Topology Reconstruction of Dynamical Networks via Constrained Lyapunov Equations
11 Topology Reconstruction of Dynamical Networksvia Constrained Lyapunov Equations
Henk J. van Waarde, Pietro Tesi, and M. Kanat Camlibel
Abstract —The network structure (or topology) of a dynamicalnetwork is often unavailable or uncertain. Hence, we considerthe problem of network reconstruction. Network reconstructionaims at inferring the topology of a dynamical network usingmeasurements obtained from the network. In this technical notewe define the notion of solvability of the network reconstructionproblem. Subsequently, we provide necessary and sufficientconditions under which the network reconstruction problemis solvable. Finally, using constrained Lyapunov equations, weestablish novel network reconstruction algorithms, applicableto general dynamical networks. We also provide specializedalgorithms for specific network dynamics, such as the well-knownconsensus and adjacency dynamics.
Index Terms —Dynamical networks, consensus, network recon-struction, topology identification, Lyapunov equation.
I. I
NTRODUCTION N ETWORKS of dynamical systems appear in many con-texts, including biological networks [1], water distribu-tion networks [2] and (wireless) sensor networks [3].The overall behavior of a dynamical network is greatly influ-enced by its network structure (also called network topology).For instance, in the case of consensus networks, the dynamicalnetwork reaches consensus if and only if the network graph isconnected [4]. Unfortunately, the interconnection structure ofdynamical networks is often unavailable. For instance, in thecase of wireless sensor networks [3] the locations of sensors,and hence, communication links between sensors is not alwaysknown. Other examples of dynamical networks with unknownnetwork topologies are encountered in biology, for instance inneural networks [1] and genetic networks [5].Consequently, the problem of network reconstruction isstudied in the literature. The aim of network reconstruction(also called topology identification) is to find the networkstructure and weights of a dynamical network, using measure-ments obtained from the network. To this end, most papersassume that the states of the network nodes can be measured.The literature on network reconstruction methods can roughlybe divided into two parts, namely methods for stochastic and deterministic dynamical networks.Methods for stochastic network dynamics include inversecovariance estimation [6], [7] and methods based on power
Henk van Waarde and Kanat Camlibel are with the Bernoulli Institute forMathematics, Computer Science and Artificial Intelligence, Faculty of Scienceand Engineering, University of Groningen, P.O. Box 407, 9700 AK Groningen,The Netherlands. Pietro Tesi is with the Engineering and Technology InstituteGroningen, Faculty of Science and Engineering, University of Groningen,9747 AG Groningen, The Netherlands. Pietro Tesi is also with the Depart-ment of Information Engineering, University of Florence, 50139 Florence,Italy. Email: [email protected], [email protected],[email protected], [email protected] spectral analysis [8]. Moreover, network reconstruction basedon compressive sensing [9] has been investigated. Furthermore,the authors of [10] consider network reconstruction using
Wiener filtering .Apart from methods for stochastic networks, network re-construction for deterministic network dynamics has beenconsidered. In the paper [11] the concept of node-knockout is introduced, and a network reconstruction method based onthis concept is discussed. The paper [12] considers the problemof reconstructing a network topology from a transfer matrixof the network. Conditions are investigated under which thenetwork structure can be uniquely determined. Furthermore,the paper [13] considers network reconstruction using a so-called response network .In this note, we consider network reconstruction for deter-ministic networks of linear dynamical systems. In contrast topapers studying network reconstruction for specific networkdynamics such as consensus dynamics [11] and adjacencydynamics [14], we consider network reconstruction for general linear network dynamics described by state matrices containedin the so-called qualitative class [15]. It is our aim to inferthe unknown network topology of such dynamical networks,from state measurements obtained from the network.The contributions of this technical note are threefold. Firstly,we rigorously define what we mean by solvability of the net-work reconstruction problem for dynamical networks. Looselyspeaking, we say that the network reconstruction problemis solvable if the measurements obtained from a networkcorrespond only with the network under consideration (andnot with any other dynamical network). Secondly, we provide necessary and sufficient conditions under which the networkreconstruction problem is solvable. Thirdly, we provide aframework for network reconstruction of dynamical networks,using constrained Lyapunov equations. We will show that ourframework can be used to establish algorithms to infer net-work topologies for a variety of network dynamics, includingLaplacian and adjacency dynamics. An attractive feature ofour approach is that the conditions under which our algorithmsreconstruct the network structure are not restrictive. In otherwords, we show that our algorithms return the correct networkstructure if and only if the network reconstruction problem issolvable.Although this note mainly focuses on continuous-timenetwork dynamics, we also show how our reconstructionalgorithms can be applied to discrete-time systems, and tosystems with sampled measurements.The organization of this technical note is as follows. First,in Section II, we introduce preliminaries and notation used inthis note. Subsequently, we give a formal problem statement in a r X i v : . [ m a t h . O C ] S e p Section III. In Section IV we discuss necessary and sufficientconditions for the solvability of the network reconstructionproblem. Section V provides our network reconstruction al-gorithms. We consider an illustrative example in Section VI.Finally, Section VII contains our conclusions.II. P
RELIMINARIES
We denote the set of natural, real, and complex numbersby N , R , and C respectively. Moreover, the set of real m × n matrices is denoted by R m × n . We denote the set ofpositive (non-negative) real numbers by R > (respectively, R ≥ ). Furthermore, the set of all symmetric n × n matri-ces is given by S n . The vector of ones is denoted by .Furthermore, for x , x , . . . , x n ∈ R , we use the notation col( x , x , . . . , x n ) ∈ R n , which denotes the n -dimensionalcolumn vector with elements x , x , . . . , x n . The image of amatrix A is denoted by im A and the kernel of A is denotedby ker A . For a given set S , the power set S is the set ofall subsets of S . Let X and Y be nonempty sets. If for each x ∈ X , there exists a set F ( x ) ⊆ Y , we say F is a set-valued map from X to Y , and we denote F : X → Y .The image of a set-valued map F : X → Y is defined as im F := { y ∈ Y | ∃ x ∈ X such that y ∈ F ( x ) } . A. Preliminaries on systems theory
Consider the linear time-invariant system ˙ x ( t ) = Ax ( t ) y ( t ) = Cx ( t ) , (1)where x ∈ R n is the state, y ∈ R p is the output, and the realmatrices A and C are of suitable dimensions. We denote the unobservable subspace of system (1) by (cid:10) ker C | A (cid:11) , i.e., (cid:10) ker C | A (cid:11) := n − (cid:92) i =0 ker (cid:16) CA i (cid:17) . The subspace (cid:10) ker C | A (cid:11) is A -invariant, that is, A (cid:10) ker C | A (cid:11) ⊆ (cid:10) ker C | A (cid:11) . Furthermore, system (1)is observable if and only if (cid:10) ker C | A (cid:11) = { } (see, e.g.,Chapter 3 of [16]). If system (1) is observable, we say thepair ( C, A ) is observable. B. Preliminaries on graph theory
All graphs considered in this note are simple, i.e., withoutself-loops and with at most one edge between any pair ofvertices. We denote the set of simple, undirected graphs of n nodes by G n . Consider a graph G ∈ G n , with vertex set V = { , , . . . , n } and edge set E . The set of neighbors N i of vertex i ∈ V is defined as N i := { j ∈ V | ( i, j ) ∈ E } .We will now define various families of matrices associatedwith graphs in G n . To this end, we first define the set-valuedmap Q : G n → S n as Q ( G ) := { X ∈ S n | for all i (cid:54) = j, X ij (cid:54) = 0 ⇐⇒ ( i, j ) ∈ E } . The set of matrices Q ( G ) is called the qualitative class of thegraph G ∈ G n [15]. The qualitative class has recently been studied in the context of structural controllability of dynamicalnetworks [17], [18]. Note that each matrix X ∈ Q ( G ) carriesthe graph structure of G , in the sense that X contains nonzerooff-diagonal entries in exactly the same positions correspond-ing to the edges in G . Furthermore, note that the diagonalelements of matrices in the qualitative class are unrestricted.Hence, examples of matrices in Q ( G ) include the well-known (weighted) adjacency and Laplacian matrices, whichare defined next. Define the set-valued map A : G n → S n as A ( G ) := { A ∈ Q ( G ) | for all i, j, A ij ≥ and A ii = 0 } . Matrices in A ( G ) are called adjacency matrices associatedwith the graph G . Subsequently, define the set-valued map L : G n → S n as L ( G ) := { L ∈ Q ( G ) | L = 0 and for all i (cid:54) = j, L ij ≤ } . Matrices in the set L ( G ) are called Laplacian matrices of G .A Laplacian matrix L ∈ L ( G ) is said to be unweighted if L ij ∈ { , − } for all i (cid:54) = j . Similarly, an adjacency matrix A ∈ A ( G ) is called unweighted if A ij ∈ { , } for all i, j . C. Preliminaries on consensus dynamics
Consider a graph G ∈ G n , with vertex set V = { , , . . . , n } and edge set E . With each vertex i ∈ V , we associate a lineardynamical system ˙ x i ( t ) = u i ( t ) , where x i ∈ R is the stateof node i , and u i ∈ R is its control input. Suppose that eachnode i ∈ V applies the control input u i ( t ) = − (cid:88) j ∈N i a ij ( x i ( t ) − x j ( t )) , where a ij = a ji > for all i ∈ V and j ∈ N i . Then, thedynamics of the overall system can be written as ˙ x ( t ) = − Lx ( t ) , (2)where x = col( x , x , . . . , x n ) , and L ∈ L ( G ) is a Laplacianmatrix. We refer to system (2) as a consensus network . Con-sensus networks have been studied extensively in the literature,see, e.g., [4] and the references therein.III. P ROBLEM FORMULATION
In this section we define the network reconstruction prob-lem. We consider a linear time-invariant network system,with nodes satisfying single-integrator dynamics. We assumethat the state matrix of the system (and hence, the networktopology) is not directly available. Moreover, we suppose thatthe state vector of the system is available for measurementduring a time interval [0 , T ] . It is our goal to find conditionson the system under which the exact state matrix can bereconstructed from such measurements. Moreover, if the statematrix can be reconstructed, we want to develop algorithmsto infer the state matrix from measurements.We will now make these problems more precise. Since wewant to consider network reconstruction for general networkdynamics (instead of specific consensus or adjacency dyna-mics), we consider any set-valued map K : G n → S n suchthat for all G ∈ G n , we have that K ( G ) ⊆ Q ( G ) (3) is a nonempty subset. The map K is specified by the availableinformation on the type of network. For example, if we knowthat we deal with a consensus network, we have K = −L . Onthe other hand, if no additional information on the commu-nication weights (such as sign constraints) is known, we let K = Q . With this in mind, we consider the system ˙ x ( t ) = Xx ( t ) for t ∈ R ≥ x (0) = x , (4)where x ∈ R n is the state, and X ∈ im K (i.e., X ∈ K ( G ) forsome network graph G ∈ G n ). In what follows, we denote thestate trajectory of (4) by x x ( · ) , where the subscript indicatesdependence on the initial condition x . We assume that X is unknown, but the state trajectory of (4) can be measuredduring the time-interval [0 , T ] , where T ∈ R > . The problemof network reconstruction concerns finding the matrix X (andthereby, the graph G ), using the state measurements x x ( t ) for t ∈ [0 , T ] . Of course, this is only possible if the statetrajectory x x ( · ) of (4) is not a solution to the differentialequation ˙ x ( t ) = ¯ Xx ( t ) for some other admissible state matrix ¯ X (cid:54) = X . Indeed, if this were the case, the state measurementscould correspond to a network described either by X or ¯ X ,and we would not be able to distinguish between the two. Thisleads to the following definition. Definition 1:
Consider system (4), and denote its statetrajectory by x x ( · ) . We say that the network reconstructionproblem is solvable for system (4) if for all ¯ X ∈ im K suchthat x x ( · ) is a solution to ˙ x ( t ) = ¯ Xx ( t ) for t ∈ [0 , T ] , (5)we have ¯ X = X . In the case that the network reconstructionproblem is solvable for system (4), we say that the networkreconstruction problem is solvable for ( x , X, K ) . Remark 1:
As the state variables of system (4) are sums ofexponential functions of t , they are real analytic functions of t .It is well-known that if two real analytic functions are equalon a non-degenerate interval, they are equal on their wholedomain (see, e.g., Corollary 1.2.5 of [19]). Consequently, thestate vector x x ( · ) of system (4) satisfies (5) for t ∈ [0 , T ] if and only if x x ( · ) satisfies (5) for t ∈ R ≥ . Therefore,Definition 1 can be equivalently stated for t ∈ R ≥ instead of t ∈ [0 , T ] .In this note we are interested in conditions on x , X ,and K under which the network reconstruction problem issolvable for ( x , X, K ) . More explicitly, we have the followingproblem. Problem 1:
Consider system (4). Provide necessary andsufficient conditions on x , X , and K under which the networkreconstruction problem is solvable for system (4).In addition to Problem 1, we are interested in solving thenetwork reconstruction problem itself. This is stated in thefollowing problem. Problem 2:
Consider system (4), and denote its state vectorby x x ( · ) . Suppose that x x ( · ) is available for measurementduring the time interval [0 , T ] , and that the network reconstruc-tion problem is solvable for (4). Provide a method to computethe matrix X . Remark 2:
Note that we assume that the states of all nodes inthe network can be measured. This assumption is necessary in the sense that the network reconstruction problem is notsolvable (in the case of Q ( G ) ) if we can only measure a partof the state vector. To see this, suppose that we only haveaccess to a p -dimensional output vector y ( t ) = Cx ( t ) , where C = (cid:0) I (cid:1) ∈ R p × n . We claim that for each X ∈ Q ( G ) and x ∈ R n there existsa graph ¯ G , a matrix ¯ X ∈ Q ( ¯ G ) \ { X } and a vector ¯ x ∈ R n such that Ce Xt x = Ce ¯ Xt ¯ x . (6)That is, we cannot distinguish between X and ¯ X on the basisof output measurements. To see that this claim is true, wewrite X as X = (cid:18) X X X X (cid:19) , where the partitioning of X is compatible with the one of C .Now we distinguish two cases. First suppose that X (cid:54) = 0 .Clearly, there exists a vector z ∈ R n − p such that z (cid:62) X (cid:54) = 0 and z (cid:62) z = 1 . Define S := (cid:18) I S (cid:19) , where S := I − zz (cid:62) = S − . Then let ¯ X := SXS and ¯ x := Sx . It is not difficult to see that ¯ X (cid:54) = X and (6) issatisfied for this choice of ¯ X and ¯ x . Secondly, consider thecase that X = 0 . Then X = 0 . We can choose ¯ X as ¯ X := (cid:18) X
00 ¯ X (cid:19) , where ¯ X (cid:54) = X . In this case, it can be shown that ¯ X and ¯ x := x satisfy (6). Hence, for the network reconstructionproblem to be solvable it is necessary to measure all nodes.IV. M AIN RESULTS : SOLVABILITY OF THE NETWORKRECONSTRUCTION PROBLEM
In this section we state our main results regarding Problem1. That is, we provide conditions on x , X , and K underwhich the network reconstruction problem is solvable. Firstly,in Section IV-A we provide necessary and sufficient conditionsfor the solvability of the network reconstruction problem in thegeneral case that K is any mapping satisfying (3). Later, weconsider the special cases in which K = Q (Section IV-B),and the cases in which K = −L or K = A (Section IV-C). A. Solvability for general K In this section, we provide a general solution to Problem1. Let G ∈ G n be a graph, and let the mapping K be as in(3). Recall that we consider the dynamical network describedby system (4). As a preliminary result, we give conditionsunder which the state trajectory x x ( · ) of system (4) is alsothe solution to the system ˙ x ( t ) = ¯ Xx ( t ) for t ∈ R ≥ x (0) = x , (7) where ¯ X ∈ K ( ¯ G ) for some graph ¯ G ∈ G n . This result is givenin the following proposition. Proposition 3:
Consider systems (4) and (7), and let x x ( · ) be the state trajectory of (4). The trajectory x x ( · ) is also thesolution to system (7) if and only if x ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) . Proof:
Suppose that the state trajectory x x ( · ) of (4) isalso the solution to system (7). This means that x x ( · ) is thesolution to both the differential equation ˙ x ( t ) = Xx ( t ) for t ∈ R ≥ , (8)and the differential equation ˙ x ( t ) = Xx ( t ) + ( ¯ X − X ) x ( t ) for t ∈ R ≥ . (9)In particular, by substitution of t = 0 , this implies that x iscontained in ker (cid:0) ¯ X − X (cid:1) . Moreover, by taking the i -th time-derivative of (8) and (9), we find that x ∈ ker (cid:0) ¯ X − X (cid:1) X i for i = 1 , , . . . , n − . Consequently, we obtain x ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) .Conversely, suppose that the initial state x of system (4)satisfies x ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) . By X -invariance of (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) , this implies that the state trajectory x x ( · ) of system (4) satisfies x x ( t ) ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) forall t ∈ R ≥ . Specifically, we have that x x ( t ) ∈ ker (cid:0) ¯ X − X (cid:1) for all t ∈ R ≥ . We conclude that x x ( · ) is the solution toEquation (9), and consequently, to Equation (7). Remark 3:
Note that a condition equivalent to the onegiven in Proposition 3 can be stated in terms of the commoneigenspaces of X and ¯ X . Such a condition was previouslyproven by Battistelli et al. [20], [21] in the case that X and ¯ X are Laplacian matrices.By combining Proposition 3 and the fact that the state variablesof (4) are real analytic functions in t (see Remark 1), we obtainTheorem 4. This theorem states a necessary and sufficientcondition under which the network reconstruction problem issolvable for ( x , X, K ) . Theorem 4:
Let G ∈ G n be a graph, and let the mapping K be as in (3). Moreover, consider a matrix X ∈ K ( G ) anda vector x ∈ R n . The network reconstruction problem issolvable for ( x , X, K ) if and only if for all ¯ X ∈ im K \ { X } ,we have x (cid:54)∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) . Remark 4:
Although Theorem 4 gives a general necessaryand sufficient condition for network reconstruction, it is notdirectly clear how to verify this condition. Especially since X is assumed to be unknown, it seems difficult to check that x (cid:54)∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) . In fact, we will show in SectionV that the condition of Theorem 4 can be checked using onlythe measurements x x ( t ) for t ∈ [0 , T ] .Note that the condition of Theorem 4 is not only given in termsof x and X , but also in terms of all other matrices ¯ X ∈ im K .In the following theorem, we provide a simple sufficient condition for the solvability of the network reconstructionproblem, which is stated in terms of x and X . Theorem 5:
Let G ∈ G n be a graph, and let the mapping K be as in (3). Moreover, consider a matrix X ∈ K ( G ) anda vector x ∈ R n . The network reconstruction problem issolvable for ( x , X, K ) if the pair ( x (cid:62) , X ) is observable. Proof:
Suppose that the pair ( x (cid:62) , X ) is observable, andassume that x ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) for some matrix ¯ X ∈ im K . We want to show that ¯ X = X . Note that by hypothesis,we have x ∈ ker( ¯ X − X ) X i , for i = 0 , , . . . , n − . As aconsequence, we obtain the equalities ¯ XX i x = X i +1 x , (10)for i = 0 , , . . . , n − . It is not difficult to see that byinduction, Equation (10) implies that X i x = ¯ X i x , (11)for i = 1 , , . . . , n . In other words, the matrix X (cid:0) x Xx . . . X n − x (cid:1) is equal to ¯ X (cid:0) x ¯ Xx . . . ¯ X n − x (cid:1) . (12)Since the pair ( x (cid:62) , X ) is observable and X is symmetric, thematrix (cid:0) x Xx . . . X n − x (cid:1) is invertible. This allowsus to conclude that X equals ¯ X (cid:0) x ¯ Xx . . . ¯ X n − x (cid:1) (cid:0) x Xx . . . X n − x (cid:1) − . However, by (11), this implies that X = ¯ X . Consequently, forall ¯ X ∈ im K\{ X } we have x (cid:54)∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) . Finally,we conclude by Theorem 4 that the network reconstructionproblem is solvable for ( x , X, K ) .In the next section, we show that for a specific mapping K ,the observability condition of Theorem 5 is necessary and sufficient. However, in general, the observability condition isnot necessary. In particular, this will be shown for consensusnetworks in Section IV-C. B. Solvability for K = Q In this subsection, we consider the case that K = Q . Thiscase corresponds to the situation where we do not have anyadditional information (such as sign constraints) on the entriesof the state matrix X . To be precise, we consider system (4),where X ∈ Q ( G ) for some network graph G ∈ G n . We willsee that the solvability of the network reconstruction problemfor ( x , X, Q ) is in fact equivalent to the observability of thepair ( x (cid:62) , X ) . This is stated in the following theorem. Theorem 6:
Consider a graph G ∈ G n , let X ∈ Q ( G ) , andlet x ∈ R n . The network reconstruction problem is solvablefor ( x , X, Q ) if and only if the pair ( x (cid:62) , X ) is observable. Proof:
Sufficiency follows immediately from Theorem 5by taking K = Q . Hence, assume that the pair ( x (cid:62) , X ) isunobservable. We want to show that the network reconstruc-tion problem is not solvable for ( x , X, Q ) . To do so, we willconstruct a matrix ¯ X (cid:54) = X such that x ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) .Let v ∈ R n be a nonzero vector such that v (cid:62) (cid:0) x Xx . . . X n − x (cid:1) = 0 . (13)Such a vector exists, as ( x (cid:62) , X ) is unobservable. Subse-quently, define the matrix ¯ X := X + vv (cid:62) . By definition of v ,we obtain ¯ X i x = X i x , for i = 1 , , . . . , n . Consequently, x ∈ (cid:10) ker (cid:0) ¯ X − X (cid:1) | X (cid:11) . It remains to be shown that ¯ X ∈ im Q , i.e., ¯ X ∈ Q ( ¯ G ) for some ¯ G ∈ G n . Define the simpleundirected graph ¯ G = ( V, E ) , where V := { , , . . . , n } , and for distinct i, j ∈ V , we have ( i, j ) ∈ E if and only if ¯ X ij (cid:54) = 0 . By definition of the qualitative class Q ( ¯ G ) , we obtain ¯ X ∈ Q ( ¯ G ) . We conclude that the network reconstructionproblem is not solvable for ( x , X, Q ) . C. Solvability for K = −L and K = A In what follows, we consider solvability of the networkreconstruction problem for consensus and adjacency networks.We will start with consensus networks. That is, we considerthe system ˙ x ( t ) = − Lx ( t ) for t ∈ R ≥ x (0) = x , (14)where x ∈ R n is the state and L ∈ L ( G ) denotes the Laplacianmatrix of a graph G ∈ G n . In this section we show bymeans of an example that observability of ( x (cid:62) , − L ) is notnecessary for the solvability of the network reconstructionproblem for ( x , − L, −L ) . In Section V we will use thisfact to establish an algorithm for network reconstruction ofconsensus networks, that does not require observability of thepair ( x (cid:62) , − L ) . Consider the star graph G and Laplacian matrix L , depicted in Figure 1.
12 34 L = − − − − − − . Fig. 1: Star graph G with Laplacian matrix L .Moreover, consider the initial condition x ∈ R given by x = col(1 , , , . We claim that the network reconstructionproblem is solvable for ( x , − L, −L ) , even though the pair ( x (cid:62) , − L ) is unobservable. Indeed, it can be verified that theunobservable subspace of ( x (cid:62) , − L ) is (cid:10) ker x (cid:62) | − L (cid:11) = im v ,where the vector v is defined as v := col(0 , , , − . Thisimplies that ( x (cid:62) , − L ) is unobservable. To prove that thenetwork reconstruction problem is solvable for ( x , − L, −L ) ,consider a Laplacian matrix ¯ L ∈ im L such that x ∈ (cid:10) ker (cid:0) L − ¯ L (cid:1) | − L (cid:11) . Following the proof of Theorem 5(Equations (11) and (12)), we find that ( L − ¯ L ) (cid:0) x − Lx . . . ( − L ) n − x (cid:1) = 0 . (15)In other words, the columns of the matrix D := L − ¯ L arecontained in the unobservable subspace of ( x (cid:62) , − L ) . Since D is symmetric and (cid:10) ker x (cid:62) | − L (cid:11) = im v , we find D = αvv (cid:62) = α −
60 2 1 − − − , (16)for some α ∈ R . If α (cid:54) = 0 , the entries D and D of thematrix D have opposite sign. Since we have L = L = 0 ,we conclude from the relation ¯ L = L − D that ¯ L and ¯ L have opposite sign. However, this is a contradiction as ¯ L isa Laplacian matrix. Therefore, we conclude that α = 0 , andhence, D = 0 . Consequently, we obtain L = ¯ L . By Theorem 4, we conclude that the network reconstruction problem issolvable for ( x , − L, −L ) . Thus, we have shown that obser-vability of the pair ( x (cid:62) , L ) is not necessary for the solvabilityof the network reconstruction problem for ( x , − L, −L ) .It can be shown that im v also equals (cid:10) ker x (cid:62) | A (cid:11) , where A ∈ A ( G ) denotes the unweighted adjacency matrix asso-ciated with the star graph G depicted in Figure 1. Then, usingthe exact same reasoning as before, we conclude that the pair ( x (cid:62) , A ) is unobservable, but the network reconstruction prob-lem is solvable for ( x , A, A ) . In other words, observabilityof ( x (cid:62) , A ) is not necessary for the solvability of the networkreconstruction problem for ( x , A, A ) .V. M AIN RESULTS : THE NETWORK RECONSTRUCTIONPROBLEM
In this section, we provide a solution to Problem 2. That is,given measurements generated by an unknown network, weestablish algorithms to infer the network topology. Similarto the setup of Section IV, we start with the most generalcase in which K is any mapping satisfying (3). For this case,we obtain a general methodology to infer X ∈ K ( G ) frommeasurements. Subsequently, we provide specific algorithmsfor network reconstruction in the case that K = Q (SectionV-B), and in the case of consensus and adjacency networks(Section V-C). A. Network reconstruction for general K Recall that we consider the system (4), where the matrix X and graph G are unknown, but the state vector x x ( · ) of (4)can be measured during the time interval [0 , T ] . In this section,we establish a method to infer the matrix X and graph G usingthe vector x x ( t ) for t ∈ [0 , T ] . Firstly, define the matrix P := (cid:90) T x x ( t ) x x ( t ) (cid:62) dt = (cid:90) T e Xt x x (cid:62) e Xt dt. (17)Note that P can be computed from the measurements x x ( t ) for t ∈ [0 , T ] . The unknown matrix X is a solution to a Lyapunov equation involving the matrix P . Indeed, we have XP + P X = (cid:90) T (cid:16) Xe Xt x x (cid:62) e Xt + e Xt x x (cid:62) e Xt X (cid:17) dt = (cid:90) T ddt (cid:16) e Xt x x (cid:62) e Xt (cid:17) dt = x T x (cid:62) T − x x (cid:62) , (18)where x T := x x ( T ) = e XT x . In other words, X satisfiesthe Lyapunov equation XP + P X = Q, (19)where Q is defined as Q := x T x (cid:62) T − x x (cid:62) . Note that we cancompute the matrix Q from the measurements x x ( t ) at time t = 0 and time t = T . Therefore, if the matrix S = X is the unique solution to the Lyapunov equation SP + P S = Q, (20)we can find X (and therefore G ), by solving (20) for S .However, it turns out that in general it is not necessary for network reconstruction that the Lyapunov equation (20) hasa unique solution S . In fact, we only need a unique solution S in the image of K . That is, the Lyapunov equation (20)may have many solutions, but if only one of these solutionsis contained in im K , we can solve the network reconstructionproblem for ( x , X, K ) . This is stated more formally in thefollowing theorem. Theorem 7:
Let G ∈ G n , and let the mapping K be as in(3). Moreover, consider X ∈ K ( G ) , x ∈ R n , and let P and Q be as defined in (17) and (19) respectively. The networkreconstruction problem is solvable for ( x , X, K ) if and onlyif there exists a unique matrix S satisfying SP + P S = Q, S ∈ im K . (21)Moreover, under this condition, we have S = X .Before we can prove Theorem 7, we need the followingproposition, which states that ker P equals the unobservablesubspace of the pair ( x (cid:62) , X ) . Proposition 8:
Let P , x and X be as in (17). Then we have ker P = (cid:10) ker x (cid:62) | X (cid:11) . Proof:
Let v ∈ ker P . We have v (cid:62) P v = (cid:90) (cid:62) (cid:16) x (cid:62) e Xt v (cid:17) dt = 0 , (22)from which we obtain x (cid:62) e Xt v = 0 for all t ∈ [0 , T ] . Since x (cid:62) e Xt v is a real analytic function, we see that x (cid:62) e Xt v = 0 forall t ∈ R ≥ (cf. Remark 1). This implies that v ∈ (cid:10) ker x (cid:62) | X (cid:11) .Conversely, suppose that v ∈ (cid:10) ker x (cid:62) | X (cid:11) . This implies that x (cid:62) e Xt v = 0 for all t ∈ R ≥ . We compute P v = (cid:90) T e Xt x x (cid:62) e Xt v dt = 0 . (23)In other words, we obtain v ∈ ker P . We conclude that ker P = (cid:10) ker x (cid:62) | X (cid:11) , which completes the proof. Proof of Theorem 7:
To prove the ‘if’ part, suppose that thenetwork reconstruction problem is not solvable for ( x , X, K ) .We want to prove that the solution to (21) is not unique.By hypothesis, there exists a matrix ¯ X ∈ im K \ { X } suchthat e Xt x = e ¯ Xt x for all t ∈ [0 , T ] . We can repeat thediscussion of Equation (18) for ¯ X , to show that ¯ X also solvesthe Lyapunov equation (21). Consequently, we conclude thatthere exists no unique solution S satisfying (21).Conversely, to prove the ‘only if’ part, suppose that there existsno unique solution to (21). Note that S = X is always asolution to (21) by Equation (19). This implies that there existsa matrix ¯ X (cid:54) = X satisfying (21). Consequently, ¯ X ∈ im K ,and ( ¯ X − X ) P + P ( ¯ X − X ) = 0 . (24)Since P is symmetric positive semidefinite, there exists anorthogonal matrix U ∈ R n × n such that P = U Λ U (cid:62) , where Λ = (cid:18) D
00 0 (cid:19) , with D a positive definite diagonal matrix. We define thematrix ˆ X := U (cid:62) ( ¯ X − X ) U . It follows from (24) that ˆ X satisfies the Lyapunov equation ˆ X Λ + Λ ˆ X = 0 . Next, we partition ˆ X as ˆ X = (cid:18) ˆ X ˆ X ˆ X ˆ X (cid:19) , where the partitioning of ˆ X is compatible with the one of Λ .Then, we rewrite ˆ X Λ + Λ ˆ X = 0 as (cid:18) ˆ X D + D ˆ X D ˆ X ˆ X D (cid:19) = (cid:18) (cid:19) . Since D is nonsingular, ˆ X = 0 . Moreover, since D and − D do not have common eigenvalues, the Lyapunov equation ˆ X D + D ˆ X = 0 has a unique solution given by ˆ X = 0 (cf. Theorem 2.5.10 of [22]). This means that Λ ˆ X = 0 .Therefore, P ( ¯ X − X ) = 0 . By Proposition 8 we have x (cid:62) X i ( ¯ X − X ) = 0 for i = 0 , , . . . , n − . By exploitingsymmetry, we obtain x ∈ (cid:10) ker( ¯ X − X ) | X (cid:11) . We concludeby Theorem 4 that the network reconstruction problem is notsolvable for ( x , X, K ) .Finally, as we have shown in Equation (19) that X ∈ im K isalways a solution to the Lyapunov equation SP + P S = Q ,it is immediate that S = X if there exists a unique solution S to (21).Theorem 7 provides a general framework for network re-construction. Indeed, suppose that the network reconstructionproblem is solvable for ( x , X, K ) . We can compute the matri-ces P and Q from the state measurements x x ( t ) for t ∈ [0 , T ] .Then, network reconstruction boils down to computing theunique solution S to the constrained Lyapunov equation (21).In the subsequent sections, we will show how this can be donefor several types of network dynamics. B. Network reconstruction for K = Q In this section, we consider network reconstruction in the casethat K is equal to Q . Based on Theorem 7 we will derive analgorithm to identify the unknown matrix X ∈ Q ( G ) usingstate measurements taken from the network.Recall from Theorem 7 that the network reconstruction prob-lem is solvable for ( x , X, Q ) if and only if there exists aunique matrix S satisfying SP + P S = Q and S ∈ im Q . Notethat im Q is equal to S n , the set of n × n symmetric matrices. Inother words, if the network reconstruction problem is solvablefor ( x , X, Q ) , the solution to the problem can be found bycomputing the unique symmetric solution to the Lyapunovequation SP + P S = Q . It is not difficult to see that thereexists a unique symmetric solution to SP + P S = Q if andonly if there exists a unique solution to SP + P S = Q . Thisyields the following corollary of Theorem 7. Corollary 9:
Let G ∈ G n be a graph, and let X ∈ Q ( G ) .Moreover, consider a vector x ∈ R n , and let P and Q be as defined in (17) and (19) respectively. The networkreconstruction problem is solvable for ( x , X, Q ) if and onlyif the Lyapunov equation SP + P S = Q admits a uniquesolution S . Under this condition, we have S = X .Based on Corollary 9, we establish Algorithm 1, which infersthe state matrix X and graph G from measurements. Recallfrom Theorem 6 that the network reconstruction problem issolvable for ( x , X, Q ) if and only if ( x (cid:62) , X ) is observable. Of course, we can not directly check observability of ( x (cid:62) , X ) since X is unknown. However, we can in fact check ob-servability of the pair ( x (cid:62) , X ) using the matrix P . Indeed,by Proposition 8, ( x (cid:62) , X ) is observable if and only if thematrix P is nonsingular. Note that this condition is similar tothe so-called persistency of excitation condition, found in theliterature on adaptive systems (cf. Section 3.4.3 of [23]). Algorithm 1
Network reconstruction for ( x , X, Q ) Input:
Measurements x x ( t ) for t ∈ [0 , T ] ; Output:
Matrix X or “No unique solution exists”; Compute the matrix P = (cid:82) T x x ( t ) x x ( t ) (cid:62) dt ; if rank P < n then return “No unique solution exists”; else Compute the matrix Q = x x (cid:62) − x T x (cid:62) T ; Solve SP + P S = Q with respect to S ; return X = S ; end if A classic method to solve the Lyapunov equation in Step 6 ofAlgorithm 1 is the
Bartels-Stewart algorithm [24]. In addition,much effort has been made to develop methods for solvinglarge-scale Lyapunov equations [25], [26]. Typically, suchmethods use the Galerkin projection of the Lyapunov equationonto a lower-dimensional Krylov subspace [26]. The resultingreduced problem is then solved by means of standard schemesfor (small) Lyapunov equations. Using these techniques, it ispossible to efficiently solve large-scale ( n > ) Lyapunovequations [26].
Remark 5:
In theory, the correctness of Theorem 7, Corollary9, and Algorithm 1 is independent of the exact choice oftime
T > . However, choosing small T results in a matrix P with high condition number, and hence numerical rankcomputation (as in line 2 of Algorithm 1) becomes inaccurate.Consequently, in practice the value of T should be sufficientlylarge. In our simulations, good results were obtained using T = 10 (see Section VI). Remark 6:
Even though the focus of this note is on continuous-time systems, we remark that Algorithm 1 can also be appliedfor network reconstruction of discrete-time networks of theform z ( k + 1) = M z ( k ) for k ∈ N z (0) = z , (25)where z ∈ R n and M ∈ im Q . In this case, we assume thatwe can measure the state z ( k ) , for k = 0 , , . . . , m , where m ≥ n . From these measurements, we compute P := m − (cid:88) k =0 z ( k ) z ( k ) (cid:62) , Q := m − (cid:88) k =0 z ( k +1) z ( k ) (cid:62) + z ( k ) z ( k +1) (cid:62) . Similar to the continuous-time case, the matrix P is nonsingu-lar if and only if ( z (cid:62) , M ) is observable. Under this condition,we can reconstruct M by computing the unique solution tothe Lyapunov equation M P + P M = Q .The above approach can also be used for the continuous-time network (4) in the case that we cannot measure the state trajectory x x ( · ) during a time interval , but only haveaccess to sampled measurements . Indeed, suppose that we canmeasure x x ( kτ ) for k = 0 , , . . . , m , where τ ∈ R > is somesampling period. We can then use the framework for discrete-time systems on z ( k ) := x x ( kτ ) to reconstruct the matrix M = e Xτ . Subsequently, we can reconstruct X by computingthe (unique) matrix logarithm of e Xτ . C. Network reconstruction for K = −L and K = A Although Algorithm 1 is applicable to general network dynam-ics described by state matrices X ∈ Q ( G ) , the observabilitycondition guaranteeing uniqueness of the solution to (20) canbe quite restrictive if the type of network is a priori known.We have already seen in Section IV-C that observability of thepair ( x (cid:62) , X ) is not necessary for the solvability of the networkreconstruction problem for adjacency or consensus networks.Therefore, in this section we focus on network reconstructionfor ( x , − L, −L ) and ( x , A, A ) .Recall from Theorem 7 that the network reconstruction prob-lem is solvable for ( x , − L, −L ) if and only if there exists aunique matrix S satisfying SP + P S = Q and S ∈ − im L .Based on the definition of L (see Section II-B), we find thefollowing corollary of Theorem 7. Corollary 10:
Let G ∈ G n be a graph, and let L ∈ L ( G ) .Moreover, consider a vector x ∈ R n , and let P and Q be as defined in (17) and (19) respectively. The networkreconstruction problem is solvable for ( x , − L, −L ) if andonly if there exists a unique solution S to SP + P S = Q, S ∈ S n , S = 0 , S ij ≥ for i (cid:54) = j. (26)Moreover, under this condition, we have S = − L .The constraint S ij ≥ for i (cid:54) = j can be stated as a linearmatrix inequality (LMI) in the matrix variable S . Indeed, S ij ≥ is equivalent to e (cid:62) i Se j ≥ , where e k denotes the k -th column of the n × n identity matrix. Consequently, byCorollary 10, network reconstruction for ( x , − L, −L ) boilsdown to finding the matrix S satisfying linear matrix equationsand linear matrix inequalities, given by (26). There is efficientsoftware available to solve such problems. See, for instance,the LMI Lab package in Matlab and Yalmip [27]. We candeduce a corollary similar to Corollary 10 for the class A ( G ) .In this case, the restrictions on the elements of S are S ii = 0 and S ij ≥ for all i ∈ V and all j (cid:54) = i .VI. I LLUSTRATIVE EXAMPLE
In this section we illustrate the developed theory by consi-dering an example of a sensor network. Specifically, consider agraph G = ( V, E ) consisting of 100 sensor nodes, monitoringa region of × (see Figure 2). It is assumed that thesensors are linked using a so-called geometric link model [28].This means that there is a connection between two nodes inthe network if and only if the distance between the two nodesis less than a certain threshold, set to be equal to
135 m inthis example. It is assumed that the sensors run consensusdynamics, that is, the dynamics of the network is given by ˙ x ( t ) = − Lx ( t ) , where x ∈ R , and L ∈ L ( G ) is the unweighted Laplacian associated with G . The components ofthe initial condition x ∈ R were selected randomly within [0 , . Moreover, for this example, measurements were usedover the time-interval [0 , , i.e., T = 10 . We compute thematrices P and Q , and solve (26) using Yalmip. The resultingidentified Laplacian matrix is denoted by L r . The relative andmaximum element-wise errors between the identified Lapla-cian L r and original Laplacian L are very small. Specifically,we obtain (cid:107) L r − L (cid:107)(cid:107) L r (cid:107) = 1 . · − , max i,j ∈ V | L ij − ( L r ) ij | = 2 . · − , where (cid:107)·(cid:107) denotes the induced 2-norm.Fig. 2: Graph G of the sensor network.VII. C ONCLUSIONS
In this technical note, we have considered the problem ofnetwork reconstruction for networks of linear dynamical sys-tems. In contrast to papers studying network reconstructionfor specific network dynamics such as consensus dynamics[11] and adjacency dynamics [14], we considered networkreconstruction for general linear network dynamics describedby state matrices contained in the qualitative class. We for-mulated what is meant by solvability of the network recon-struction problem. Subsequently, we provided necessary andsufficient conditions under which the network reconstructionproblem is solvable. Using constrained Lyapunov equations,we established a general framework for network reconstructionof networks of dynamical systems. We have shown thatthis framework can be used for a variety of network types,including consensus and adjacency networks. Finally, we haveillustrated the theory by reconstructing the network topologyof a sensor network. R
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