Graph signal aware decomposition of dynamic networks via latent graphs

Andrei Buciulea; Antonio G. Marques; Bishwadeep Das; Elvin Isufi

REVIEW 3 major objections 5 minor 49 references

Graph signal aware decomposition of dynamic networks via latent graphs

T0 review · 3 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper proposes modeling a dynamic network's adjacency tensor as a weighted sum of a few latent graphs, and claims smooth node signals make that decomposition recover unobserved edges with a provably convergent alternating estimator.

desk verdict A useful signal-aware extension of the authors' ICASSP latent-graph decomposition, but the convergence proof as written does not cover the implemented ADMM steps and the real-data evaluation is partly self-confirming; worth reviewing with major revision. read the letter →

arxiv 2506.08519 v1 pith:QZCT75EM submitted 2025-06-10 eess.SP cs.SI

Bishwadeep Das , Andrei Buciulea , Antonio G. Marques , Elvin Isufi This is my paper

classification eess.SPcs.SI MSC 05C8215A6990C2694A12

keywords dynamicgraphlearninglatentdecompositiontemporalnetworkssignalsmoothnesstensoralternatingminimizationpartialobservationsnetworkreconstruction

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the evolution of a dynamic network can be represented as a small set of latent graphs whose weights change over time, and that node signals observed alongside the topology help recover those components when most edges are unobserved. The proposed Dynamic Graph Decomposition (DGD) writes the time-varying adjacency tensor as a sum of outer products of latent adjacency matrices and temporal signature vectors, then estimates both by alternating convex updates. If the claim is right, partially observed social, sensor, and climate networks can be reconstructed by exploiting the coupling between topology and signal smoothness, and the recovered components are directly interpretable as graphs. The numerical study reports that DGD outperforms standard tensor decompositions and signal-only topology inference, with the largest gains when only 20–40 percent of the topology is observed.

What carries the argument

At the core is the outer-product decomposition $\boldsymbol{\mathcal A}=\sum_{r=1}^R \boldsymbol A_r \diamond \boldsymbol c_r$ and its time-slice form $\boldsymbol A_{:,:,t}=\sum_r [\boldsymbol c_r]_t \boldsymbol A_r$, in which the dynamic topology is a weighted sum of $R$ latent graph adjacency matrices, with $\diamond$ building the tensor whose $t$-th slice is $[\boldsymbol c_r]_t \boldsymbol A_r$. The workhorse objective is Problem (11), which balances five terms: the masked least-squares fit, the $\ell^1$ sparsity of each latent graph, the signal-smoothness cost $\sum_t\sum_r [\mathbf C]_{t,r}\operatorname{tr}(\boldsymbol A_r \mathbf Z_t)/2$ with $\mathbf Z_t$ encoding pairwise squared signal differences, the orthogonality penalty $\sum_{r\ne \bar r}\operatorname{tr}(\boldsymbol A_r^\top \boldsymbol A_{\bar r})$, and the smoothness penalty $\|\mathbf D\mathbf C\|_F^2$ on the temporal signatures, together with a positivity/degree constraint. Alternating minimization exploits the block multiconvexity of this objective: each $\boldsymbol A_r$ update and the $\mathbf C$ update is strongly convex when the other blocks are fixed, and ADMM performs projected gradient steps, giving the convergence guarantee of Proposition 1 via block-coordinate descent theory.

What would settle it

Run DGD and its no-signal variant NSDGD on a synthetic dynamic network whose node signals are deliberately generated to be non-smooth with respect to the true graph (for example, white noise projected onto the high-pass Laplacian eigenvectors), keeping 20–40 percent of the edges observed; if DGD does not beat NSDGD in masked-edge relative error, the reported gains come from the smoothness prior rather than from the decomposition structure itself.

Watch

Extended reading notes

Core claim

Dynamic Graph Decomposition (DGD) models the adjacency tensor $\boldsymbol{\mathcal A}\in\mathbb{R}^{N\times N\times T}$ as $\boldsymbol{\mathcal A}=\sum_{r=1}^R \boldsymbol A_r \diamond \boldsymbol c_r$, so the $t$-th time slice is $\boldsymbol A_{:,:,t}=\sum_{r=1}^R [\boldsymbol c_r]_t\boldsymbol A_r$: each observed topology is a non-negative linear combination of $R$ sparse, undirected latent graphs scaled by their temporal signatures. The estimation problem minimizes a masked least-squares fit to the observed topology plus sparsity on each $\boldsymbol A_r$, a quadratic-variation penalty that rewards smoothness of the observed node signals on the reconstructed topology, a penalty that keeps the supports of different latent graphs disjoint, a temporal-difference penalty on the signatures $\boldsymbol c_r$, and a per-node degree lower bound that prevents trivial zero solutions. Algorithm 1 alternates between $R$ convex $\boldsymbol A_r$-subproblems and one convex $\mathbf C$-subproblem, each solved with the alternating direction method of multipliers; Proposition 1 proves that the sequence converges to a stationary point (a block-coordinate minimizer) under a mild condition on the observed mask and signatures. Experiments on a synthetic network that switches between stochastic block models and on sea-surface temperature, US temperature, and face-to-face contact datasets report that DGD reconstructs masked edges with lower relative error than CPD, BTD, SICA, smooth-graph learning, and the same method without the signal prior, with the largest advantage at low observation percentages.

Load-bearing premise

The load-bearing premise is that the node signals are smooth on the true dynamic graph at every time step, so that connected nodes carry similar values; the paper relies on this assumption both to steer the latent graphs toward correct edges and, for the SeaSurf and USTemp datasets, to construct the ground-truth graphs it evaluates against.

Editorial extensions

If this is right

With 20–40 percent of the topology observed, DGD reconstructs the unobserved edges with lower relative error than the compared tensor-decomposition, signal-only, and no-signal baselines on the tested datasets.
Every latent graph is useful: no single component reconstructs the network as well as their combination, because the temporal signatures of different components peak during different time windows.
The signal prior contributes most when observations are scarce, since the gap between DGD and its no-signal counterpart narrows as the observed fraction of the topology grows.
The alternating estimator can be run to a stationary point, and in the experiments the objective converges within about twenty iterations for most datasets.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper leaves implicit that the recovered latent graphs are a compressed, interpretable summary of the network's evolution; a natural next step is to feed them into link prediction or anomaly detection, tasks the introduction cites as motivations but does not evaluate.
Because each block subproblem is convex, a streaming extension is plausible: update latent graphs and temporal signatures incrementally as new time slices arrive, which the authors list as future work.
A testable extension is to run DGD on a dataset with independently recorded edge labels while making the node signals deliberately non-smooth; that would isolate the contribution of the signal prior from the contribution of the latent-graph decomposition itself.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Take a look at this if you work on dynamic graph learning. The core idea is a two-way tensor decomposition A = sum_r A_r ⋄ c_r into latent adjacency matrices and temporal signatures, which the authors introduced at ICASSP 2024. What's new here is the signal smoothness coupling (Eq. 6), the non-overlap and degree constraints, and the application to reconstructing unobserved topology from partial edges plus smooth node signals. That is a reasonable and useful extension.

What the paper does well: the experiments are fairly solid. DGD beats CPD, BTD, SICA, and the no-signal variant NSDGD across datasets, and the gap grows when observations are scarce. The Contact dataset is an honest independent check—ground truth comes from real interactions, not from the same smoothness prior—and DGD does well there. The authors also report when SGL beats them on SwDyn and give a sensible explanation. That is good scientific behavior.

The soft spots are real, though. Proposition 1 claims convergence to a stationary point, but Appendix B only checks multiconvexity, feasibility, and continuity; it never verifies that the implemented ADMM updates—single projected-gradient steps on the augmented Lagrangian—actually solve the block subproblems, which is what the cited Xu-Yin theorem requires. With finite K (as used everywhere), the inner loop does not produce exact minimizers; with K→∞, there is no proof that this inexact scheme converges to the subproblem solution. So the analyzed sequence is not the one the algorithm runs. This is a load-bearing gap in the abstract's central claim, and it should be fixed or the claim downgraded to empirical convergence.

The evaluation on SeaSurf and USTemp also builds the ground-truth graphs by selecting edges where the signal is smooth—the same criterion the method uses as a prior. The temporal train/test split is a plus, but the ground truth itself is not independent, which cheats the comparison in the method's favor. I'd like to see the authors either add another independent ground-truth dataset or explicitly show results on Contact as the primary evidence. The smoothness assumption itself is not stress-tested; if the signals aren't smooth, the prior can pull edges the wrong way.

Who should read this? Researchers in graph signal processing and dynamic network inference. The method is plausible and the empirical comparisons are useful. I would send it to review with the expectation of major revision, mainly to fix or weaken the convergence claim and to de-confound the real-data evaluation.

Referee Report

3 major / 5 minor

Summary. The paper proposes Dynamic Graph Decomposition (DGD), a model in which the time-varying adjacency tensor of a dynamic network is represented as a sum of R latent graph adjacency matrices A_r modulated by temporal signature vectors c_r. The formulation combines a masked least-squares fit of the observed topology with priors for graph sparsity, signal smoothness, non-overlapping latent supports, temporal smoothness, and a minimum-degree constraint. The authors solve the resulting nonconvex problem by alternating minimization, using inner projected-gradient/ADMM loops for the A_r and C blocks, and claim in Proposition 1 that the algorithm converges to a block-coordinate minimizer (stationary point) of the problem. They also present experiments on synthetic and three real datasets, reporting that DGD outperforms tensor-decomposition baselines and signal-only baselines in reconstructing unobserved edges, especially under limited observations.

Significance. If the formulation and convergence claims were correct, the paper would contribute a useful bridge between tensor decomposition and graph signal processing for dynamic network completion: the latent graphs are interpretable, and the signal-smoothness prior offers a principled way to use nodal observations when topology is partially missing. The synthetic experiments with known latent graphs are a good idea, and the comparison against CPD, BTD, SGL, and a no-signal ablation is informative. However, the paper's central technical claims are not currently supported: the convergence proof does not cover the implemented algorithm, and the gradients used in the algorithm do not match the stated Hadamard-masked objective. These issues are load-bearing, because they affect both the theoretical guarantee and the validity of the empirical evaluation. The core decomposition idea is promising and potentially salvageable, which is why I am not recommending rejection outright, but the manuscript requires substantial revision.

major comments (3)

[§IV-A and Appendix B (Proposition 1)] The proof of Proposition 1 does not apply to Algorithm 1 as implemented. Algorithm 1 updates A_r and C via single projected-gradient steps, Eqs. (17) and (26), inside finite-K inner loops; it does not compute an exact minimizer of the block subproblems (14) and (23). Appendix B cites Theorem 2.3 and Assumptions 1–2 of Xu and Yin [46] and verifies block multiconvexity, convexity of feasible sets, finiteness of minima, and continuity of set maps, but it never verifies the key assumption that each block update is an exact (or sufficiently accurate) minimizer. With finite K, the generated sequence is not a block-coordinate minimization sequence; with K tending to infinity, the paper provides no proof that the projected-gradient ADMM loop converges to the solution of the convex subproblem. Moreover, Proposition 1's phrase 'as K→∞' does not address the outer iterations I→∞. The result therefore does not establish convergence of the sequence actually produced by Algorithm 1 to a stationary point of Problem (11).
[§III-B, §III-C, and Appendix C (gradients)] The updates in Algorithm 1 do not minimize the objective stated in Eq. (11). For the C-subproblem, the loss in Eq. (22) is the Hadamard-masked term 1/2||M0∘(A_vec − A0 C^T)||_F^2, whose gradient with respect to C is −A0^T(M0∘(A_vec − A0 C^T)). Equation (39) instead uses F(−A_vec^T A0 + C A0^T A0) with F=diag(1^T_{N^2} M0), which is the gradient of a row-weighted full-observation loss 1/2||(A_vec − A0 C^T)F^{1/2}||_F^2 and agrees with the true gradient only when the mask is constant across entries. The same problem appears in Eq. (13), where the per-entry mask is summarized by the scalars 1^T_{N^2}m_t rather than applied entry-wise; the quadratic and cross terms in A_r are therefore not those of Eq. (11). As written, the implemented Algorithm 1 solves a different objective, so the convergence proof, even if repaired, would not apply to the reported algorithm.
[§V-A (SeaSurf and USTemp ground truth)] The real-data evaluation is partly self-confirming for the signal-smoothness prior. For SeaSurf and USTemp, the ground-truth graphs are constructed by selecting the top edges where the associated signal is smooth, using odd time measurements; DGD's prior g(X,A,C) in Eq. (6) enforces exactly this smoothness property on the estimated topology, and the method is then evaluated on even time measurements. The target is thus generated under the same smoothness model that the method optimizes. This does not make the comparison logically circular, since the held-out even measurements are separate, but it means the experiments cannot test the validity of the smoothness assumption, and it systematically favors DGD over methods that do not use signal smoothness. The claim that DGD recovers the 'true' latent graphs on these datasets should therefore be qualified.

minor comments (5)

[Eqs. (10) and (37)] The temporal difference matrix D is described as T×(T−1), but it is used as a (T−1)×T operator in ||DC||_F^2 and in D^T D; please correct the dimensions and notation.
[Eq. (26)] Equation (26) writes C^k = P_{S_A}(...), but the projection should be onto the nonnegative orthant S_C, not onto S_A.
[Abstract and §V-C] The abstract states that DGD outperforms 'signal-based topology identification methods,' but Figure 8 reports that SGL outperforms DGD on the SwDyn dataset for 10–40% observed topology. The abstract should be qualified to reflect this exception.
[Algorithm 1] The inner loops use fixed iteration count K and outer loop count I, but the paper does not specify a stopping criterion or convergence tolerance; please state how K and I are chosen in the experiments and how the results depend on them.
[General] The paper refers to a code repository but does not provide a URL or data-availability statement; since the experimental protocol depends on the exact implementation of the mask and the ADMM loops, a public code link would improve reproducibility.

Circularity Check

1 steps flagged · score 4.0 of 10

Evaluation of SeaSurf and USTemp is partly self-confirming because the ground-truth graphs are built from the same signal-smoothness criterion that the DGD objective optimizes; the core factorization and optimization are not circular.

self definitional [Section V.A, SeaSurf/USTemp ground-truth construction; Eq. (6) signal-smoothness prior]
"Since there is no ground-truth graph, we estimate one for each time window. This estimation is based on half of the available temporal measurements (those corresponding to odd time instants, τ = 1, 3, 5, ...), selecting the top 4N edges where the associated signal is smooth. [...] The signal-incorporating prior g() is the sum of the signal smoothness function for each time instant [...] This promotes the creation of new links from the signals in X that are smooth in the original temporal network when portions of A are not observed due to the mask."

On SeaSurf and USTemp the 'true' unobserved edges are defined as the top edges by signal smoothness, while the DGD objective contains exactly the same smoothness measure: Eq. (6), g(X,A,C)=Σ_t Σ_r [C]_{t,r} tr(A_r Z_t)/2, with the paper stating it 'promotes the creation of new links from the signals in X that are smooth.' Thus the masked entries that DGD is evaluated on are labels produced by the same quadratic-variation criterion the model optimizes. The reported reconstruction accuracy on these two datasets is therefore partially forced and cannot independently validate the graph-signal coupling; the gap over NSDGD is inflated. The synthetic, SwDyn, and Contact evaluations use independently defined ground truths and retain evidentiary value.

full rationale

The optimization derivation is self-contained: (1) is an estimation model, and (11) is a well-posed objective whose variables are estimated by the alternating updates (17)-(19) and (26)-(28); no constant or parameter in the derivation is defined in terms of the target output. The self-citations ([1], [45]) are not load-bearing: [1] is a preliminary conference version and [45] is only a pointer to the definition of block-coordinate minimizer. The main circularity is evaluative. For SeaSurf and USTemp the 'ground-truth' graphs are created by selecting the top edges where the associated signal is smooth, which is precisely the quadratic-variation prior used in Eq. (6) and in the DGD objective (11). Consequently, DGD's reconstruction of masked entries on those datasets is scored against labels generated by the model's own prior, and the reported advantage over the no-signal variant NSDGD is inflated there. The SwDyn, Contact, and synthetic experiments are not constructed this way, so the core factorization and the convergence claim retain independent content. The Proposition 1 proof gap (inner ADMM inexactness versus exact block minimization assumed by [46]) is a correctness concern, not a circular reduction, and is not scored here.

Assumptions & free parameters 8 free parameters · 5 assumptions · 1 invented entities

The central approximation depends on R latent graphs and five or more regularization hyperparameters, all chosen by hand; the smoothness prior is the key domain assumption; no new physical entities are introduced.

free parameters (8)

number of latent graphs R = 10 for main results
Model order chosen by the user; error decreases with R and the paper notes optimal R is non-trivial to identify (Sec. V.B).
sparsity weight gamma
Hyperparameter in objective (11) controlling sparse latent graphs; selection procedure not reported.
signal smoothness weight delta
Hyperparameter in objective (11) controlling how strongly node signals influence the latent graphs; central to the claimed signal-awareness.
non-overlap weight beta
Hyperparameter penalizing shared support across latent graphs; authors note higher beta can separate edges (Sec. V.B).
temporal smoothness weight mu
Hyperparameter on ||D C||_F^2 in objective (11).
regularization weight rho
Hyperparameter on ||C||_F^2, needed for strong convexity in C (Appendix A).
minimum degree zeta
Lower bound on per-node degree in constraint (8), set to avoid trivial zero solutions; value not specified.
ADMM and outer iteration counts lambda, lambda_c, K, I
Algorithm hyperparameters; the convergence proof requires K to infinity, but finite K is used in practice.

assumptions (5)

domain assumption Graph signals are smooth on the dynamic graph at each time t
Assumed in Eq. (6) via quadratic variation; if violated, the signal prior biases the latent graphs.
domain assumption The dynamic adjacency tensor admits a representation as sum_{r=1}^R A_r diamond c_r with R much less than T
Low-rank linear-combination model in Eq. (1); any real network may need R close to T, weakening the reconstruction claim.
domain assumption Latent graphs satisfy nonnegativity, symmetry, zero diagonal, and degree lower bound
Constraint set S_A in Problem (11); plausible for social and infrastructure networks but restrictive for signed or directed edges.
standard math The block coordinate descent convergence theorem of Xu and Yin [46] applies to the implemented algorithm
Invoked in Appendix B; the paper does not fully verify its assumptions for the ADMM inner loop.
domain assumption Assumption 1: for all r, sum_t [C]_{t,r}^2 1^T m_t > 0
Stated in Section IV-A; needed for strong convexity in A_r and may fail if a latent graph has zero temporal signature.

invented entities (1)

Latent adjacency matrices A_1,...,A_R independent evidence
purpose: Hidden factor matrices whose linear combination reconstructs the dynamic topology and which are meant to be interpretable as recurring interaction patterns.
They are validated indirectly by reconstruction of held-out entries (F1 and RE), but they are not uniquely identifiable without further constraints.

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Cite this review

Pith. "Pith review of Graph signal aware decomposition of dynamic networks via latent graphs." pith.science (2026). https://pith.science/paper/QZCT75EM

@misc{pith2026250608519,
  author       = {Pith},
  title        = {Pith review of: Graph signal aware decomposition of dynamic networks via latent graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QZCT75EM}},
  note         = {Machine review of arXiv:2506.08519}
}

read the original abstract

Dynamics on and of networks refer to changes in topology and node-associated signals, respectively and are pervasive in many socio-technological systems, including social, biological, and infrastructure networks. Due to practical constraints, privacy concerns, or malfunctions, we often observe only a fraction of the topological evolution and associated signal, which not only hinders downstream tasks but also restricts our analysis of network evolution. Such aspects could be mitigated by moving our attention at the underlying latent driving factors of the network evolution, which can be naturally uncovered via low-rank tensor decomposition. Tensor-based methods provide a powerful means of uncovering the underlying factors of network evolution through low-rank decompositions. However, the extracted embeddings typically lack a relational structure and are obtained independently from the node signals. This disconnect reduces the interpretability of the embeddings and overlooks the coupling between topology and signals. To address these limitations, we propose a novel two-way decomposition to represent a dynamic graph topology, where the structural evolution is captured by a linear combination of latent graph adjacency matrices reflecting the overall joint evolution of both the topology and the signal. Using spatio-temporal data, we estimate the latent adjacency matrices and their temporal scaling signatures via alternating minimization, and prove that our approach converges to a stationary point. Numerical results show that the proposed method recovers individually and collectively expressive latent graphs, outperforming both standard tensor-based decompositions and signal-based topology identification methods in reconstructing the missing network especially when observations are limited.

Figures

Figures reproduced from arXiv: 2506.08519 by the authors.

**Figure 1.** Decomposition of the temporal topology evolution in terms of the component adjacency tensors. (left) an evolving topology over [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗

**Figure 2.** Convergence of the DGD algorithm for different datasets [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 4.** Visualization of the components obtained from the DGD algorithm on USTemp dataset for [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗

Figures from the paper (6 more)

**Figure 5.** Figure 5: Since each latent graph has a temporal signature that [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]

**Figure 5.** Figure 5: Visualization of the components obtained from the DGD algorithm on Contact data for [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: Temporal signatures associated with each one of the latent [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: Reconstruction potential of R = 3 individual latent graphs (blue, orange and yellow) for each dataset compared to that of the their model-based combination (purple). Here we show RE for (left) SwSyn, (centre-left) SeaSurf, (centre-right) USTemp, and (right) Contact dat…

**Figure 8.** Figure 8: Relative error (RE) for different approaches as a function of the percentage of observed adjacency tensor A for (left) SwDyn, (centre-left) SeaSurf, (centre-right) USTemp, and (right) Contact datasets, respectively. The legend in [PITH_FULL_IMAGE:figures/full_fig_p011…

**Figure 9.** Figure 9: F1 of each of the R = 3 latent graphs (blue, orange and yellow) for each dataset compared to that of the their model-based combination (purple) REFERENCES [1] B. Das and E. Isufi, “Tensor graph decomposition for temporal networks,” in IEEE Intl. Conf. Acoustics, Speech…

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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