Reciprocity Can Erase Direction:tructural Boundaries and Controlled Utility of Magnetic Graph Networks

Dawei Han; Zicheng Wang; Yaoxing Ren; Yi Lv; Rongqian Wu; Fumin Wang

cs.LGNov 3, 2026

Reciprocity Can Erase Direction:tructural Boundaries and Controlled Utility of Magnetic Graph Networks

Authors: Dawei Han, Zicheng Wang, Yaoxing Ren, Yi Lv, Rongqian Wu, Fumin Wang

Organizations: National Local Joint Engineering Research Center for Precision Surgery & Regenerative Medicine, The First Affiliated Hospital of Xi’an Jiaotong University, Xi’an, PR China · School of Future Technology, Xi’an Jiaotong University, Xi’an, PR China · Office of Global promotion, Division of International Cooperation and Exchanges, Xidian University · Arts and Science Department, Cornell University, New York 14850, USA ·

Abstract

Magnetic graph networks encode direction through phase, but a learned graph can erase the intended directional mechanism before prediction. We characterize this failure for support-based magnetic Laplacians. Exact identities relate imaginary energy and distance from zero phase to amplitude-weighted reciprocity. Top- $k$ selection forces complete collapse when the observed-node count satisfies $m\leq k+1$ , and dense support imposes a lower bound on reciprocal-arc fraction. We then construct paired propagation examples with identical node features and symmetric amplitudes but opposite labels, giving a provable chance-level limit for direction-erasing representations. In 72 fixed neural fits, support phase achieved test AUROC 1.0000 on a unilateral ring but 0.5000 on reciprocal ring and complete supports; continuous score phase and a real directed comparator achieved 1.0000 throughout. All models solved the direction-free control. These are oracle-direction mechanism tests, not evidence of learned clinical direction. In the preserved public DeepDRiD development study, continuous phase was active but its robustness difference from zero phase was only $+0.0036$ (descriptive 95% interval $-0.0014$ to $+0.0095$ ) on validation patients used for checkpoint selection. The contribution is a structural and experimental framework for distinguishing erased direction from unestablished task utility. It provides a precise circumstance in which magnetic phase preserves useful information, without asserting universal or clinical superiority.

Figures & tables

Figure 1: Research roadmap. a, A complete four-node support illustrates the structural collapse boundary for support-based phase. b, Direction-reversed synthetic pairs share node features and symmetric amplitudes but have opposite propagation labels;

s

denotes the source and

t

the target. Direction is supplied in edge scores. c, The public DeepDRiD cohort tests application-specific utility. The small interval shows the continuous-minus-zero mean-readout contrast on 100 validation patients used for checkpoint selection:

\Delta R=+0.0036

, descriptive 95% interval

[-0.0014,+0.0095]

. The squares depict the view/region organization, not patient photographs. The arrows summarize the argument, not a transfer of data between experiments.

Figure 2: Analytic structural boundaries. a, The reciprocal-arc fraction is bounded below by

\max\{0,2-(m-1)/k_{\mathrm{eff}}\}

, with

k_{\mathrm{eff}}=\min(k,m-1)

. Complete reciprocal support occurs on and above the orange boundary, where support phase collapses. The pale region indicates a zero lower bound, not guaranteed asymmetry. b, Exact normalized operator distance for four fixed charges. Its energy-weighted reciprocity

r_{E}

differs from the count-based quantity in panel a. These are analytic values, not fitted experimental measurements.

$k$ / task	Zero phase	Support phase	Continuous	Real directed
1	0.5000	1.0000	1.0000	1.0000
2	0.5000	0.5000	1.0000	1.0000
7	0.5000	0.5000	1.0000	1.0000
1 (feature)	1.0000	1.0000	1.0000	1.0000
2 (feature)	1.0000	1.0000	1.0000	1.0000
7 (feature)	1.0000	1.0000	1.0000	1.0000

Table 1: Synthetic test AUROC, averaged across all three initialization seeds. Top rows: direction-dependent propagation. Bottom rows: direction-free feature control. Test graphs are held out from training and checkpoint selection. All seed-level results and accuracies appear in the appendix.

Figure 3: Controlled directional utility. a, An illustrative source–target pair on an eight-node cycle. Reversal changes whether the source pulse reaches the adjacent target in one step, while preserving node features and symmetric amplitudes. b,c, Complete test AUROC matrices for the propagation task and direction-free feature control. Each cell is the mean of three initialization seeds on 128 held-out pairs; all 72 fits are retained. Columns correspond to unilateral (

k=1

), reciprocal-ring (

k=2

), and complete (

k=7

) support. Color encodes AUROC, not model identity. Values are rounded to four decimals; unrounded per-seed values, including the small float32 departures from 0.5, are retained in the source data and appendix. Directional scores are supplied inputs.

Figure 4: How continuous phase preserves reciprocal score information. a, An illustrative pair has

s_{ij}=2

and

s_{ji}=1

with reciprocal binary support. At

q=0.25

and

\tau=1

, support phase is zero whereas score-based phase is

\theta_{ij}=1.1963

radians; the reverse entry has the opposite phase. b, Measured operator distance on the paired benchmark, normalized by

\|A\|_{F}

. Reciprocal support erases support phase at

k=2

and 7, while continuous phase retains the asymmetry. Numeric labels are rounded; plotted values use the full recorded precision. Nonzero distance is a structural property, and the separate propagation experiment establishes utility on its designed task.

	$R$		Full-view AUROC
Model	Mean	Relational	Mean	Relational
Support phase	0.9351	0.9367	0.9504	0.9548
Continuous phase	0.9403	0.9399	0.9568	0.9540
Zero phase	0.9367	0.9367	0.9552	0.9508
Shuffled phase	0.9387	0.9356	0.9512	0.9532
Real directed	0.9356	0.9342	0.9540	0.9488

Table 2: Preserved DeepDRiD development results on 100 validation patients used for checkpoint selection. Each estimate uses a three-seed probability ensemble. These are not independent test results.

Figure 5: Preserved DeepDRiD development contrasts. a,b, Continuous-phase minus comparator differences under mean and relational readouts, respectively. Colors and marker shapes identify the comparator consistently; both panels use the same horizontal scale. Points are three-seed ensemble estimates, with descriptive 95% paired patient-bootstrap intervals; the dashed line is zero difference. All eight contrasts are shown. Checkpoint selection and evaluation used the same 100 validation patients, and intervals are unadjusted for selection or multiplicity. These selected development results do not establish independent clinical superiority.

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

Task	$k$	Model	A21	A22	A23	C21	C22	C23
Direction	1	Zero phase	0.50000	0.50000	0.50000	0.5000	0.5000	0.5000
Direction	1	Support phase	1.00000	1.00000	1.00000	1.0000	1.0000	1.0000
Direction	1	Continuous phase	1.00000	1.00000	1.00000	1.0000	1.0000	1.0000
Direction	1	Real directed	1.00000	1.00000	1.00000	1.0000	1.0000	1.0000
Direction	2	Zero phase	0.50000	0.50000	0.50000	0.5000	0.5000	0.5000
Direction	2	Support phase	0.50000	0.50000	0.50000	0.5000	0.5000	0.5000

Appendix

Table 3: Complete synthetic test results. A21–A23 are AUROC and C21–C23 are accuracy for seeds 20260921–20260923. AUROC is shown to five decimals to expose the small finite-precision deviation described in the main text.

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Spectral positional encodings (PEs) for directed graphs face two obstacles: magnetic Laplacians require an

O(n^3)

Hermitian eigendecomposition per potential, and their complex eigenvectors are defined only up to unitary gauge, which prior work handles with basis-invariant architectures. We propose learnable spectral PEs of the form

h_θ(A_q)\,R

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A_q

is a normalized magnetic operator,

h_θ

a learnable scalar spectral response, and

R

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k = O(\log(1/\varepsilon))

block steps suffice uniformly over heat--resolvent response families, and give a covering-number argument for why low-dimensional structured families generalize where free per-eigenvalue weights overfit. On a directed SBM whose symmetrization is uninformative by construction, direction-blind PEs stay at chance while magnetic Krylov PEs converge to the exact-eigendecomposition oracle as the depth grows. The same probes yield gauge-invariant pairwise features with

1/\sqrt{s}

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q{=}0

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Jiaqing Xie, Yuxin Wang

Fudan University

Positional Encoding Spectral Methods Invariant Representation Learning Krylov Subspace Methods Graph Positional Encoding Spectral Graph Embedding

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λ

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λ>0

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H^1

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0.2\%

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Mohammad Forouhesh

Amirkabir University of Technology, Iran

Oversmoothing Inverse Problems Graph Regularization

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