{"id":"32d17bc0-7dad-40ec-acf1-8eaaaceba4c0","arxiv_id":"2412.18492","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A Koopman-based two-step method identifies both the interaction topology and the local nonlinear dynamics of continuous-time networks with external inputs.","lead":"This paper proposes a two-step algorithm for recovering the structure and local equations of nonlinear networks from input-state data, using Koopman operator theory to work with nonlinear dynamics through linear lifted models. It matters because reverse-engineering network dynamics from measurements is central to biology, power grids, and traffic systems, and the method is designed to scale to large sparse networks.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 1's zero-mean condition is violated by the numerical examples' node functions {x_i, x_i^2}, so the edge-detection guarantee does not apply to the demonstrated results.","rationale":"The reader's verdict identifies the zero-mean violation in the numerical examples as the core weakness; my stress-test agrees and sharpens it. The exact projection calculation shows that the violation is not a mere technicality: at the level of exact L2 projections, a non-existent edge 2->1 yields a nonzero Λ_12 when node functions {x, x^2} are used. This means Theorem 1's guarantee is not merely unproven in the examples; it is false for those node functions. The method may still work in practice because the Lasso regression and threshold δ suppress small spurious coefficients, but the paper does not provide an alternative theoretical justification. The second-step dictionary-invariance issue is real but secondary, since the paper explicitly assumes the true dynamics belongs to the chosen dictionary (24), so parameter recovery reduces to standard EDMD. Therefore the central claim of complete network identification with guaranteed edge detection is not supported in the demonstrated setting. The appropriate verdict remains CONDITIONAL: the method is plausible and empirically promising, but the theoretical claim requires either corrected node functions (e.g., centered monomials satisfying (25)) or a revised theorem. No other concern supersedes this one.","tokens_in":20210,"tokens_out":12229,"duration_ms":109621,"concrete_test":"Compute the exact L2 projection of F_1(x_1)=x_1^2 onto span{x_2, x_2^2} on X=[-1,1]^2. The coefficient of x_2^2 is (∫ x_1^2 dx_1)(∫ x_2^2 dx_2)/(∫ x_2^4 dx_2 · ∫ 1 dx_1) = (2/3 · 2/3)/(2/5 · 2) = 5/9 ≠ 0, so the projection yields Λ_12 > 0 for a non-edge. This settles that the zero-mean condition is necessary for Theorem 1 and that the paper's demonstrated node functions violate it.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing concern is that Theorem 1, the theoretical basis for the first (topology identification) step, requires the node functions φ_kl and ψ_kl to satisfy the zero-mean conditions (25)-(26). Those conditions make the subspaces F_x,k mutually orthogonal, which is exactly what forces Λ_ik=0 for non-neighbors k in the proof via Lemma 1. In all numerical examples (Section 5.1 and 5.2), the node functions are monomials {x_i, x_i^2} and {u_i, u_i^2} on [-1,1]. While x_i has zero mean, x_i^2 does not: ∫_{-1}^{1} x^2 dx = 2/3 ≠ 0. Hence F_x,k ⊥ F_x,l fails. A direct calculation shows the failure is consequential: for a two-node system on [-1,1]^2 with F_1 = x_1^2 and no edge 2->1, the L2 projection of F_1 onto span{x_2, x_2^2} has component (5/9)x_2^2, so Λ_12 = 5/9 > 0 even though H_12=0. Thus, under the paper's stated node-function choice, exact orthogonal projection produces false positives, and Theorem 1's zero-false-positive guarantee is void. The numerical success of Section 5 therefore rests on thresholding (δ) and Lasso regularization rather than on the proved result. Because the first step is the foundation of the two-step complete-identification claim (if the neighbor sets are not exactly recovered, the local identification in step two is corrupted), this gap directly undermines the central claim as demonstrated.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript develops a two-step identification method for continuous-time nonlinear networked dynamics with heterogeneous local dynamics, external inputs, and node-dependent coupling functions. In the first step, the authors extend the dual Koopman method to non-autonomous systems and use it to estimate the vector field, then regress it on 'node functions' to determine each node's neighbor set and input set (Boolean topology reconstruction). In the second step, they build a local finite-dimensional Koopman lifted dynamics for each node from the estimated neighbors and recover the coefficients of local and coupling dictionary functions. The main theoretical result, Theorem 1, states that under zero-mean conditions on the node functions, the projection weights Lambda_ik are nonzero exactly when there is an edge k to i. Numerical experiments on Erdos-Renyi graphs, non-polynomial vector fields, and Hindmarsh-Rose neuronal networks support scalability (up to 1500 nodes) and accuracy, and the paper also constructs a modular lifted representation of the network dynamics.","tokens_in":20555,"tokens_out":9154,"duration_ms":85313,"significance":"If the stated guarantees held for the demonstrated configurations, this would be a valuable contribution to nonlinear network identification: it relaxes the homogeneity assumptions of earlier Koopman-based network identification, provides a data-efficiency argument for sparse networks, and yields a modular lifted model that can be reused for control or analysis. The paper includes a broad set of numerical experiments and comparisons with the dual method of [18]. However, the gap between the assumptions of Theorem 1 and the node functions used in Sections 5.1-5.3, together with the unstated invariance assumption in the local identification step, means that the current manuscript does not yet rigorously support the central claim as demonstrated. The core idea is promising and the necessary fixes are identifiable, so the issues are reparable within the scope of the paper.","major_comments":[{"comment":"The edge-detection guarantee is void for the node functions used in the experiments. Equations (25)-(26) require all node functions to have zero mean, and the proof of Theorem 1 uses this to conclude that F_x,k is orthogonal to F_x,l and to F_u,l. However, in all numerical sections the node functions are {x_i, x_i^2} and {u_i, u_i^2} on [-1,1], and x_i^2 and u_i^2 have mean 2/3, not zero. For a two-node example with no edge 2 to 1 and F_1 = x_1^2, the L2 projection onto span{x_2, x_2^2} has coefficient 5/9 on x_2^2, so the analogue of Lambda_12 is nonzero. Thus the no-false-positive statement does not cover the demonstrated results; the reported AUROC values are obtained through thresholding and Lasso rather than through Theorem 1. The authors should either use zero-mean node functions (for example, shifted or Legendre polynomials) in the experiments, or extend Theorem 1 to general node functions with explicit correction or threshold bounds.","section":"Theorem 1, Section 4.2.2 and Sections 5.1-5.2"},{"comment":"The theorem assumes that Lambda_ik is computed from the exact orthogonal projection in (27), but Algorithm 1 computes the coefficients from the Lasso problem (33), which is a different estimator. Even when (25)-(26) hold, a regularized regression does not coincide with the projection, so Theorem 1 does not by itself justify the statement that the estimated Lambda_ik are the projection weights used in the proof. The relation between the Lasso solution and the projection coefficients needs to be quantified, or Theorem 1 needs to be restated for the finite-sample regularized estimator, including the effect of the threshold delta in (34).","section":"Section 4.2.2, Eqs. (27)-(33)"},{"comment":"The local parameter recovery assumes that the finite-dimensional dictionary span F_i is invariant, or nearly invariant, under the Koopman operator or its generator, so that the discrete lifted matrices in (40) describe an exact finite-dimensional representation and log(Abar_i) in (43) equals the generator matrix restricted to that span. This invariance assumption is not stated. For general dictionaries such as the monomials, sinusoids, and exponentials used in Section 5.2, the span is not invariant, and then the coefficient estimates in (47) are not guaranteed to converge to the true alpha, beta, and gamma. A bound on the effect of the projection error on (47), or a precise approximation assumption, is needed to make the central identification claim rigorous.","section":"Section 4.3.2, Eqs. (40)-(44) and (47)"}],"minor_comments":[{"comment":"The proof drops O(T_s^2) terms before taking the limit; a uniform remainder bound over the finite set of test-function choices would be needed to justify the interchange of the limit and the argmin.","section":"Section 3.2, Proposition 1"},{"comment":"The shift operator T^t is called a left shift in Section 2.1 and a right shift in Section 4.1, although both definitions are the same; please make the terminology consistent.","section":"Sections 2.1 and 4.1"},{"comment":"The symbol Mbar_i is used both for the estimated input set and for the total number of input dictionary functions; please use distinct notations for these two objects.","section":"Section 4.3.1"},{"comment":"The acronym 'RSME' appears in figure captions and Table 1; it should be 'RMSE' for consistency with the definition in Section 5.","section":"Throughout Section 5"},{"comment":"The expressions 'sin(xti)' and 'exti' are typeset in a way that makes it difficult to distinguish the variable x_{t_i} from the exponential e^{x_{t_i}}; please clarify the notation and define all variables in the piecewise dynamics.","section":"Section 5.2, Eq. (50)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a plausible extension of the authors' earlier dual method in [18], and the numerical study is substantial. The main concern is theoretical: the theorem that carries the topology-identification guarantee does not apply to the node functions used in the experiments, and the local identification step relies on an unstated invariance assumption. These are fixable by either adjusting the experiments to satisfy the theorem's conditions or by extending the theory to cover the implemented estimator; I therefore recommend major revision rather than rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper does extend the Koopman identification toolbox: a non-autonomous dual method, a clean orthogonality argument for topology detection under zero-mean node functions, and a modular local identification step. The numerics are genuinely convincing—networks up to 1500 nodes, several nonlinear examples, comparison against the prior dual method. If the method works as intended, it's a useful step toward scalable reverse engineering of heterogeneous networks.\n\nThat said, the paper's theoretical guarantees don't cover its own experiments. Theorem 1 relies on the node functions satisfying the zero-mean conditions (25)–(26), which make the subspaces F_{x,k} pairwise orthogonal. In every numerical example, the node functions are monomials {x_i, x_i^2} and {u_i, u_i^2} on [-1,1]. The squares have mean 2/3, so the orthogonality fails. An explicit two-node calculation shows the exact L2 projection of x_1^2 onto span{x_2, x_2^2} yields a nonzero coefficient for x_2^2. So under the paper's stated choices, the no-false-positive result is void. The numeric success must be coming from Lasso regularization and thresholding, not from Theorem 1. That's not a fatal flaw, but it changes what the paper has shown: a heuristic that works nicely in examples, not a guaranteed reconstruction.\n\nThe other soft spot is the local parameter recovery. Equations (43)–(47) assume the dictionary span for a node's states, neighbors, and inputs is approximately invariant under the Koopman generator. That's a standard Koopman assumption, but it's not stated, and the paper gives no error bound. Proposition 1 is also heuristic—fine as a test-function-selection rule, not a theorem.\n\nThe sparsity data-efficiency claim is repeated in the abstract but never formalized. I don't think any of these are unfixable. The orthogonality condition can be met by using centered polynomials or other zero-mean bases, and the invariance assumption could be tested or stated explicitly. As it stands, I would send this to a serious referee, but expect a major revision. The core idea is worth developing; the paper currently overstates the guarantee.\n\nFor your own work: the dual method extension and the modular network lifting are worth citing, but I wouldn't rely on the topology-detection guarantee yet.","headline":"A solid two-step Koopman network identification method with strong numerics, but the edge-detection guarantee is void for the experiments shown.","tokens_in":21077,"tokens_out":2905,"would_cite":true,"duration_ms":26093,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["93B30"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper claims that a two-step Koopman-operator method recovers both the interaction graph and the local dynamics of heterogeneous nonlinear networks from input-state snapshots, with data cost set by sparsity rather than dictionary size.","keywords":["Koopman operator","network identification","Boolean network reconstruction","nonlinear systems","lifted dynamics","system identification","sparse networks","dynamic mode decomposition"],"falsifier":"Take a two-node network with no true edge and node functions {$x_k, x_k^2$} on $[-1,1]$; since $x_k^2$ has nonzero mean, the orthogonality behind Theorem 1 fails, so a noise-free run that still yields $\\Lambda_{ik}$ above threshold for the non-neighbor would show that the zero-mean condition is doing the theoretical work and must be enforced in practice.","tokens_in":19991,"feed_emoji":"🕸️","tokens_out":7307,"duration_ms":60917,"temperature":0.7,"pith_summary":"This paper develops a two-step, data-driven method for reconstructing both the wiring and the local dynamics of a continuous-time nonlinear network whose nodes may have different internal dynamics, external inputs, and pairwise coupling functions. The first step recovers, for every node, which other nodes and inputs affect it, using a Koopman-operator approximation of the vector field and per-node functions whose orthogonal projections produce an edge weight. The second step restricts estimation to the discovered neighbors and recovers the coefficients of the local dynamics and couplings from a lifted linear model. If the method works as claimed, a sparse network can be fully identified with far fewer measurement samples than the number of dictionary functions needed to describe the dynamics, which is why the approach scales to networks with over a thousand nodes.","feed_headline":"Koopman lifting identifies nonlinear networks from sparse samples","feed_subtitle":"Edges first, then local dynamics, with data cost set by sparsity rather than dictionary size.","key_machinery":"The machinery is the dual Koopman method: approximate the Koopman operator and its generator on the finite space of samples rather than on a chosen dictionary, use Gaussian radial-basis test functions selected to minimize the vector-field approximation error, and read off the vector-field values at the sample points. Edge detection then uses per-node subspaces $F_{x,k}=\\mathrm{span}\\{\\phi_{kl}\\}$ and $F_{u,k}=\\mathrm{span}\\{\\psi_{kl}\\}$; the zero-mean conditions (25)-(26) make these subspaces mutually orthogonal, so the projection weight $\\Lambda_{ik}$ of node $i$'s vector field onto node $k$'s subspace is nonzero exactly when node $k$ influences node $i$ (Theorem 1). Once neighborhoods are known, the local identification step fits lifted matrices $A_i, E_i, B_i$ and converts them back to the unknown coefficients through the continuous-time formulas (43)-(44) and the extraction rule (47).","core_discovery":"The central claim is that the Boolean interaction graph and the full local dynamics can both be recovered from input-state snapshots without solving a regression over the entire network dictionary. Theorem 1 guarantees that weights $\\Lambda_{ik}$ and $\\Delta_{ik}$, computed from orthogonal projections of each node's vector-field component onto the spans of node functions, vanish exactly when the corresponding coupling or input term is absent, provided the node functions integrate to zero and the true terms are not orthogonal to those spans. Under that guarantee, a sparse regression on the estimated vector-field samples yields per-node neighbor sets; a second, local Koopman lift then estimates the parameters of $F_i$, $H_{ik}$, and $G_{ik}$. The paper also produces a modular linear lifted model of the whole network, assembled blockwise from the local fits.","pith_inferences":["A testable extension is to center and orthogonalize node functions empirically against the sampling distribution before computing projections, which could preserve the no-false-positive property for polynomial dictionaries like the monomials used in the numerical demonstrations.","Because Theorem 1's converse also requires the true coupling to be non-orthogonal to the node-function subspace, a dictionary that misses a mode of the coupling will misclassify that edge; a diagnostic would be to inspect the residual norm after each projection.","The modular lifted model opens a route to distributed prediction and control, since each node's block is constructed independently from local neighbors once the graph is known.","A natural stress test is to run the two-step method on partially observed networks, where hidden nodes would likely be absorbed into the recovered local dynamics and could create spurious edges; the method's guarantees appear to require full-state measurement."],"forward_implications":["Boolean network reconstruction and local dynamics identification can be performed from the same dataset, so no separate topological probing experiment is needed.","Under network sparsity, the number of samples needed scales with the number of neighbors actually present rather than with the total dictionary size, making large sparse networks tractable.","Heterogeneous nodes are handled explicitly: each node gets its own local dictionary, so internal dynamics, coupling functions, and input effects may all differ across nodes.","A by-product linear lifted model of the whole network, assembled blockwise from local fits, can be used for analysis and control design.","Sampling time $T_s$ directly controls accuracy because the continuous-time parameter formulas come from a zero-order-hold approximation of the lifted dynamics."],"supporting_citations":[{"why":"Defines the Koopman operator, the linear lifting at the heart of the method.","marker":"[13]"},{"why":"Supplies the dual and main identification techniques that this paper extends to non-autonomous and networked settings.","marker":"[18]"},{"why":"Shows how to lift systems with control inputs via dynamic mode decomposition with control, used for the constant-input sampling scheme.","marker":"[28]"},{"why":"Extends the Koopman operator to non-autonomous systems with a shift operator, the formulation adopted for external inputs.","marker":"[14]"},{"why":"Prior Koopman-based network identification in discrete time with homogeneous dynamics and identical couplings, relaxed here.","marker":"[21]"},{"why":"Provides the semigroup and generator background justifying the infinitesimal-generator approximation used throughout.","marker":"[20]"}],"fun_headline_variants":["Two-step Koopman method reveals network edges and node dynamics","Sparse-data Koopman identification of nonlinear network structure and dynamics","Koopman identifies Boolean graph and local dynamics without full dictionary","Edges first, then dynamics: Koopman-based network identification","Koopman network ID: detect interactions, then fit local dynamics"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The edge-detection guarantee relies on the node functions integrating to zero over the state and input domains so that different nodes' subspaces are orthogonal, and the parameter estimates rely on the chosen local dictionaries being nearly invariant under the Koopman generator; the numerical examples use monomials such as $x_i^2$ and $u_i^2$ that do not integrate to zero, so the no-false-positive guarantee is not operative in those demonstrations.","fun_headline_variants_meta":{"raw":{"variants":["Two-step Koopman method reveals network edges and node dynamics","Sparse-data Koopman identification of nonlinear network structure and dynamics","Koopman identifies Boolean graph and local dynamics without full dictionary","Edges first, then dynamics: Koopman-based network identification","Koopman network ID: detect interactions, then fit local dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000468,"raw_usage":{"total_tokens":2278,"prompt_tokens":836,"completion_tokens":1442,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":452,"completion_tokens_details":{"reasoning_tokens":1353}},"tokens_in":452,"tokens_out":1442,"duration_ms":10446,"temperature":1.0,"reasoning_tokens":1353,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T04:42:00.375466+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take a two-node network with no true edge and node functions {$x_k, x_k^2$} on $[-1,1]$; since $x_k^2$ has nonzero mean, the orthogonality behind Theorem 1 fails, so a noise-free run that still yields $\\Lambda_{ik}$ above threshold for the non-neighbor would show that the zero-mean condition is doing the theoretical work and must be enforced in practice.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the Koopman operator, the linear lifting at the heart of the method."},{"cited_title":"IEEE Trans","cited_arxiv_id":null,"evidence_quote":"Supplies the dual and main identification techniques that this paper extends to non-autonomous and networked settings."},{"cited_title":"SIAM Journal on Applied Dynamical Systems 15(1), 142–161 (2016)","cited_arxiv_id":null,"evidence_quote":"Shows how to lift systems with control inputs via dynamic mode decomposition with control, used for the constant-input sampling scheme."},{"cited_title":"Automatica 93, 149–60 (2018)","cited_arxiv_id":null,"evidence_quote":"Extends the Koopman operator to non-autonomous systems with a shift operator, the formulation adopted for external inputs."},{"cited_title":"Nonlinear Theory and Its Applications, IEICE 13(2) (2022)","cited_arxiv_id":null,"evidence_quote":"Prior Koopman-based network identification in discrete time with homogeneous dynamics and identical couplings, relaxed here."},{"cited_title":"Springer (2020)","cited_arxiv_id":null,"evidence_quote":"Provides the semigroup and generator background justifying the infinitesimal-generator approximation used throughout."}],"review_version":1}