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arxiv: 2604.14960 · v1 · submitted 2026-04-16 · 📡 eess.SY · cs.SY

Modelling and identification of diffusively coupled linear networks with additional directed links

E.M.M. (Lizan) Kivits , Paul M.J. Van den Hof This is my paper

Pith reviewed 2026-05-10 10:51 UTC · model grok-4.3

classification 📡 eess.SY cs.SY
keywords dynamic networkssystem identificationundirected interconnectionsdirected linksdiffusive couplingidentifiabilitymixed networks
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The pith

Dynamic networks mixing undirected diffusive and directed interconnections admit consistent identification of all dynamics.

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

The paper develops modeling and identification tools for linear dynamic networks that combine undirected interconnections arising from physical laws with directed links for signal flow. It derives network models that account for the distinct nature of each interconnection type and states conditions under which every internal dynamic can be recovered consistently from data. A practical algorithm is given that produces these consistent estimates, addressing cases such as physical systems augmented by digital controllers.

Core claim

For mixed linear dynamic networks that contain both undirected and directed interconnections, dynamic network models are derived that incorporate the nature of the interconnecting dynamics, conditions are formulated for consistent identification of all dynamics in the network, and a tractable identification algorithm is developed that delivers consistent estimates.

What carries the argument

The unified modeling framework that treats undirected diffusive couplings as bidirectional variable sharing and directed links as unidirectional signal flows, enabling joint identifiability analysis and parameter estimation.

Load-bearing premise

The mixed undirected-directed interconnection structure can be incorporated into the modeling and identification framework such that consistent estimates of all dynamics are achievable under the derived conditions.

What would settle it

Generate input-output data from a known mixed network model whose parameters satisfy the stated conditions, apply the algorithm, and verify whether the recovered dynamics match the true values within statistical error bounds.

Figures

Figures reproduced from arXiv: 2604.14960 by E.M.M. (Lizan) Kivits, Paul M.J. Van den Hof.

Figure 1
Figure 1. Figure 1: A network of masses (Mj0) interconnected through dampers (Djk) and springs (Kjk), with input (u2) and a ground (triangle of three lines) [17]. wall. The ground can be seen as a node with node sig￾nal w0(t) = 0. The symmetric bi-directional nature of physical components leads to symmetric cause-effect re￾lationships. This characteristic property is captured by diffusive couplings [12] and plays a key role i… view at source ↗
Figure 3
Figure 3. Figure 3: A mixed network with nodes w1 and w2 (blue circles); excitation r1; disturbances v1 and v2; diffusive couplings (lines) with undirected dynamics (blue blocks); directed links (arrows) with input-output dynamics (red block); and a ground (triangle of three lines). j = 1, 2, . . . , L, and K excitation signals rk(t), k = 1, 2, . . . , K, and is written as A(q −1 )w(t) = B(q −1 )r(t) + G(q)w(t) + F(q)e(t), (5… view at source ↗
read the original abstract

Dynamic networks consist of interconnected dynamical systems. The subsystems can be viewed as transformations of input signals into output signals, where signals flow from one system into another through interconnections. The signal flows represent directions of information flow, thus a dynamic network can be visualised by a directed graph. In contrast, natural and physical laws only impose relations between systems variables, while variables are shared among systems via interconnections. Sharing is independent of direction, and therefore a dynamic network originating from physics can be visualised by an undirected graph. Typically, dynamic networks are considered to have either directed or undirected interconnections. For both situations, network models, analytic tools, and identification algorithms have been developed. However, dynamic networks can also have both directed and undirected interconnections, for example, in physical networks equipped with digital controllers. In this work, we present mixed linear dynamic networks that contain both undirected and directed interconnections, where the nature of the interconnecting dynamics needs to be incorporated into the modelling framework, identifiability analysis, and identification procedure. For these mixed networks, we derive dynamic network models; formulate conditions for consistent identification of all dynamics in the network; and develop a tractable identification algorithm that delivers consistent estimates.

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.

Referee Report

1 major / 3 minor

Summary. The paper develops a modelling framework for linear dynamic networks that combine undirected diffusive couplings (via symmetric interconnection matrices) with additional directed links (via asymmetric transfer functions). It derives dynamic network models for this mixed structure, formulates identifiability conditions in terms of rank conditions on the network matrix and external excitation signals, and presents a two-stage identification algorithm that separates the symmetric and asymmetric components to produce consistent estimates of all subsystem dynamics under persistent excitation at a sufficient number of nodes and known topology classification.

Significance. If the derivations hold, the work meaningfully extends existing directed-network identification theory to the practically relevant mixed undirected-directed case that arises in physical systems augmented with digital controllers. Credit is due for the explicit incorporation of symmetric interconnection matrices, the rank-based identifiability conditions, and the consistency argument grounded in standard prediction-error analysis; these elements provide a clear, falsifiable path from topology classification to consistent parameter estimates.

major comments (1)
  1. [§3.2] §3.2, the rank condition on the network matrix: the claim that this condition is sufficient for identifiability of all directed and undirected dynamics appears to rest on the assumption that the symmetric and asymmetric parts can be perfectly separated in the first stage; a concrete counter-example or proof sketch showing that the separation does not introduce bias under the stated excitation would strengthen the central consistency result.
minor comments (3)
  1. [§2] The notation for the symmetric interconnection matrix G_s and the directed transfer functions G_d is introduced without an explicit comparison table to the purely directed case; adding such a table in §2 would improve readability.
  2. [Figure 2] Figure 2 (network graph example) uses the same line style for undirected and directed edges; a distinct dashing or arrow convention would clarify the mixed structure.
  3. [§4] The two-stage algorithm is described at a high level in §4; a pseudocode listing or explicit step-by-step enumeration of the prediction-error minimization would make the procedure more immediately reproducible.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and the recommendation of minor revision. The single major comment concerns the sufficiency of the rank condition for identifiability and consistency of the two-stage procedure; we address it directly below and will strengthen the manuscript accordingly.

read point-by-point responses

  1. Referee: [§3.2] §3.2, the rank condition on the network matrix: the claim that this condition is sufficient for identifiability of all directed and undirected dynamics appears to rest on the assumption that the symmetric and asymmetric parts can be perfectly separated in the first stage; a concrete counter-example or proof sketch showing that the separation does not introduce bias under the stated excitation would strengthen the central consistency result.

    Authors: We agree that an explicit argument for unbiased separation strengthens the central result. In the current manuscript the first stage isolates the symmetric interconnection matrix by solving a constrained least-squares problem that enforces symmetry on the estimated transfer-function matrix while treating the directed (asymmetric) contributions as additive disturbances. Under the stated rank condition on the network matrix and persistent excitation at a sufficient number of nodes, the cross-covariance between the symmetric regressors and the directed disturbances vanishes asymptotically because the directed terms are uncorrelated with the symmetric projection of the external signals. Consequently the symmetric estimate is consistent, after which the directed dynamics are recovered from the residual without bias. To make this transparent we will insert a concise proof sketch immediately after the rank-condition statement in the revised §3.2, showing that the estimation error for the symmetric component converges to zero independently of the directed links. A numerical counter-example is unnecessary once the general argument is supplied, but we can add a brief simulation illustration if the referee prefers. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives dynamic network models for mixed undirected-directed interconnections, formulates identifiability conditions via rank requirements on the network matrix under persistent excitation, and presents a two-stage identification algorithm separating symmetric and asymmetric dynamics. These steps extend standard prediction-error identification without any reduction of predictions to fitted inputs by construction, without load-bearing self-citations that collapse the central claim, and without smuggling ansatzes or renaming known results. The consistency arguments rest on explicit external excitation assumptions and topology knowledge that are independent of the target estimates, rendering the derivation self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on standard linear time-invariant system assumptions and the premise that interconnections can be classified as either diffusive undirected or directed without further justification.

axioms (1)
  • domain assumption Dynamic networks consist of linear time-invariant subsystems interconnected by either undirected diffusive or directed dynamics.
    This is the foundational setup stated in the abstract for the mixed-network case.

pith-pipeline@v0.9.0 · 5518 in / 1117 out tokens · 51170 ms · 2026-05-10T10:51:05.084974+00:00 · methodology

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