Computing and Communicating Functions in Disorganized Wireless Networks

Fangzhou Wu; F. Richard Yu; Guo Wei; Li Chen; Nan Zhao; Yunfei Chen

REVIEW 3 major objections 3 minor 30 references

Computing and Communicating Functions in Disorganized Wireless Networks

T0 review · 3 major / 3 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper shows that a disorganized wireless network can be reorganized into layers so that the fusion center's function-computation rate is the minimum of simple per-hop CoMAC rates, generalizing over-the-air computation and orthogonal…

desk verdict A promising hierarchical CoMAC framework undermined by a load-bearing entropy normalization error and an invalid Jensen step; the special cases are right but the general rate formula does not hold as stated. read the letter →

arxiv 1908.05030 v1 pith:YCP6UZPB submitted 2019-08-14 cs.IT math.IT

Fangzhou Wu , Li Chen , Nan Zhao , Yunfei Chen , F. Richard Yu , Guo Wei This is my paper · ORCID

classification cs.ITmath.IT

keywords functioncomputationCoMAChierarchicalnetworksdisorganizedwirelessachievableratetimeallocationpowercontrolnestedlatticecodes

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

The paper is trying to prove that function computation in wireless data aggregation can be extended from single-hop networks to arbitrary multi-hop, disorganized topologies without losing the efficiency of over-the-air computation. It proposes multi-layer function computation, which reorganizes the topology into layers of subgroups and groups, computes subgroup functions by concurrent transmissions (CoMAC), combines them with orthogonal communication, and reconstructs the desired function hop by hop. The central result is a closed-form computation rate equal to the minimum over all hops of the per-subgroup rate, so the worst subgroup in the worst layer governs the whole network. If the result holds, network designers get a single formula for allocating time and power, and existing CoMAC and orthogonal communication schemes become special cases of one general scheme.

What carries the argument

The central object is the layered hierarchical reorganization: every layer's nodes are partitioned into groups, each group into subgroups; within a subgroup all nodes transmit simultaneously so CoMAC computes a subgroup function, and within a group the subgroup functions are forwarded over orthogonal time slots so the node reconstructs the group function. The load-bearing recurrence is Eq. (18), which states that the number of function values available at a node is the minimum of what its parents supply and what its own subgroups compute; Theorem 2 is the repeated minimum of these per-hop rates. The physical-layer engine that makes each per-subgroup rate achievable is a sequence of nested lattice codes.

What would settle it

Simulate or build a three-layer network with four source nodes, two second-layer relays, and one fusion center under i.i.d. Rayleigh fading, and measure the maximum number of desired-function values per channel use with optimal time allocation; if the measured rate disagrees with Eq. (29), the paper's central claim is wrong.

Watch

Extended reading notes

Core claim

The paper establishes that any disorganized network whose nodes each have a single transmission destination can be reorganized into an $L$-layer hierarchy, and that in this hierarchy the fusion center can compute the desired function at rate $$R=\min_{l\in[2:L]}\min_{k\in\mathcal{K}_l}\alpha_{l,k}\min_{c\in\mathcal{C}_{\mathcal{N}_{l,k}}}\$beta^{{(c)}}$_{l,k}\mathbb{E}\left[C_+\left(\frac{1}{$K^{{(c)}}$_{\mathcal{N}_{l,k}}}+\min_{i\in\mathcal{K}^{(c)}_{\mathcal{N}_{l,k}}}|$h^{{i\to k}}$_{l-1}|^$2P^{{i\to k}}$_{l-1}\right)\right],$$ where $C_+(x)=\max\{\frac{1}{2}\log x,0\}$, $\alpha_{l,k}$ is the fraction of channel uses given to node $N_{l,k}$, and $\beta^{(c)}_{l,k}$ is the fraction given to subgroup $c$. The proof is a recurrence: the number of desired-function values that survive to a node is the minimum of the numbers its parent groups supply and the numbers its own subgroups compute, so the overall rate is the worst per-hop subgroup rate. The paper then derives closed-form solutions for time allocation with fixed power and with adaptive power, and notes that with one subgroup per layer the rate reduces to CoMAC while with one node per subgroup it reduces to orthogonal time-sharing.

Load-bearing premise

The formula assumes that every disorganized network can be reorganized into a layered hierarchy in which each node has exactly one transmission destination and every intermediate subgroup and group function carries the same entropy as the desired function; if either condition fails, the overall rate is not simply the minimum over per-hop rates.

Editorial extensions

If this is right

In a two-layer network, setting the number of subgroups per group to one recovers the classical CoMAC rate, and setting it to the number of source nodes recovers the orthogonal time-sharing rate.
Because the overall rate is the minimum across subgroups and layers, the optimal time allocation equalizes the products $\alpha_{l,k}\beta^{(c)}_{l,k}$ for all subgroups, with the closed-form optimum given by the reciprocal-sum expression in Eq. (29).
Adaptive power control improves the achievable rate over fixed power control, and its optimal allocation is expressed through the Lambert W function.
Adding layers lowers the computation rate because each layer consumes channel uses, while increasing the number of groups can support a larger number of source nodes with a slower rate decline.

Reading between the lines

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

The paper does not spell out that its min-of-rates formula turns function computation into a tree resource-allocation problem, so standard scheduling and network-utility methods could be adapted to choose group sizes and routes that maximize the bottleneck rate.
If a subgroup function has a different entropy from the desired function, the rate normalization by $H(f(\mathbf{b}_v))$ in Definition 2 would need to be layer-specific; testing this with non-symmetric functions would show where the formula needs correction.
The reorganization requires each node to have exactly one destination; extending the result to broadcasting or multi-destination nodes would likely require a min-cut-like bound over several per-node rates rather than a single minimum.
The simulations suggest that adding groups slows the rate loss as the number of sources grows; a natural next step is to prove a scaling law as both the number of sources and the number of groups grow.

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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

Colleague,

Short version: this paper has a sensible framing but a load-bearing error in the proof of its main theorem. I do not think the rate formulas are valid as stated, and the resource allocation sections inherit the problem.

What is genuinely new: the idea of reorganizing a "disorganized" network into a layered hierarchy of subgroups and groups, then running CoMAC within subgroups and orthogonal communication within groups, is a natural and useful extension of the relay-free CoMAC framework. The reductions to existing results (L=2, C=1 recovers [21]; orthogonal time-sharing is also recovered) are correct and show the authors know the prior art. The closed-form time/power allocation is a nice exercise and would be valuable if the underlying rate expression were correct. The paper is also clearly written and the simulations are matched to the formulas.

The soft spots are not minor. In the proof of Theorem 1, Eq. (16), the number of subgroup-function values obtained in |T| channel uses is written as U = R|T| / H(f(bv)) for every subgroup. But Definition 2 defines the rate of a function relative to that function's own entropy. For a subgroup function f^(c) that depends on a subset of sources, H(f^(c)) is generally smaller than H(f(bv)). Using the same denominator for all subgroups makes the bottleneck "min_c U^(c)" meaningless; a subgroup with high entropy but high rate can be the bottleneck, and the min in Eq. (14) does not follow. Theorem 2 chains these group rates, so the general rate formula is unsupported. I checked the stress-test counterexample (two subgroups with H=1 and H=10); it works. This is not a constant-factor artifact; it flips which subgroup is the bottleneck.

There is also a second, independent error in Eq. (21): Jensen's inequality is applied to C+(x)=max(0,0.5 log x), which is not concave on x>0. That step is used to move the expectation inside C+, so the fixed-power optimization and the claimed closed form are not justified.

Remark 3's equivalence claim (any disorganized network can be represented as such a hierarchy) is asserted without proof and seems too strong for arbitrary topologies.

In short: the framework and the special cases are right, but the general rate theorem and the optimization results built on it do not stand. This needs a real revision before it can be trusted. If I were editing, I would send it to review rather than desk-reject, because the idea is relevant and the errors are specific enough to fix. But the current version should not be accepted.

Who is this for? Researchers working on CoMAC and hierarchical data aggregation will find the model and the special-case checks useful, and future work might fix the entropy issue. I would not cite the rate formula as it stands.

Best.

Referee Report

3 major / 3 minor

Summary. The paper considers computation of a desired function of distributed sources in a wireless network whose topology may be disorganized. The authors propose reorganizing such a network into a layered hierarchy of groups and subgroups, and then introduce multi-layer function computation (ML-FC) in which each subgroup function is computed by CoMAC and each group function is reconstructed by orthogonal communication over the subgroup outputs. The main theoretical claim is a closed-form computation rate for the desired function, expressed as a minimum over layers, groups, and subgroups of time-sharing factors times per-subgroup CoMAC rates (Theorems 1 and 2). The paper then formulates time-allocation and power-control problems and derives closed-form solutions, with simulation results comparing fixed and adaptive power control and optimal versus equal time allocation. The central claim is that this framework generalizes both classical CoMAC and orthogonal communication schemes to multi-hop settings.

Significance. If the rate formula and optimization results were correct, the paper would provide a useful design rule for multi-hop over-the-air computation and would extend CoMAC beyond single-hop relay-free networks. The problem is timely, and the hierarchical decomposition of the network into subgroups and groups is a natural and appealing conceptual step. The manuscript also gives closed-form resource-allocation expressions and simulations over Rayleigh fading channels, which would be of practical interest. However, the derivation of the main rate formula contains a load-bearing error in the conversion from computation rates to numbers of computed values, and the optimization section applies Jensen's inequality to a function that is not concave. These issues invalidate the central claims as stated, so the current contribution is not established.

major comments (3)

[Section IV-B, Eq. (16)] The proof of Theorem 1 mishandles the relation between computation rate and the number of computed function values. Definition 2 states R = lim T_d/(n H(f)), so the number of values obtained in |T| channel uses is T_d = R |T| H(f), not R |T| / H(f) as written for U^(c)_l,k. The displayed equation for U^(c)_l,k is therefore dimensionally inconsistent, and the cancellation leading to the final expression in Eq. (16) is invalid. Independently, the normalization should use the entropy H(f^(c)_l,k) of the subgroup function, not H(f(bv)) of the desired function; these entropies differ in general, since a subgroup function typically depends on only a subset of sources. For example, with two one-source subgroups computing f=(S1,S2), H(S1)=1 bit, H(S2)=10 bits, equal per-subgroup rates R0 and beta=0.5, the paper's Eq. (14) gives 0.5 R0, while the entropy-corrected computation gives a rate bottlenecked by the first subgroup and equal to R0/22. The error changes which subgroup is the bottleneck, so it is not a constant-factor issue. Since Theorem 2 and all of Section V are built on Eq. (16), the claimed general rate and closed-form allocations are unsupported.
[Section V-A, Eq. (21)] The Jensen step in Eq. (21) is invalid because C+(x) = max{0, 0.5 log x} is not concave on the positive reals. A concrete counterexample is X=0.5 with probability 0.5 and X=1.5 with probability 0.5: E[X]=1, so C+(E[X])=0, while E[C+(X)] = 0.25 log(1.5) > 0. Thus the asserted inequality E[C+(1/K + min |h|^2 P)] <= C+(1/K + E[min |h|^2] P) is false. Consequently Problem 1 and Problem 2 do not optimize the actual computation rate, but rather a rate expression obtained through an invalid upper bound. The subsequent closed-form solutions and the simulation comparisons based on those solutions therefore do not establish the claimed performance of the proposed time allocation. The same non-concavity also casts doubt on the convexity argument for Problem 4.
[Section III-A, Remark 3] Remark 3 asserts without proof that every disorganized network can be equivalently represented as a layered hierarchy in which each node has exactly one transmission destination. In a general wireless network a node's transmission may be received by multiple relays, and a node may have multiple intended destinations, but the reorganization in Fig. 3 does not preserve such connectivity. As stated, the result applies only to networks with a single-destination routing structure. The authors should either prove the equivalence under explicit conditions or restrict the claims about 'disorganized networks' accordingly. This is a scope issue for the paper's main title and motivation, and it is not addressed in the system model.

minor comments (3)

[Section IV-A, Algorithm 1] The step in Algorithm 1 that says 'The given channel uses for the group {N1,1,...,N1,5} belongs a set T2,1' should say 'belong to a set T2,1' and should specify whether the group includes all five nodes or only those in the depicted subgroup.
[Eq. (14) and Eq. (17)] The equality between the sample average (1/|T^(c)|) sum over m in T^(c) and an expectation E[.] is only justified as an ergodic limit as the block length grows. It would be clearer to state this limit explicitly rather than writing the equality for a finite set of channel uses.
[Section V-B, Eq. (34)] The denominator E[ min_i |h|^2 / |h|^2 ] uses h as a representative coefficient without defining its distribution or its relation to the channels in the subgroup. The expression would be clearer if written with explicit indices, for example E[ |h_i->k|^2 / |h_j->k|^2 ] under the stated symmetry assumption.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity found: the rate derivation composes an external CoMAC theorem with min-rate chaining; special-case reductions are consistency checks, not assumed inputs.

full rationale

The central claim is not assumed as an input anywhere in the derivation. Lemma 1 obtains the per-subgroup computation rate directly from the external CoMAC result [21, Theorem 3 and Section IV-A], whose authors do not overlap with the present paper. Theorem 1 then composes these external rates by taking minima over subgroup time fractions, and Theorem 2 chains the group rates across layers by repeated minima; this is a rate-composition argument, not a self-referential one. The optimization sections maximize that composed rate, so they are not renaming a fitted parameter as a prediction. The special-case reductions in Remarks 5 and 6, and the simulation checks in Section VI, instantiate the general formula with L=2 and particular subgroup counts and then recover [21] and time-sharing; recovering a known benchmark from a more general formula is a consistency check rather than circularity. Remark 3's equivalence between disorganized and hierarchical networks is a modeling assumption about topology, not a conclusion derived from the rate formula. The cited [22] includes some of the same authors, but it is not load-bearing: the proof of Lemma 1 and the CoMAC-rate steps cite [21], not [22]. One potential mathematical gap is that Eq. (16) normalizes every subgroup-function count by H(f(bv)) instead of the subgroup function's own entropy; this is a correctness or modeling concern, not circularity, because the rate expression is not assumed as an input. Thus the paper shows no significant circularity.

Assumptions & free parameters 0 free parameters · 5 assumptions · 2 invented entities

The paper rests on prior CoMAC rate theorems, a strong topology reorganization assumption, an implicit equal-entropy assumption, and an invalid concavity assumption. The central rate formula inherits the equal-entropy issue, and the fixed-power optimization inherits the Jensen error. There are no data-fitted constants, so the free-parameter list is empty.

assumptions (5)

domain assumption Relay-free CoMAC achievable rate formula (Eq. 3) from [21, Theorem 3] holds for each subgroup.
Lemma 1 directly invokes this result for each subgroup; the multi-layer proof does not rederive it.
ad hoc to paper Any disorganized network can be reorganized into a hierarchical network with groups and subgroups, preserving routing paths (Remark 3).
This equivalence is asserted without proof; it constrains topology to a layered tree with single destinations per node.
ad hoc to paper The desired function can be divided into subgroup functions and reconstructed by a function g_l,k (Eqs. 6 and 7).
For general functions this decomposition is not guaranteed; the paper only gives examples such as arithmetic sum and type functions.
ad hoc to paper The entropy of each subgroup function equals H(f(bv)), the entropy of the desired function.
Used implicitly in Eq. (16) without statement or justification. It is false for typical cases, e.g., a sum over a subset of sources.
ad hoc to paper C+(x)=max(0,0.5 log x) is concave so Jensen's inequality can be applied in Eq. (21).
The function is not concave over its domain; the inequality is invalid for K>1.

invented entities (2)

Hierarchical network reorganization
purpose: Convert arbitrary disorganized topology into layered groups and subgroups to enable CoMAC analysis.
No independent falsifiable handle; it is a modeling assumption whose equivalence to real topologies is asserted, not proven.
Multi-layer function computation (ML-FC)
purpose: Protocol that combines CoMAC and orthogonal communication across layers.
A protocol rather than a physical entity; its performance is claimed but not independently verified.

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Pith. "Pith review of Computing and Communicating Functions in Disorganized Wireless Networks." pith.science (2026). https://pith.science/paper/YCP6UZPB

@misc{pith2026190805030,
  author       = {Pith},
  title        = {Pith review of: Computing and Communicating Functions in Disorganized Wireless Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YCP6UZPB}},
  note         = {Machine review of arXiv:1908.05030}
}

read the original abstract

For future wireless networks, enormous numbers of interconnections are required, creating a disorganized topology and leading to a great challenge in data aggregation. Instead of collecting data individually, a more efficient technique, computation over multi-access channels (CoMAC), has emerged to compute functions by exploiting the signal-superposition property of wireless channels. However, the implementation of CoMAC in disorganized networks with multiple relays (hops) is still an open problem. In this paper, we combine CoMAC and orthogonal communication in the disorganized network to attain the computation of functions at the fusion center. First, to make the disorganized network more tractable, we reorganize the disorganized network into a hierarchical network with multiple layers that consists of subgroups and groups. In the hierarchical network, we propose multi-layer function computation where CoMAC is applied to each subgroup and orthogonal communication is adopted within each group. By computing and communicating subgroup and group functions over layers, the desired functions are reconstructed at the fusion center. The general computation rate is derived and the performance is further improved through time allocation and power control. The closed-form solutions to optimization are obtained, which suggest that existing CoMAC and orthogonal communication schemes can be generalized.

Figures

Figures reproduced from arXiv: 1908.05030 by the authors.

**Figure 2.** The classical CoMAC for the relay-free network [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Reorganization of the disorganized network [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: Procedure of ML-FC a group consisting of several subgroups. The data of Nl,k is denoted as sl,k. We describe the procedure of ML-FC as Algorithm 1. With the help of the example shown in [PITH_FULL_IMAGE:figures/full_fig_p015_4.png]

**Figure 5.** Figure 5: Computation rates of CoMAC with different schemes with respect to the number of source nodes [PITH_FULL_IMAGE:figures/full_fig_p025_5.png]

**Figure 6.** Figure 6: Computation rates of ML-FC with different schemes with respect to the number of subgroups [PITH_FULL_IMAGE:figures/full_fig_p026_6.png]

**Figure 7.** Figure 7: Computation rates of ML-FC with different schemes with respect to the number of subgroups [PITH_FULL_IMAGE:figures/full_fig_p027_7.png]

**Figure 8.** Figure 8: Computation rates of ML-FC with respect to the number of groups [PITH_FULL_IMAGE:figures/full_fig_p028_8.png]

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Reviewed August 14, 2026 · model on record in the stance chip above.

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Computation over Gaussian networks with orthogonal components,

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