Topological Neural Networks over the Air

Claudio Battiloro; Paolo Di Lorenzo; Simone Fiorellino

REVIEW 3 major objections 7 minor 39 references

Topological Neural Networks over the Air

T0 review · 3 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Neural networks defined on cell complexes can be trained to perform their convolutional filtering over a wireless channel, absorbing fading and noise into the filter itself, and this makes them more robust than graph-based or…

desk verdict Reasonable incremental extension of over-the-air GNNs to cell complexes, with a clean formulation but an empirical robustness claim that needs sharper causal evidence. read the letter →

arxiv 2502.10070 v1 pith:Q3CURSYU submitted 2025-02-14 cs.IT cs.LGmath.IT

Simone Fiorellino , Claudio Battiloro , Paolo Di Lorenzo This is my paper · ORCID

classification cs.ITcs.LGmath.IT

keywords topologicalneuralnetworksover-the-aircomputationcellcomplexessignalprocessingfadingchannelsedgeflowssourcelocalizationwirelessdistributedlearning

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 argues that topological neural networks can be made robust to real wireless communication by putting the channel inside the filter rather than treating it as an external disturbance. It introduces AirTNN, a cell-complex neural network whose convolutional filtering is executed over the air: every message is faded and noised by the channel, and the network learns its filter weights under those same impairments. The claim is that this design keeps accuracy close to the ideal noiseless setting while fading strength grows and signal-to-noise ratio drops, beating both over-the-air graph networks and communication-agnostic graph and topological baselines. The reason to care is that edge-flow data collected by distributed sensors (traffic, hydraulic, or communication network monitoring) must in practice be exchanged over wireless links, and this is a first proposal that builds that exchange into the learning architecture.

What carries the argument

The over-the-air topological shift is the load-bearing object: lower and upper neighborhood aggregation is performed by wireless transmission, so instead of the fixed matrices $\mathbf{S}^{(d)}$ and $\mathbf{S}^{(u)}$ the network uses random shift matrices $\mathbf{S}_{\mathrm{air}}^{(d,p)}$ and $\mathbf{S}_{\mathrm{air}}^{(u,p)}$ whose nonzero entries are the channel gains $h_{ij}^{(d,p)}$ and $h_{ij}^{(u,p)}$. The topological filter over the air is the shift-and-sum operator $y = \sum_p w_p^{(d)} \mathbf{x}^{(d,p)} + \sum_p w_p^{(u)} \mathbf{x}^{(u,p)}$, which extends cell-complex FIR convolution to noisy random shifts, and AirTNN layers are banks of such filters followed by pointwise nonlinearities. Training treats the channel as part of the computation graph by sampling fading and noise afresh each step, so the learned weights are robust to those impairments at inference.

What would settle it

Run the same source-localization experiment but force the lower and upper neighborhood transmissions to share a single physical wireless topology with correlated fading, instead of the assumed independent networks: if AirTNN's accuracy collapses to the level of AirGNN or the communication-agnostic baselines, the two-topology independence assumption is the load-bearing part of the claimed robustness. A second, cheaper check is to keep fading gains fixed across all $p$-hop shifts instead of re-sampling them per transmission round, which removes the diversity the architecture relies on.

Watch

Extended reading notes

Core claim

AirTNN replaces the ideal cell-complex shift operators with over-the-air shifts whose matrix entries are channel fading coefficients and whose outputs carry additive Gaussian noise. A one-hop shift of an edge signal is $\mathbf{x}^{(d,1)} = \mathbf{S}_{\mathrm{air}}^{(d,1)} \mathbf{x} + \mathbf{n}^{(d,1)}$ over lower neighborhoods and $\mathbf{x}^{(u,1)} = \mathbf{S}_{\mathrm{air}}^{(u,1)} \mathbf{x} + \mathbf{n}^{(u,1)}$ over upper neighborhoods, and $p$-hop shifts are formed by products of these random matrices with accumulated noise. The paper shows that a topological filter composed of such noisy shifts reduces exactly to a standard cell-complex FIR filter when the channel is perfect, and that training by sampling channel coefficients and noise at each step yields filter weights matched to the channel statistics. In experiments on Stochastic Block Model cell complexes with diffused edge-flow signals, AirTNN retains high source-localization accuracy as the Rayleigh fading scale and noise grow, and it outperforms AirGNN as well as communication-agnostic GNN and TNN baselines.

Load-bearing premise

The whole mechanism depends on being able to build two separate wireless communication networks, one for lower-neighbor and one for upper-neighbor exchanges, with independent fading across links and transmission rounds; if those two topologies cannot be deployed or synchronized in practice, the over-the-air topological aggregation never materializes.

Editorial extensions

If this is right

AirTNN generalizes standard cell-complex topological filters, because perfect communication reduces its over-the-air shifts to the ordinary shift operators and recovers the textbook filter.
Because the same noisy channel model is present in training and inference, no retraining or channel estimation is needed at test time for robustness to the channel statistics seen in training.
The architecture uses two distinct communication topologies, one per neighborhood type, whose partially redundant links can make the overall distributed processing more resilient to link failures.
For edge-flow source localization, which matters for anomaly detection in infrastructure networks, the paper's experiments indicate that a wireless-aware topological model beats both graph-only over-the-air models and models trained under ideal-communication assumptions.
This is the first over-the-air topological neural network, extending the over-the-air computation idea from graph neural networks to cell complexes and higher-order data.

Reading between the lines

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

The paper assumes pre-designed resource allocation for the two communication topologies; jointly learning power control or beamforming together with the filter weights is a natural extension that could improve accuracy beyond the reported results.
If the lower and upper neighborhood transmissions share physical links or correlated fading in a real deployment, the diversity that the paper credits for robustness would be reduced, and the performance gap over AirGNN might shrink.
The same over-the-air shift construction applies in principle to signals on higher-order cells (polygons, $k=2$) and to simplicial complexes, where the upper and lower neighborhood structure is richer; the paper does not test those cases.
A matched training distribution is what delivers robustness: if test-time fading or noise follows a different distribution than the training samples, the architecture's advantage would likely degrade, suggesting a distribution-shift experiment as a sharp test.

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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

Short take: this is a legitimate incremental extension of AirGNN to regular cell complexes, and I think the architecture is worth building on. But the paper's central empirical claim—that the dual-neighborhood design itself confers channel robustness—is not actually established by the figures as presented.

The genuinely new piece is the formulation in Eqs. (4)-(12): defining topological shift operations with per-round fading and noise for both lower and upper neighborhoods of a cell complex, then building an AirTF and AirTNN layer that reduce to standard cell-complex filters under perfect channels and to AirGNN when only the lower neighborhood is ke5pt. That reduction is explicitly noted, which I appreciate. The training scheme (sampling channels per training step) follows the AirGNN recipe and is sensible. The paper is clearly written, the synthetic experiment at least demonstrates feasibility, and they ship a code repository.

The main soft spot is that the robustness story is not causally identified. Figures 2 and 3 plot absolute accuracy; the 'Ideal setting' is a single horizontal line, not per-model ideal baselines. AirTNN has two filter branches and more parameters than AirGNN, so its superior absolute accuracy could simply reflect extra capacity or additional upper-neighborhood information rather than higher resilience to channel fading and noise. To make the claimed mechanism stick, the paper needs degradation curves from each model's own ideal-setting baseline, or an ablation that removes the upper branch, or at least matched parameter budgets. Without that, the text's attribution—that two neighborhoods experiencing different channel conditions drive the gain—is an interpretation, not a result.

A second issue is the premise that the two distinct communication topologies (lower and upper) can be realized in practice with independent fading, separate time slots, and proper resource allocation. This is assumed rather than discussed. If the two topologies share physical links or require synchronization that is not feasible, the whole mechanism does not materialize. A paragraph acknowledging this would go a long way.

Minor but worth noting: Eqs. (10)-(12) contain typos in the recursion index and a stray noise term in the product. The math is conceptually sound, but it needs a cleanup before publication.

Overall, this is a small, honest step. The central idea—train with sampled channels and use both neighborhoods—is plausible and consistent with prior work on AirGNN and AirNet. The experiments are too thin to fully support the robustness claim as stated, but the paper deserves serious referee time. I would send it to review, expecting major revisions: add degradation-based evaluation, an upper-neighborhood ablation, and a discussion of the communication-topology assumption. As it stands, I would cite it as an extension of AirGNN to cell complexes, but not yet as evidence for the robustness mechanism.

Referee Report

3 major / 7 minor

Summary. The paper proposes AirTNN, a topological neural network architecture for edge signals over regular cell complexes that incorporates over-the-air wireless communication into the topological filtering operation. The filter aggregates information over lower and upper neighborhoods through channel-dependent shift operators with Rayleigh fading and AWGN, and the network is trained by sampling channel coefficients and noise at each training step. Numerical experiments on a synthetic edge-flow source localization task compare AirTNN with AirGNN, GNN, and TNN baselines across fading severity δ and SNR, reporting higher accuracy for AirTNN and attributing this to the exploitation of two distinct neighborhoods with different channel conditions.

Significance. The paper extends over-the-air graph neural networks to topological neural networks, which is a natural and useful step for distributed processing of edge-flow data over wireless networks. The architectural definitions are explicit, the training procedure is simple to reproduce (a code link is provided), and the synthetic data generation is described in detail. The claim that the architecture is robust to channel impairments is plausible and consistent with prior work on training with channel randomness, but the empirical support is incomplete: the reported figures compare absolute accuracies without per-model ideal baselines or ablations, so the causal attribution of the performance gain to channel robustness is not yet established. If the authors supply the missing baselines and an ablation, the paper would be a solid contribution.

major comments (3)

[§4, Figs. 2–3 and experimental setup] The central claim that AirTNN 'consistently demonstrates enhanced robustness against disturbances introduced by wireless communication' is not established by the reported experiments. Figs. 2 and 3 plot absolute accuracy, not the degradation from each model's ideal (noiseless/fading-free) operating point; the 'Ideal setting' line is unlabeled and no per-model ideal accuracies are reported. Because AirTNN in Eq. (13) uses two filter banks (lower and upper neighborhoods) while AirGNN uses a single graph shift, AirTNN has more trainable parameters and additional topological information; a higher-capacity model can achieve higher absolute accuracy while degrading equally or more under channel randomness. To support the robustness attribution, the authors should report per-model accuracy under ideal communication and the accuracy drop versus δ and SNR, include error bars or multiple seeds, and add an ablation that removes the upper-neighborhood branch (or matches the parameter budget) to show that the dual-neighborhood design, rather than expressivity, is responsible for the resilience.
[§4, 'Experimental setup'] The training distribution of the channel parameters is not specified. The figures sweep δ at fixed SNR=20 dB and SNR at δ=1, but the paper does not state over which δ and SNR ranges the stochastic channel coefficients and noise are sampled during training for AirTNN (or for AirGNN). Without this information, the comparison might reflect a mismatch in train/test channel distributions rather than a property of the proposed architecture. The authors should state the training channel model (e.g., Rayleigh scale and SNR values used per mini-batch) and ensure that AirGNN is trained under identical conditions.
[§3, before Eqs. (4)–(5)] The architecture rests on the assumption that 'proper resource allocation strategies (e.g., power control, analog beamforming, etc.)' can realize two distinct wireless communication topologies corresponding to the lower and upper neighborhoods, with independent fading across links and transmission rounds. No design, feasibility argument, or reference is provided for this assumption. Since the simulation simply instantiates the assumed independent channels, the numerical results do not validate the wireless feasibility. The authors should add a discussion of how such a resource allocation could be implemented (e.g., time/frequency separation, interference management) or at least state clearly that this is an idealization and discuss its practical limitations.

minor comments (7)

[§3, Eqs. (10)–(12)] There are typos in the recursion and noise terms. In Eq. (10), the inner shift should be S_air^(d,p-1), not S_air^(d,P-1), and the final noise term 'n(d,ρ)' should be a specifically indexed term such as n(d,i) or n(d,p). Eqs. (11) and (12) have analogous problems. These equations define the AirTF, so they should be corrected.
[§3, after Eq. (12)] The filter weights are written as {w_p^{(d)}}_{p=1}^P and {w_p^{(u)}}_{p=1}^P, but the sums in Eqs. (12)-(14) run from p=0 to P; align the indexing and explicitly define w_0.
[§4, Figs. 2–3] The 'Ideal setting' line is not defined in the captions or the text. Specify which model (or which ideal-communication configuration) it represents and its numerical value.
[§4, 'Dataset design'] The dataset paragraph gives the generative model but not the values of η and ψ, the number of training samples, or the number of independent runs; provide these for reproducibility.
[§4, 'Experimental setup'] State explicitly which of the five models listed in the setup are represented by the curves labeled 'TNN' and 'GNN' in Figs. 2 and 3; as written, the reader cannot tell whether these are the ideal-communication baselines or the models with channel impairments at test time.
[§3, 'AirTF'] The notation T_air(S(d), S(u)) uses S(d), S(u) for both the ideal shift operators from Eq. (2) and the channel-dependent operators in Eqs. (8)-(9); use distinct notation (e.g., S_air^(d,p), S_air^(u,p)) in the AirTF definition to avoid ambiguity.
[Abstract and §1] The phrase 'no previous works investigated TNNs over realistic wireless channels' is a strong claim; it is preceded by 'to the best of our knowledge' in Section 1, but the abstract omits this qualifier.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proposed AirTNN architecture is constructed directly from the wireless channel model and is evaluated empirically against external baselines.

full rationale

The paper makes no analytical claim that reduces by construction to a fitted parameter or to a self-citation. The AirTNN layer is defined by explicitly inserting stochastic channel fading and noise into the topological shift operations in Eqs. (4)-(11), and the resulting filter in Eq. (12) is a linear combination of those noisy shifted signals. There is no hidden equivalence between the definition of the architecture and the reported robustness result: the robustness claim is an empirical outcome of training with sampled channels and testing against AirGNN, TNN, and GNN baselines. The only self-citations (e.g., refs. [11], [13], [14], [16]) support background concepts such as cell attention and simplicial attention networks, but they are not load-bearing for the over-the-air construction, which is new. The use of the training approach from AirGNN [26] is transparent and the comparison is against that external method, not reduced to it. The skeptical concern that Figs. 2-3 show absolute accuracy rather than degradation from each model's own ideal baseline is a valid experimental-design critique, but it is not a circularity: the claim of enhanced robustness is not derived from the architecture definition; it is an empirical assertion that could in principle be refuted by a better-controlled experiment. No step in the derivation chain equates an input to an output by definition, and no fitted parameter is renamed as a prediction. Therefore, the circularity score is 0.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim is empirical. The architecture is built on standard topological signal processing plus a standard wireless channel model. The main free parameters are unstated experimental hyperparameters and data generation settings, which affects reproducibility rather than indicating circular fitting. No new entities are postulated. The key assumptions are the feasibility of two independent communication topologies and the realism of the synthetic data generation.

free parameters (3)

Filter length P = not reported
The AirTF order P in Eq. (12) sets the aggregation radius; its value is not given for the experiments, so sensitivity to this architectural choice cannot be assessed.
Training hyperparameters (learning rate, epochs, batch size, number of features) = not reported
These affect the reported accuracy; omitting them makes the numerical comparison difficult to reproduce.
Synthetic data parameters (psi, tau, eta) = not reported
Spike intensity variance psi, diffusion order tau, and number of spikes eta are chosen by hand in the dataset design but not specified, which changes task difficulty.

assumptions (4)

standard math Cell complex FIR filters defined in Eq. (2) are valid shift-and-sum operators over regular cell complexes.
Adopted from [5,7,28] in Section 2; the paper does not re-derive the theory.
domain assumption Wireless channels are slow-fading, independent across links, constant within a communication round, and i.i.d. across rounds, with AWGN.
Stated in Section 3 before Eqs. (4)-(5); this is the channel model used in training and testing.
domain assumption Proper resource allocation can create two separate communication topologies (lower and upper neighborhoods) with independent fading.
Assumed in Section 3; the entire AirTNN architecture depends on this feasibility.
ad hoc to paper The synthetic flow generation model x1 = B1^T x0 + B2 x2 and diffusion S^tau (x1 + delta_c) + n represent realistic edge-flow data.
Defined in Section 4, Dataset design; the empirical evaluation relies on this model's realism.

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Cite this review

Pith. "Pith review of Topological Neural Networks over the Air." pith.science (2026). https://pith.science/paper/Q3CURSYU

@misc{pith2026250210070,
  author       = {Pith},
  title        = {Pith review of: Topological Neural Networks over the Air},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/Q3CURSYU}},
  note         = {Machine review of arXiv:2502.10070}
}

read the original abstract

Topological neural networks (TNNs) are information processing architectures that model representations from data lying over topological spaces (e.g., simplicial or cell complexes) and allow for decentralized implementation through localized communications over different neighborhoods. Existing TNN architectures have not yet been considered in realistic communication scenarios, where channel effects typically introduce disturbances such as fading and noise. This paper aims to propose a novel TNN design, operating on regular cell complexes, that performs over-the-air computation, incorporating the wireless communication model into its architecture. Specifically, during training and inference, the proposed method considers channel impairments such as fading and noise in the topological convolutional filtering operation, which takes place over different signal orders and neighborhoods. Numerical results illustrate the architecture's robustness to channel impairments during testing and the superior performance with respect to existing architectures, which are either communication-agnostic or graph-based.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

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

[1] [1]

Telecommunications of the Future

INTRODUCTION In the last few years, there has been a large interest in develop- ing methodologies to analyze, process, and learn from data defined over the vertices of a graph [1, 2]. To this aim, several processing tools have been designed hinging on different graph shift operators (e.g., adjacency, Laplacian, etc.), thus leading to key contributions suc...

work page

[2] [2]

Topological Neural Networks over the Air

BACKGROUND This section reviews some useful basics of TSP over cell complexes. Regular Cell Complex. A regular cell complex is a topological arXiv:2502.10070v1 [cs.IT] 14 Feb 2025 space X having a partition {Xσ}σ∈PX of subspaces Xσ of X called cells, where PX is the indexing set of X , such that [4]:

work page Pith review arXiv 2025

[3] [3]

For each c ∈ X, every sufficient small neighborhood of c inter- sects finitely many Xσ

work page

[4] [4]

For all τ and σ we have that Xτ ∩ X σ ̸= ∅ iff Xτ ⊆ X σ, where X σ is the closure of the cell

work page

[5] [5]

Every Xσ is homeomorphic to Rk for some k

work page

[6] [6]

Condition 2 implies that the indexing set PX has a poset structure, given by τ ≤ σ iff Xτ ⊆ Xσ, and we say that τ bounds σ

For every σ ∈ PX there is a homeomorphism ϕ of a closed ball in Rk to X σ such that the restriction of ϕ to the interior of the ball is a homeomorphism onto Xσ. Condition 2 implies that the indexing set PX has a poset structure, given by τ ≤ σ iff Xτ ⊆ Xσ, and we say that τ bounds σ. This is known as the face poset of X . The regularity condition 4 implie...

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[7] [7]

TOPOLOGICAL NEURAL NETWORKS OVER THE AIR We consider a scenario in which data defined over the edges of the complex are collected by sensors, which can wirelessly exchange information according to two distinct communication topologies en- coded by the lower and upper connectivity induced by the complex structure. In particular, given the cell complex stru...

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[8] [8]

APPLICA TIONS AND NUMERICAL RESULTS Our framework can be applied in scenarios where it is of interest to learn from topological data (e.g., defined over edges and/or polygons of a cell complex), which are collected by sensors that communicate over wireless channels to enable distributed processing. Several ex- amples can be found in applications related t...

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