A nonlinear graph-based theory for dynamical network observability
aa r X i v : . [ n li n . C D ] M a r A nonlinear graph-based theory for dynamical network observability
Christophe Letellier , Irene Sendi˜na-Nadal , & Luis A. Aguirre Normandie Universit´e CORIA, Campus Universitaire du Madrillet, F-76800 Saint-Etienne du Rouvray, France Complex Systems Group & GISC, Universidad Rey Juan Carlos, 28933 M´ostoles, Madrid, Spain Center for Biomedical Technology, Universidad Polit´ecnica de Madrid, 28223 Pozuelo de Alarc´on, Madrid, Spain and Departamento de Engenharia Eletrˆonica, Universidade Federal de MinasGerais – Av. Antˆonio Carlos 6627, 31.270-901 Belo Horizonte MG, Brazil (Dated: March 7, 2018)A faithful description of the state of a complex dynamical network would require, in principle, themeasurement of all its d variables, an unfeasible task for systems with practical limited access andcomposed of many nodes with high dimensional dynamics. However, even if the network dynamicsis observable from a reduced set of measured variables, how to reliably identifying such a minimumset of variables providing full observability remains an unsolved problem. From the Jacobian matrixof the governing equations of nonlinear systems, we construct a pruned fluence graph in which thenodes are the state variables and the links represent only the linear dynamical interdependencesencoded in the Jacobian matrix after ignoring nonlinear relationships. From this graph, we identifythe largest connected sub-graphs where there is a path from every node to every other node andthere are not outcoming links. In each one of those sub-graphs, at least one node must be measuredto correctly monitor the state of the system in a d -dimensional reconstructed space. Our procedureis here validated by investigating large-dimensional reaction networks for which the determinant ofthe observability matrix can be rigorously computed. When dealing with large complex systems, observabil-ity becomes a key concept that addresses the ability of ex-amining the system dynamics from a reduced set of mea-surements collected in a finite time. Indeed, to properlyunderstand the functioning of many biological or techno-logical networks, it is fundamental to be able to retrievethe complex behavior emerging from the local interac-tions of dynamical units when just a limited amount ofinformation is available.The idea of observability was first introduced byKalman for linear systems [1] which was further extendedto nonlinear systems by several other researchers, e.g. [2].That now classical way of investigating observability pro-vides a yes-or-no answer, that is, the system is either fullyobservable or not through a given set of measurements.In order to bypass this binary classification of observabil-ity, Friedland proposed the use of a conditioning numberbetween 0 (non-observable) and 1 (fully observable) toquantify the observability of linear systems [3]. Lateron, Aguirre showed that observability depends on thechosen coordinate set to describe the system dynamics[4]. This work led to the introduction of the observabil-ity coefficients to characterize the observability of manylow-dimensional chaotic systems [5–8].Recently, an attempt to apply those coefficients tosmall dynamical networks was reported [9] but, it waspointed out that such an assesment is out of scope forlarge dynamical systems due to the impossibility of calcu-lating the determinant of the corresponding observabilitymatrix [10]. One way to tackle this drawback is by intro-ducing symbolic observability coefficients [10, 11] thatallow treating larger dimensional systems although thenumber of variable combinations to investigate increasesexponentially with the system dimension.In order to avoid the use of a brute-force search for a minimum sensor set, observability is addressed in [12]by means of graph-theoretic methods. Mainly based onperforming a linearization of the system (all dynamicalinterdependences between variables are considered con-stant as in linear systems), such a technique reveals thatsparse networks with heterogeneous degree distributionsare less observable while the observability of denser andhomogeneous networks relies on just a few nodes [12].However, these latter results may not hold for nonlinearsystems as the presence of nonlinearities are one of themain causes of observability loss [7, 10]. Some variantsof this graphical approach were developed in [13, 14] byconsidering the effect of connection types in the result-ing topologies and in the change of the number of thenecessary sensors. However, none of them actually takesinto account the nonlinear nature of the dynamical in-terdependence between the state variables. Moreover, itwas recently shown that Liu and coworkers’ graphical ap-proach may not provide the right reduced set of variablesto measure (see the supplement material in Ref. [15] andRef. [16]).Our goal is therefore to address the observability ofa complex system to identify a minimum set of vari-ables providing access to the rest of state variables, fol-lowing an analogous graph-theoretic approach as in Liuand coworkers (and inspired in structured system the-ory [17, 18]), but properly handling the effect of nonlin-ear dynamical interdependences among variables in thesystem’s observability. We show the correctness of thisapproach by using benchmark reaction networks comingfrom biology or physics and by comparing the obtainedresults with rigorous algebraic computations of the deter-minant of the observability matrix. Our results contra-dict the conclusions drawn in [12] evidencing importantdiscrepancies mainly resulting from treating linear andnonlinear interdependences on an equal footing [15].Let us start by considering a dynamical system whosevariables x i , i = 1 , , . . . , d evolve according to˙ x i = f i ( x ) , (1)where x ∈ R d is the state vector, and f i is the i th com-ponent of the vector field f . The dynamical system (1)is said to be state observable at time t if the initial state x (0) can be uniquely determined from the knowledge ofa variable s = h ( x ) ∈ R m , with m < d , measured inthe inverval [0; t ] [18]. In practice, the observability of(1) through s is assessed by computing the rank of theobservability matrix O s ( x ) = d h ( x )d L fh ( x )...d L d − f h ( x ) , (2)where d ≡ ∂∂ x and L fh ( x ) is the Lie derivative of h alongthe vector field f . This is thus the Jacobian matrix ofthe Lie derivatives of s [2]. The system (1) is said to bestate observable if and only if the observability matrixhas full rank, that is, rank( O s ) = d [26]. Notice that,the full observability of a system is determined by thespace spanned not only by the measured variables butalso by their appropriate Lie derivatives [19].A systematic check of all the possible combinationsturns out to be a daunting task for large d . Therefore, itbecomes crucial to furnish methods to unveil a tractableset of variables providing full observability of a system.A first attempt was reported in Liu et al. [12] usinga graphical representation of the functional relationshipamong the system variables. We follow such an approachby choosing as the network representation of the system(1) its corresponding “fluence graph” where a directedlink x j → x i is drawn whenever x j appears in the dif-ferential equation of x i , that is, if the element J ij of theJacobian matrix of the Eq. (1) is non-zero [27].An illustrative example is provided in Fig. 1(a) for theR¨ossler system ( x ≡ ( x, y, z ) and f ≡ ( − y − z, x + ay, b + z ( x − c ))). In this procedure, a link from x j to x i (with i = j ) is present whenever J ij = 0 independently on thelinear (solid lines) or nonlinear (thick dashed lines) na-ture of the functional dependence. At this point is wherewe deviate from Liu and coworkers approach as it ignoresthe fact that a lack of observability most often originatesfrom the nonlinear relationship between variables [7]. Inorder to correct this shortcoming, we propose to distin-guish linear from nonlinear couplings [10, 11] by pruningfrom the fluence graph all nonlinear links and keepingonly those associated with the constant elements in theJacobian matrix of the system. We call this reduced flu-ence graph, the pruned fluence graph (Fig. 1(b)) that wetake as the minimum graph containing the information y zx y zx (a) Fluence graph (b) Pruned fluence graphFIG. 1: Fluence graphs of the R¨ossler system. (a) Full fluencegraph where an edge is plotted between variables x i and x j whenever J ij = 0. The thick dashed line indicates a nonlinearterm in the Jacobian matrix. (b) The same as in (a) butedges nonlinearly relating two variables are removed from thefull fluence graph. A dashed circle surrounds a root stronglyconnected component (SCC). In both graphs, edges x i → x i are omitted since they do not contribute to the determinationof the SCC. flow that will allow us to select the minimum set of sen-sors to ensure observability of the whole system whileworking in a d -dimensional reconstructed space. (a) 5D rational system (b) 9D RB system (c) 13D DNA systemFIG. 2: Pruned fluence graphs of (a) the 5D rational systemfor the circadian oscillations in the Drosophila period protein,(b) the 9D Rayleigh-B´enard convection, and (c) the 13D re-action network for the replication of fission yeast. Numbers i label the variable x i of the models, continuous lines from i to j represent that variable x j is linearly influencing variable x i and variables surrounded by a dashed circle are part of anSCC without outcoming edges. In the following, the graph analysis described in [12] toisolate the minimum set of sensors still holds. Namely, anode in the pruned fluence graph is a sensor if it belongsto a root strongly connected component (SCC) of thegraph (a subgraph in which there is a directed path fromeach node to every node in the subgraph) and with no outcoming links, that is, such an SCC is either an isolatedsubgraph or a root (a sink) of information flowing fromany other subgraphs in the network [28]. By measuringat least one of the nodes in each subgraph classified asa root SCC in the pruned fluence graph, we make theconjecture that such a selection is not only minimal butalso provides a good observability.In order to validate our hypothesis, we applied theabove procedure to several nonlinear dynamical systemswidely known in the physical and biological scientificcommunity. For each of them we confirm that a candi-date set of variables to be measured actually provides fullobservability by checking that the determinant Det O s of the analytical observability matrix O s as defined inEq. (2), is always nonzero [6, 20]. Note that for dimen-sions larger than 4, the determinant cannot always becomputed due to its complexity (Maple software fails tocompute some observability matrices for a 5D rationalsystem [10]). To deal with this difficulty, a symbolic for-malism was introduced in [10, 11] that allows to quantifythe observability of a given measure by means of a sym-bolic observability coefficient η = 1, if the observabilityis full, η > .
75 if good, and poor otherwise [21]. Briefly,it is based on a symbolic Jacobian matrix ˜ J ij whose el-ements can be either 1 , ¯1 and ¯¯1 which encode, respec-tively, constant, nonlinear and rational terms (whose de-nominators contain variables x j ) of the Jacobian matrix J ij = ∂f i /∂x j [10].It turns out that the symbolic observability coefficientsare inversely proportional to the complexity of the de-terminant of the observability matrix [6]. Therefore, ingeneral, in those cases in which the sensor set is provid-ing full observability, the determinant Det O s can thusbe analytically computed. We used this property as avalidation of our hypothesis, and check whether a nonvanishing determinant is obtained for a given set of vari-ables potentially providing full observability. The R¨ossler system.
Let us now consider the R¨osslersystem whose Jacobian matrix J Ros = − − a z x − c (3)has two non-constant terms and whose symbolic form ac-counting only for the linear dynamical interdependenciesis reduced to ˜ J linRos = . (4)The two nonlinear terms, J and J are, therefore,not considered for constructing the pruned fluence graphshown in Fig. 1(b). On the other hand, when all terms inthe Jacobian matrix are considered equivalent indepen-dently of their linear or nonlinear nature as in [12], theresulting graph is the one shown in Fig. 1(a) except thatthe thick dashed line will be drawn as a thin line. For thislatter case, the decomposition in SCC singles out a rootSCC composed by the three variables, suggesting thatany of the three variables can be measured to achieve ob-servability. However, it is known that measuring variable z alone provides poor observability ( η z, ˙ z, ¨ z = 0 .
44) of theR¨ossler dynamics [5, 10]. The picture changes completelyif only the linear dependencies are considered as depictedin the pruned fluence graph (Fig. 1(b)). In this case, thisgraph contains a single root SCC composed only of vari-ables x and y for which η x, ˙ x, ¨ x = 0 .
84 and η y, ˙ y, ¨ y = 1,respectively. Variable z is therefore no longer part of therecommended set of measurements when taking into ac-count the pruned fluence graph, in full agreement withthe symbolic observability coefficients [15].Moreover, and not surprisingly confirming these re-sults, when looking at the determinants of the corre-sponding analytical observability matrices we find thatDet O y ˙ y ¨ y = 1, whose constant value means full observ-ability, Det O x ˙ x ¨ x = x − ( a + c ), meaning that observabilityis good as long as the determinant is not vanishing andit depends on variable x (order 1), and Det O z ˙ z ¨ z = z ,indicating that observability is very poor as it dependson the square of z (order 2). A 5D model for the circadian oscillations of theDrosophila.
Let us now consider a 5D rational model forthe circadian oscillations in the
Drosophila period pro-tein [22]. Dynamical equations and the correspondingsymbolic Jacobian matrix reduced to just the constantelements are reported in the Supplemental Material forall the systems considered in the subsequent part of thisletter. The pruned fluence graph (Fig. 2a) presents threeroot SCC, thus suggesting that just measuring variables x and x and either x or x , is sufficient to fully andefficiently account for the dynamics of the whole system.This prediction about the appropriate set of variables tomonitor is confirmed by the symbolic observability coeffi-cients of the reconstructed spaces { x x x } and { x x x } [29] whose values are both equal to 1, and by the con-stant analytical determinants Det O x x x = − k s k andDet O x x x = k s k , where k , k , , and k s are parametersof the model [15]. A 9D Rayleigh-B´enard convection model.
Let us nowconsider a 9D system describing the Rayleigh-B´enardconvection in a square platform [23]. Its pruned fluencegraph shown in Fig. 2b exhibits six root SCC, suggestingthat a good observability might be obtained by measur-ing variables x , x , x , and x , and either one between x or x , and another between x or x . Therefore, atleast 6 variables are needed in this 9D reaction networkto effectively ensure full observability. This selection ofvariables is in agreement with the fact that most of thecombinations whose observability coefficient is equal to1, do not involve variables x , x , x , x , and x [15]. Forinstance, Det O x x x x x x = − b σ / b and σ are parameters of the model, confirming that a full ob-servability can be indeed obtained with this reduced setof measured variables. A 13D model for the DNA replication in fission yeast.
A more challenging dynamical system, that is also anal-ysed in [12], is the model for cell cycle control in fis-sion yeast governing the concentrations of the state vari-ables [24]. The corresponding pruned fluence graph
TABLE I: Minimum set of variables that are needed to measure according to i) Liu and coworkers’ inference graph, and ii) theproposed pruned fluence graph. The cardinality of both minimum sensor sets m and m is reported in the column next toeach case. Last column indicates the required number of variables to measure for getting full observability according to exactanalytical calculations [15]. ∨ ≡ or and ∧ ≡ and.Model Inference graph m Pruned fluence graph m m ( η = 1)3D x ∨ y ∨ z x ∨ y x ∨ x ∨ x ∨ x ∨ x x ∧ x ∧ ( x ∨ x ) 3 39D x ∧ ( x ∨ x ∨ x ∨ x ∨ x ∨ x ∨ x ∨ x ) 2 ( x ∨ x ) ∧ x ∧ x ∧ x ∧ x ∧ ( x ∨ x ) 6 613D ( x ∨ x ∨ x ∨ x ∨ x ∨ x ∨ x ∨ x ) 5 ( x ∨ x ) ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x ∧ x (Fig. 2c) presents 9 root SCCs, meaning that, at least9 variables — one from each SCC — must be mea-sured. The detection of the SCCs of the pruned flu-ence graph tells us immediately that variables x , x and x can be discarded from the minimum sensor setas well as that either variable x or x , can be ex-cluded but not simultaneously. Indeed, the combina-tions providing good observability ( η = 0 .
93) with 9 vari-ables measured do not involve the sets { x , x , x , x } or { x , x , x , x } . For instance, the reconstructed spacespanned by { x x x x x x x x x } yields a η = 0 . O = ( k r + k + k ′ )( k r + k )( x − k r + k ) K mc +1 − x k c β , thatis, constant over the whole state space but the plane x = 1. This indicates that the developed frameworkto determine the minimum sensor set is not guaranteeingfull observability in a d -dimensional reconstructed spacebut its observation is mandatory to ensure good observ-ability. In fact, as reported in [15], an extra variablemust be added for getting a full observability. For in-stance, η x x x x x x x x x x = 1 is associated withDet O s = ( k r + k ) ( k r + k ) which never vanishes.These results are in full disagreement with those reportedin [12] where the authors build a fluence graph treatingall dynamical interdependences as linear. Just for illus-tration, their analysis gives rise to the existence of twoSCCs: { x } and another one with the rest of the 12 vari-ables which is a root SCC. Therefore, their conclusion isthat by just monitoring any variable in the root SCC,the system is observable as verified by the Sedoglavic’salgorithm [25], an algorithm that certifies the local (notglobal) observation in a probabilistic way.A more thorough comparison of the two approaches isgiven in Table I, where the minimum sensor set is re-ported in each case for the four nonlinear dynamical sys- tems considered in this Letter. As observed in the lastcolumn of the table, where the minimum number m ofvariables needed to get full observability according to ex-act analytical calculations, Liu and coworker’s approachtends to underestimate the number of variables.Complex networks are large dimensional systems forwhich it is not possible to measure all the variables re-quired for a full description of any of their states.Considering that nonlinear links are generally respon-sible for the lack of local observability, we proposed con-structing a pruned fluence graph considering only linearlinks – corresponding to the constant non zero terms ofthe Jacobian matrix of the network. We showed thatidentifying the root SCCs of the pruned fluence graphallowed to correctly identify the reduced set of measure-ments providing a good observability of the network dy-namics. This technique was validated with the use ofsymbolic observability coefficients and the analytical de-terminants of observability matrices. We thus presentedan easy-to-implement technique for selecting the vari-ables to be measured for reconstructing a d -dimensionalspace of a reaction network. The extension to networksof dynamical systems is straightforward as long as theJacobian matrix describes the node dynamics and theirconnectivity. Acknowledgments
ISN acknowledges partial support from the Ministeriode Econom´ıa y Competitividad of Spain under proj ectFIS2013-41057-P and from the Group of Research Ex-celence URJC-Banco de Santander. LAA acknowledgesCNPq. [1] R. Kalman, IRE Transactions on Automatic Control ,110 (1959).[2] R. Hermann and A. Krener, IEEE Transactions on Au-tomatic Control , 728 (1977).[3] B. Friedland, Journal of Dynamic Systems, Measure- ment, and Control , 444 (1975).[4] L. Aguirre, IEEE Transactions on Education , 33(1995).[5] C. Letellier, J. Maquet, L. L. Sceller, G. Gouesbet, andL. A. Aguirre, Journal of Physics A: Mathematical and General , 7913 (1998).[6] C. Letellier and L. A. Aguirre, Chaos: An Interdisci-plinary Journal of Nonlinear Science , 549 (2002).[7] C. Letellier and L. A. Aguirre, Phys. Rev. E , 056202(2005).[8] C. Letellier, L. A. Aguirre, and J. Maquet, Phys. Rev. E , 066213 (2005).[9] A. J. Whalen, S. N. Brennan, T. D. Sauer, and S. J.Schiff, Phys. Rev. X , 011005 (2015).[10] E. Bianco-Martinez, M. S. Baptista, and C. Letellier,Phys. Rev. E , 062912 (2015).[11] C. Letellier and L. A. Aguirre, Phys. Rev. E , 066210(2009).[12] Y.-Y. Liu, J.-J. Slotine, and A.-L. Barab´asi, Proceedingsof the National Academy of Sciences , 2460 (2013).[13] B. Wang, L. Gao, Y. Gao, Y. Deng, and Y. Wang, Sci-entific Reports , 5399 (2014).[14] D. Leitold, ´A. Vathy-Fogarassy, and J. Abonyi, ScientificReports , 151 (2017).[15] C. Letellier, I. Sendi˜na-Nadal, E. Bianco-Martinez, andM. S. Baptista, Scientific Reports (2018).[16] A. Haber, F. Molnar, and A. E. Motter, ArXiv e-prints(2017), 1706.05462.[17] H. H. Rosenbrock, State-space and multivariable theory (Wiley, New York, 1970), ISBN 9780177810022.[18] T. Kailath,
Linear Systems , Information and System Sci-ences Series (Prentice-Hall, 1980), ISBN 9780135369616.[19] L. A. Aguirre and C. Letellier, Journal of Physics A:Mathematical and General , 6311 (2005). [20] M. Frunzete, J.-P. Barbot, and C. Letellier, Phys. Rev.E , 026205 (2012).[21] I. Sendi˜na Nadal, S. Boccaletti, and C. Letellier, Phys.Rev. E , 042205 (2016).[22] A. Goldbeter, Proceedings of the Royal Society of Lon-don B: Biological Sciences , 319 (1995), ISSN 0962-8452.[23] P. Reiterer, C. Lainscsek, F. Sch¨urrer, C. Letellier, andJ. Maquet, Journal of Physics A: Mathematical and Gen-eral , 7121 (1998).[24] B. Novak and J. J. Tyson, Proceedings of the NationalAcademy of Sciences , 9147 (1997), ISSN 0027-8424.[25] A. Sedoglavic, Journal of Symbolic Computation , 735(2002), ISSN 0747-7171.[26] The observability matrix O s corresponds in fact to theJacobian matrix of the change of coordinates Φ s : x → X where X ∈ R d is the reconstructed state vector from the m measured variables and their adequately chosen d − m Lie derivatives [8].[27] In [12] they flip the direction of each edge using the “in-ference diagram” by drawing a directed link x i → x j if x j appears in x i ’s differencial equation.[28] Notice that, since in [12] they flip the edge direction ofthe edges with respect to our choice for the graph rep-resentation of the dynamical system, their definition ofroot SCC is reversed here.[29] The notation { x ji } is equivalent to the vector { x i , ˙ x i , ¨ x i , ... } up to the jj