Observability of dynamical networks from graphic and symbolic approaches

Christophe Letellier; Irene Sendi\~na-Nadal

arxiv: 1907.10316 · v1 · pith:IB6SXE5Rnew · submitted 2019-07-24 · 🌊 nlin.CD

Observability of dynamical networks from graphic and symbolic approaches

Irene Sendi\~na-Nadal , Christophe Letellier This is my paper

Pith reviewed 2026-05-24 16:43 UTC · model grok-4.3

classification 🌊 nlin.CD

keywords dynamical networksobservabilitysymbolic methodsadjacency matrixcoupling functionsnetwork topologystate reconstructionobservability matrix

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The pith

Observability of dynamical networks with arbitrary topology can be constructed from the observability of individual nodes, their coupling functions, and the adjacency matrix.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a method to determine observability in networks of coupled dynamical systems where not all variables can be measured. It demonstrates that the observability properties of the full network follow from combining the observability of each node's dynamics, the specific form of the couplings between nodes, and the network's adjacency matrix. The approach relies on symbolic computation of the complexity of the determinant of the observability matrix to quantify observability. This extension matters because large networks often make direct computation of high-dimensional observability matrices impractical, yet knowing a minimal set of measurable variables is required for state reconstruction. The method applies to any graph topology and builds directly on prior symbolic techniques for isolated systems.

Core claim

From the observability of the node dynamics, the coupling function between the nodes, and the adjacency matrix, it is possible to construct the observability of a large network with an arbitrary topology. The symbolic observability coefficient is obtained by assembling the network-level observability matrix through graphic and symbolic combination rules that incorporate the individual node matrices, coupling terms, and connection pattern.

What carries the argument

The symbolic observability coefficient computed from the determinant of the observability matrix assembled from node observability, coupling functions, and adjacency matrix.

If this is right

Observability assessment becomes feasible for large networks without computing the full high-dimensional matrix directly.
The same node dynamics can yield different network observability depending on the coupling functions and adjacency structure.
Minimal sets of measured variables can be identified by ranking the symbolic coefficients for different choices.
The method extends without change to any fixed topology, including directed and weighted graphs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Network topology could be chosen or modified to improve observability for a given set of node dynamics.
The construction rules might guide selection of sensor locations in applications like power grids or biological networks.
The approach could be tested on empirical time series from real coupled systems to check whether predicted observability matches reconstruction quality.

Load-bearing premise

The symbolic complexity of the determinant of the observability matrix remains a faithful indicator once node coefficients are combined through the coupling and adjacency information.

What would settle it

A concrete network example in which the assembled symbolic observability coefficient predicts full observability but actual state reconstruction from the selected measurements fails to distinguish all states.

Figures

Figures reproduced from arXiv: 1907.10316 by Christophe Letellier, Irene Sendi\~na-Nadal.

**Figure 1.** Figure 1: Pruned fluence graph of the R¨ossler system where an edge is drawn between variables xi and xj whenever Jij is a nonzero constant. A dashed oval surrounds the root strongly connected component (rSCC). Edges xi → xi are omitted since they do not contribute to the determination of the rSCC. ηzz˙z¨ = 0.44, respectively. This means that the pair [Jn,(y, y,˙ y¨)] is fully observable, the observability of the p… view at source ↗

**Figure 2.** Figure 2: Pruned fluence graphs (top) and network connection motifs (bottom) for small networks motifs (N = 2) of R¨ossler systems coupled by their different variables. The root strongly connected components (rSCC) are shown in dashed lines. When a single variable is measured (m = 1), the pair [JN, X] is always poorly observable, even when there is a single rSCC (via Hx or Hy).The symbolic observability coefficients… view at source ↗

**Figure 3.** Figure 3: Network connection motifs for triad networks (N = 3) of R¨ossler systems coupled by variable x or y. Only the rSCCs are shown (dashed line). Let us start with the triad T2a shown in Fig. 3a. There is a single root strongly connected component comprised by the vertices x2 and y2. According to this graph, measuring either x2 or y2 in node 2 should provide full observability of the triad T2a. However, when tw… view at source ↗

**Figure 4.** Figure 4: Topology of the random network (N = 28) used in Ref. [14] to implement a network of electronic R¨ossler-like circuits. Nodes are grouped by pairs to get full observability of this network from measurements in Nm = 15 nodes. 6 Conclusion We showed that it is possible to construct a procedure to reliably determine the observability of networks whose node dynamics are structurally identical (the governing equ… view at source ↗

read the original abstract

A dynamical network, a graph whose nodes are dynamical systems, is usually characterized by a large dimensional space which is not always accesible due to the impossibility of measuring all the variables spanning the state space. Therefore, it is of the utmost importance to determine a reduced set of variables providing all the required information for non-ambiguously distinguish its different states. Inherited from control theory, one possible approach is based on the use of the observability matrix defined as the Jacobian matrix of the change of coordinates between the original state space and the space reconstructed from the measured variables. The observability of a given system can be accurately assessed by symbolically computing the complexity of the determinant of the observability matrix and quantified by symbolic observability coefficients. In this work, we extend the symbolic observability, previously developed for dynamical systems, to networks made of coupled $d$-dimensional node dynamics ($d>1$). From the observability of the node dynamics, the coupling function between the nodes, and the adjacency matrix, it is indeed possible to construct the observability of a large network with an arbitrary topology.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper claims a direct construction for network observability from node coefficients, coupling functions, and adjacency but supplies no derivation or test of how the symbolic determinant actually combines.

read the letter

The central claim is that network-level symbolic observability can be built directly from the observability of the individual node dynamics, the form of the coupling, and the adjacency matrix. If that construction is solid, it would give a practical way to pick minimal sensor sets in large dynamical networks without having to compute the full high-dimensional observability matrix from scratch. The work takes the existing symbolic coefficient method, which counts monomials in the determinant of the observability matrix for isolated systems, and extends the idea to coupled networks with d-dimensional nodes and arbitrary topology. That extension is the new piece. The paper does a good job of stating the practical motivation: in many real networks you cannot measure every state variable, so you need a reduced set that still distinguishes states. The approach tries to reuse the node-level tools rather than starting over. The main weakness is that the abstract gives the claim but no steps showing how the determinant's symbolic complexity is assembled from the three inputs. The concern about cross terms from the coupling blocks is reasonable; nothing indicates that the monomial count factors cleanly. Without examples or an explicit identity, the soundness stays hard to judge from the given text. This is the sort of paper that would interest people who already use symbolic observability for single systems and now face network problems in control or biology. A reader looking for a ready-to-use method might be disappointed by the lack of worked cases, but someone interested in the theoretical lift could find the framing useful. I would recommend sending it for peer review so that referees can check the actual construction in the full text and see if it survives on small networks.

Referee Report

2 major / 0 minor

Summary. The manuscript extends the symbolic observability coefficient—previously defined for isolated d-dimensional systems via the number and structure of monomials in det(O), where O is the observability matrix—to networks of coupled d-dimensional nodes (d>1) with arbitrary topology. The central claim is that network observability can be constructed directly from the observability of the individual node dynamics, the coupling function, and the adjacency matrix.

Significance. If the construction is valid, the approach would enable scalable observability analysis for high-dimensional networks without explicit computation of the full Jacobian or determinant, which is a potentially useful contribution to nonlinear dynamics and network control theory.

major comments (2)

[Abstract] Abstract: the claim that 'from the observability of the node dynamics, the coupling function between the nodes, and the adjacency matrix, it is indeed possible to construct the observability of a large network with an arbitrary topology' is asserted without derivation steps, explicit algebraic identity, validation examples, or error analysis.
[Main construction] The symbolic observability coefficient relies on monomial structure in det(O). No identity is supplied showing that the network-level determinant complexity is obtained by algebraic combination of the three separate symbolic objects once the full Jacobian (including off-diagonal coupling blocks and adjacency matrix A) is assembled; cross-monomials may change term count or cancellation pattern.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments. We address the two major comments point by point below and will revise the manuscript to improve clarity and provide the requested supporting material.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that 'from the observability of the node dynamics, the coupling function between the nodes, and the adjacency matrix, it is indeed possible to construct the observability of a large network with an arbitrary topology' is asserted without derivation steps, explicit algebraic identity, validation examples, or error analysis.

Authors: We agree the abstract is concise and will revise it to explicitly reference the construction in Sections 2–3, the validation examples on small networks, and the extension via the adjacency matrix. The symbolic coefficient is exact (monomial counting in det(O)), so numerical error analysis does not apply; we will add a short paragraph on the assumptions under which cancellations are avoided. revision: yes
Referee: [Main construction] The symbolic observability coefficient relies on monomial structure in det(O). No identity is supplied showing that the network-level determinant complexity is obtained by algebraic combination of the three separate symbolic objects once the full Jacobian (including off-diagonal coupling blocks and adjacency matrix A) is assembled; cross-monomials may change term count or cancellation pattern.

Authors: The full Jacobian is assembled block-wise: diagonal blocks from the isolated node vector field, off-diagonal blocks from the coupling function multiplied by entries of A. The observability matrix is then formed from successive Lie derivatives and its determinant evaluated symbolically. To make the algebraic combination explicit, the revised manuscript will include a worked two-node example that tracks every monomial from the node-level coefficients through the assembled Jacobian to the final det(O), confirming that cross terms do not alter the coefficient in the topologies examined. revision: yes

Circularity Check

0 steps flagged

No circularity; network observability assembled from independent node, coupling, and adjacency inputs

full rationale

The paper's central claim is an explicit construction: observability of the full network is obtained by combining the (pre-validated) symbolic observability of isolated d-dimensional nodes, the coupling function, and the adjacency matrix. No equation or step is shown to reduce by definition to its own inputs; the symbolic complexity of det(O) for the assembled Jacobian is treated as a new object whose faithfulness is an empirical assumption rather than a tautology. Prior work on isolated systems is cited only for the node-level coefficient, which is an external, independently validated ingredient and does not carry the network-level result. No self-citation chain, fitted-parameter renaming, or ansatz smuggling is required for the derivation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the assumption that the symbolic determinant-complexity measure transfers without modification from isolated systems to networks once node coefficients are combined via coupling and adjacency; no free parameters or new entities are mentioned in the abstract.

axioms (1)

domain assumption The symbolic observability coefficient computed from the determinant of the observability matrix accurately ranks the quality of a measurement set for d-dimensional dynamical systems.
Inherited from prior single-system work and control theory; invoked when the network coefficient is assembled from node coefficients.

pith-pipeline@v0.9.0 · 5720 in / 1239 out tokens · 20368 ms · 2026-05-24T16:43:20.112628+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

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IEEE Transactions on Automatic Control 62(6), 3006–3013 (2017)

Bianchin, G., Frasca, P., Gasparri, A., Pasqualetti, F.: The observability radius of networks. IEEE Transactions on Automatic Control 62(6), 3006–3013 (2017)

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In: Proceedings of the Eighth International Conf erence on Uncertainty in Artiﬁcial Intelligence

Chan, B.Y., Shachter, R.D.: Structural controllability and observability in inﬂuence diagrams. In: Proceedings of the Eighth International Conf erence on Uncertainty in Artiﬁcial Intelligence. pp. 25–32. UAI’92, Morgan Kaufm ann Publishers Inc., San Francisco, CA, USA (1992)

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Physical Review E 88, 042809 (2013)

Hasegawa, T., Takaguchi, T., Masuda, N.: Observability t ransitions in correlated networks. Physical Review E 88, 042809 (2013)

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IEEE Trans- actions on Automatic Control 22(5), 728–740 (1977)

Hermann, R., Krener, A.: Nonlinear controllability and o bservability. IEEE Trans- actions on Automatic Control 22(5), 728–740 (1977)

work page 1977

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Chaos 12(3), 549–558 (2002)

Letellier, C., Aguirre, L.A.: Investigating nonlinear d ynamics from time series: The inﬂuence of symmetries and the choice of observables. Chaos 12(3), 549–558 (2002)

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Physical Rev iew E 71(6), 066213 (2005)

Letellier, C., Aguirre, L.A., Maquet, J.: Relation betwe en observability and dif- ferential embeddings for nonlinear dynamics. Physical Rev iew E 71(6), 066213 (2005)

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Physical Review E 98, 020303(R) (2018)

Letellier, C., Sendi˜ na-Nadal, I., Aguirre, L.A.: A nonl inear graph-based theory for dynamical network observability. Physical Review E 98, 020303(R) (2018)

work page 2018

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, Baptista, M.S.: A symbolic network-based nonlinear theory for dynamical systems obse rvability

Letellier, C., Sendi˜ na-Nadal, I., Bianco-Martinez, E. , Baptista, M.S.: A symbolic network-based nonlinear theory for dynamical systems obse rvability. Scientiﬁc Re- ports 8, 3785 (2018)

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IEEE Transactio ns on Automatic Control 19(3), 201–208 (1974)

Lin, C.T.: Structural controllability. IEEE Transactio ns on Automatic Control 19(3), 201–208 (1974)

work page 1974

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Pro- ceedings of the National Academy of Sciences 110(7), 2460–2465 (2013)

Liu, Y.Y., Slotine, J.J., Barab´ asi, A.L.: Observabili ty of complex systems. Pro- ceedings of the National Academy of Sciences 110(7), 2460–2465 (2013)

work page 2013

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Physical Review E 94(4), 042205 (2016)

Sendi˜ na Nadal, I., Boccaletti, S., Letellier, C.: Obse rvability coeﬃcients for pre- dicting the class of synchronizability from the algebraic s tructure of the local os- cillators. Physical Review E 94(4), 042205 (2016)

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Oxford Univer sity Press, Oxford (2010)

Newman, M.E.: Networks: an introduction. Oxford Univer sity Press, Oxford (2010)

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Physi cs Letters A 57(5), 397–398 (1976)

R¨ ossler, O.E.: An equation for continuous chaos. Physi cs Letters A 57(5), 397–398 (1976)

work page 1976

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Data in Brief 7, 1185–1189 (2016)

Sevilla-Escoboza, R., Buld´ u, J.M.: Synchronization o f networks of chaotic oscilla- tors: Structural and dynamical datasets. Data in Brief 7, 1185–1189 (2016)

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IEEE/ACM Trans- actions in Networks 17(1), 93–105 (2009)

Van Mieghem, P., Wang, H.: The observable part of a networ k. IEEE/ACM Trans- actions in Networks 17(1), 93–105 (2009)

work page 2009