed by integrating the survival function a perturbation that disconnects the graph prevents observability. Z ∞ Theorem 4.1: (Bound on expected network observability radius) E[δ] = Pr (δ ≥ t) dt, Consider a network with graph G = (V, E), weighted adjacency 0 matrix A = [aij ], and sensor nodes O ⊆ V. Let the weights aij be which leads us to obtain, by suitable changes of variables, √ !ω independent random variables uniformly distributed in the interval Z k k √ Z 1 ω x k [0, 1]. Define the minimal observability-preventing perturbation as E[δ] ≤ 1 − √ dx = k 1 − t dt 0 k 0 1 δ = min k∆kF, (27) Z 1 n n×n 1 ω −1 1 Γ(1/k)Γ(ω + 1) λ∈C,x∈C ,∆∈R = √ (1 − z) z k dz = √ , k 0 k Γ(ω + 1/k + 1) s.t. (A + ∆)x = λx, where the last equality follows from the definition of the Beta func- tion, B(x, y) = R 1 tx−1(1 − t)y−1dt for Real(x) > 0, Real(y) > 0, kxk = 1, 0 2 Γ(x) Γ(y) and its relation with the Gamma function, B(x, y) = Γ(x+y) . COx = 0, We now use Theorem 4.1 to investigate the asymptotic behavior of the expected observability radius on sequences of networks of increasing ∆ ∈ AG . cardinality n. In order to emphasize the dependence on n, we shall Let Ωk(O) be a collection of disjoint cuts of cardinality k, where write E[δ(n)] from now on. As a first step, we can apply Wendel’s each cut disconnects a non-empty subset of nodes from O. Let inequalities [21] to find ω = |Ωk(O)| be the cardinality of Ωk(O). Then, 1 Γ(ω + 1) (ω + 1 + 1/k)1−1/k ≤ ≤ . Γ(1/k) Γ(ω + 1) (ω + 1)1/k Γ(ω + 1 + 1/k) (ω + 1) E[δ] ≤ √ . k Γ(ω + 1 + 1/k) If in a sequence of networks ω grows to infinity and k remains constant, then the ratio between the lower and the upper bounds goes ¯ Proof: Let E ∈ Ωk(O). Notice that, after removing the edges to one, yielding the asymptotic equivalence E¯, the nodes are partitioned as V = V ∪ V , where V ∩ V = ∅, 1 2 1 2 Γ(1/k) Γ(ω + 1) Γ(1/k) 1 O ⊆ V1, and V2 is disconnected from V1. Reorder the network [δ(n)] ≤ √ ∼ √ . E (ω + 1)1/k nodes so that V1 = {1,..., |V1|} and V2 = {|V1| + 1,..., |V|}. k Γ(ω + 1 + 1/k) k yielding a perturbation with minimum norm. In fact, if ai∗−1,i∗ = min{a12, . . . , an−1,n}, then all eigenvalues of the submatrix of A with rows/columns in the set {i∗, . . . , n} are unobservable, and thus minimizers in (27). Both Theorems 4.1 and 4.2 are based on constructing perturbations by disconnecting the graph. This strategy, however, suffers from performance limitations and may not be optimal in general. The next example shows that different kinds of perturbations, when applicable, (a) Line network (b) Star network may yield a lower cost. Fig. 2. Line and star networks with self-loops. Sensor nodes are marked in (Star network) Let G be a star network with n nodes and one sensor black. Note that, self-loops are not shown. node as in Fig. 2. The adjacency and output matrices read as a11 a12 a13 ··· a1n This relation implies that a network becomes less robust to pertur- a21 a22 0 ··· 0 . bations as the size of the network increases, with a rate determined .. .. . A = a31 0 . . . , by k. In the rest of this section we study two network topologies (29) . . with different robustness properties. In particular, we show that line . . 0 an−1,n−1 0 networks achieve the bound in Theorem 4.1, proving its tightness, an1 0 0 0 ann whereas star networks have on average a smaller observability radius. CO = 1 0 0 ··· 0 . (Line network) Let G be a line network with n nodes and one sensor Differently from the case of line networks, star networks are not node as in Fig. 2. The adjacency and output matrices read as strongly structurally observable, so that different perturbations may a11 a12 0 ··· 0 result in unobservability of some modes. Theorem 4.3: (Structured perturbation of star networks) Con- a21 a22 a23 ··· 0 . . . . . sider a star network with matrices as in (29), where the weights A = ...... , (28) aij are independent random variables uniformly distributed in the 0 ··· an−1,n−2 an−1,n−1 an−1,n interval [0, 1]. Let δ(n) be the minimal cost defined as in (27). Let 0 ··· 0 an,n−1 ann |aii − ajj | C = 1 0 0 ··· 0 . γ = min √ . O i,j∈{2,...,n},i6=j 2 We obtain the following result. Then, Theorem 4.2: (Structured perturbation of line networks) Con- sider a line network with matrices as in (28), where the weights δ(n) = min{a12, a13, . . . , a1n, γ}, and aij are independent random variables uniformly distributed in the 1 1 √ ≤ E[δ(n)] ≤ √ . interval [0, 1]. Let δ(n) be the minimal cost defined as in (27). Then, 2 n(n − 1) 2 n(n − 2) 1 δ(n) = min{a , . . . , a }, and [δ(n)] = . Proof: Partition the network matrix A in (29) as 12 n−1,n E n a A Proof: It is known that line networks, when observed from one A = 11 12 , A A of their extremes, are strongly structurally observable, that is, they 21 22 1,n−1 n−1,1 n−1,n−1 are observable for every nonzero choice of the edge weights [22]. where A12 ∈ R , A21 ∈ R , A22 ∈ R . Accord- T T Consequently, for the perturbed system to feature an unobservable ingly, let x = [x1 x2 ] . The condition COx = 0 implies x1 = 0. eigenvalue, the perturbation ∆ must be such that δi,i+1 = −ai,i+1 Consequently, for the condition (A + ∆)x = λx to be satisfied, for some i ∈ {2, . . . , n − 1}. Thus, a minimum norm perturbation we must have (A12 + ∆12)x2 = 0 and (A22 + ∆22)x2 = λx2. is obtained by selecting the smallest entry ai,i+1. Since the ai,i+1 Notice that, because A22 is diagonal and ∆ ∈ AG , the condition are independent and identically distributed, δ(n) = min ai,i+1 is a (A22 + ∆22)x2 = λx2 implies that λ = aii + δii for all indices i n−1 random variable with survival function Pr(δ(n) ≥ x) = (1 − x) such that i ∈ Supp(x2), where Supp(x2) denotes the set of nonzero for 0 ≤ x ≤ 1, and Pr(δ(n) ≥ x) = 0 otherwise. Thus, entries of x2. Because kxk = 1, |Supp(x2)| > 0. We have two cases: 1 Z 1 Case |Supp(x2)| = 1: Let Supp(x) = {i}, with i ∈ {2, . . . , n}. [δ(n)] = Pr(δ(n) ≥ x)dx = . E Then, the condition (A12+∆12)x2 = 0 implies δ1,i = −a1,i, and the 0 n condition (A22 + ∆22)x2 = λx2 is satisfied with ∆22 = 0, λ = aii, and x = e , where e is the i-th canonical vector of dimension n. Theorem 4.2 characterizes the resilience of line networks to i i Thus, if |Supp(x )| = 1, then δ(n) = min a . structured perturbations. We remark that, because line networks are 2 i∈{2,...,n} 1,i strongly structurally observable, structured perturbations preventing Case |Supp(x2)| > 1: Let S = Supp(x2). Then, δii = λ − aii. observability necessarily disconnect the network by zeroing some Notice that the condition (A22 +∆22)x2 = λx2 is satisfied for every network weights. Consistently with this remark, line networks achieve x2 with support S and, particularly, for x2 ∈ Ker(A12). Thus, we the upper bound in Theorem 4.1, being therefore maximally robust let ∆12 = 0. Notice that s to structured perturbations. In fact, for O = {1} and a cut size k = 1 X 2 δ(n) = min (λ − aii) , we have Ω1(O) = {a12, . . . , an−1,n} and ω = n − 1. Thus, λ,S i∈S Γ(1)Γ(n) (n − 1)! 1 E[δ(n)] ≤ √ = = , and that δ(n) is obtained when S = {i, j}, for some i, j ∈ 1Γ(n + 1) n! n {2, . . . , n}, and λ = (aii + ajj )/2.√ Specifically, for the indexes which equals the behavior identified in Theorem 4.2. Further, {i, j}, we have k∆kF = |aii − ajj |/ 2. Thus, if |Supp(x2)| > 1, Theorem 4.2 also identifies an unobservable eigenvalue then δ(n) = γ, which concludes the proof of the first statement. Star V. CONCLUSION Line 10-1 In this work we extend the notion of observability radius to network systems, thus providing a measure of the ability to maintain observability of the network modes against structured perturbations of )] n ( the edge weights. We characterize network perturbations preventing [ 10-2 E observability, and describe a heuristic algorithm to compute pertur- bations with smallest Frobenius norm. Additionally, we study the observability radius of networks with random weights, derive a funda- mental bound relating the observability radius to certain connectivity 10-3 properties, and explicitly characterize the observability radius of line 5 10 15 20 25 n and star networks. Our results show that different network structures Fig. 3. Expected values E[δ(n)] for the two network topologies in Fig. 2 exhibit inherently different robustness properties, and thus provide as functions of the network cardinality n. Dotted lines represent upper and guidelines for the design of robust complex networks. lower bounds in Theorems 4.2 and 4.3. Solid lines show the mean over 100 networks of the Frobenius norm of the perturbations obtained by Algorithm 1. REFERENCES [1] G. Bianchin, P. Frasca, A. Gasparri, and F. 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Mesbahi, “On symmetry and controllability of decrease implies that star networks are structurally less robust to multi-agent systems,” in IEEE Conference on Decision and Control, perturbations than line networks. Crucially, unobservability in star Los Angeles, CA, USA, Dec. 2014, pp. 625–630. networks may be caused by two different phenomena: the deletion [17] F. Pasqualetti and S. Zampieri, “On the controllability of isotropic and anisotropic networks,” in IEEE Conference on Decision and Control, of an edge disconnecting a node from the sensor node (deletion of the Los Angeles, CA, USA, Dec. 2014, pp. 607–612. smallest among the edges {a12, a13, . . . , a1n}), and the creation of [18] T. Kailath, Linear Systems. Prentice-Hall, 1980. a dynamical symmetry with respect to the sensor node by perturbing [19] L. N. Trefethen and D. Bau III, Numerical linear algebra. SIAM, 1997. two diagonal elements to make them equal in weight. It turns out that, [20] G. H. Golub and C. F. Van Loan, Matrix computations. JHU Press, on average, creating symmetries is “cheaper” than disconnecting the 2012. [21] J. G. Wendel, “Note on the Gamma function,” American Mathematical network. The role of network symmetries in preventing observability Monthly, pp. 563–564, 1948. and controllability has been observed in several independent works; [22] H. Mayeda and T. Yamada, “Strong structural controllability,” SIAM see for instance [16], [17]. Finally, the comparison of line and star Journal on Control and Optimization, vol. 17, no. 1, pp. 123–138, 1979. networks shows that Algorithm 1 is a useful tool to systematically [23] H. A. David and H. N. Nagaraja, Order statistics. John Wiley & Sons, 2003. investigate the robustness of different topologies. 0}; which leads to update D according to (10); 2 2 i ← i + 1 b22 + b33 − 2λ< = 0, and b22b33 − b23b32 − λ< − λ= = 0. (22) until convergence or i > maxiter; return (φ + µ, z) or fail if i = maxiter; The Lagrange function of the minimization problem with cost ∗ 2 P3 P3 2 function k∆ kF = i=2 j=2(bij − aij ) and constraints (22) is