Recovering link-weight structure in complex networks with weight-aware random walks

Adilson Vital Jr.; Diego R. Amancio; Filipi N. Silva

REVIEW 3 major objections 6 minor 56 references

Recovering link-weight structure in complex networks with weight-aware random walks

T0 review · 3 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A weight-biased random walk preserves edge-weight relationships in node embeddings far better than unweighted or strength-based walks, with synthetic correlations above 0.9.

desk verdict A useful, well-executed empirical comparison with a predictable headline result, undermined by an unvalidated similarity metric and no error bars; worth citing and refereeing but not a landmark. read the letter →

arxiv 2508.07489 v1 pith:PUGG6RDX submitted 2025-08-10 cs.SI physics.data-an

Adilson Vital Jr. , Filipi N. Silva , Diego R. Amancio This is my paper · ORCID

classification cs.SIphysics.data-an

keywords weightednetworksnodeembeddingsweight-awarerandomwalkedge-weightpreservationcosinesimilaritygraphembeddingnetworkpruninglink-weightrecovery

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 asks whether the weights on a network's edges survive the trip into low-dimensional node embeddings, and which random-walk rule carries them best. Its answer is that a weight-aware random walk — one that moves from node $u$ to neighbor $v$ with probability $w_{uv}/\sum_w w_{uw}$ — preserves edge-weight information far better than a uniform or strength-biased walk, reaching correlations above 0.9 between original weights and embedding similarity on synthetic networks. The result matters because edge weights encode domain meaning — travel cost, contact intensity, similarity — and embeddings that keep those relationships let downstream tasks use weights without re-engineering the graph. The paper also shows the recovery is not universal: real-world networks give lower, more variable correlations, scale-free topologies with strong hubs distort the signal, and pruning low-weight edges helps only up to an intermediate threshold before over-sparsification damages the embeddings.

What carries the argument

The load-bearing object is the weight-aware random walk (WRW): a walk that steps from $u$ to neighbor $v$ with probability $P(u \to v) = w_{uv} / \sum_{w \in N(u)} w_{uw}$, so high-weight edges are traversed more often and therefore appear more frequently in the skip-gram training contexts. The evaluation machinery is equally central: the Pearson correlation between each original edge weight and the cosine similarity of the embeddings of its two endpoints, applied across synthetic models, eleven real networks, and complete graphs pruned by weight percentiles. This metric is what turns walk behavior into a measurable claim about weight preservation.

What would settle it

On a synthetic Erdős–Rényi graph with exponential edge weights and roughly 2,000 nodes, compute WRW embeddings and compare the Pearson correlation between original weights and embedding cosine similarity with Spearman's rank correlation. If the rank correlation is much lower than the reported ~0.9 Pearson value, the apparent preservation is an artifact of the correlation measure rather than genuine recovery of weight order.

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Extended reading notes

Core claim

The paper's discovery is that the random-walk rule determines whether edge weights reappear in node embeddings. Using transition probability $P(u \to v)=w_{uv}/\sum_w w_{uw}$, the cosine similarity of connected nodes' embedding vectors tracks the original edge weight — above 0.9 Pearson correlation on Erdős–Rényi, stochastic-block, and Waxman models — while unweighted walks stay near zero unless topology already encodes weight. Strength-based walks help on degree-correlated scale-free networks. In real graphs the best WRW correlation is 0.446, and shuffled-weight controls show part of the signal is structural. Low-weight-edge pruning first improves correlation, then past roughly the 30–60th

Load-bearing premise

Every conclusion rests on the assumption that the correlation between original edge weights and the similarity of embedding vectors is a fair, topology-independent measure of how well weights are preserved.

Editorial extensions

If this is right

For any downstream task that needs edge weights, the cheapest fix is to replace the walk's transition rule with WRW's $P(u \to v)\propto w_{uv}$; no other part of the embedding pipeline needs to change.
On random and community-structured synthetic graphs, the reported correlations above 0.9 mean embedding cosine similarity can stand in for original edge weights almost linearly.
On scale-free networks, the hub distortion warns that a single weighted-walk recipe will not work; strength-based walks can be a better fallback when degree and weight are tightly coupled.
Pruning experiments give a practical rule of thumb: removing low-weight edges up to about the 30–60th percentile can clean the weight signal, but heavier pruning should be avoided.
Because real-world results are heterogeneous, practitioners should validate the chosen walk on a shuffled-weight null model before trusting that recovered similarity is weight-driven.

Reading between the lines

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

A direct extension the authors do not state: if embedding cosine similarity tracks edge weights, the learned vectors can be used as an unsupervised edge-weight predictor — the dot product of two endpoint vectors estimates the strength of their link without any labeled training.
The consistent rise-plateau-drop curve suggests an automatic denoising heuristic: sweep the pruning percentile, measure weight-correlation on a held-out edge set, and keep the threshold at the start of the plateau; this could become a cheap preprocessing step for weighted graph learning.
Because the evaluation metric is a single global correlation, it cannot distinguish weights recovered in rank order from weights recovered linearly; a rank-based metric or a direct weight-regression test would tell whether WRW preserves exact values or just ordering.
The heterogeneity across real networks invites a topology-adaptive rule — select the walk strategy from measurable graph features such as degree heterogeneity, weight skewness, or modularity rather than assuming WRW is always best; some networks in the paper even benefit from shuffled weights.

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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 version: the paper is a solid, clearly written empirical comparison of three random walk strategies—unweighted, strength-biased, and weight-aware—for how well the resulting node embeddings preserve edge weights. The headline result is not surprising: the walk that actively uses edge weights preserves them best, with correlations above 0.9 in synthetic models. What earns the paper a careful read is the systematic evaluation: four synthetic generators, eleven real networks, a shuffled-weight null model, and a pruning analysis showing an optimal threshold before over-sparsification hurts. The authors are appropriately careful in the conclusion, noting that the weight-aware walk is 'generally the best' but not universal.

The strongest methodological element is the null model: shuffling weights while preserving the distribution lets them separate genuine weight-signal from topology-driven correlation. That is good practice and the right control for a study like this.

Now the soft spots. The main one is the evaluation metric. The paper defines 'weight preservation' as the Pearson correlation between original edge weights and the cosine similarity of connected node-pair embeddings, and never validates that this similarity measure is a faithful proxy. The stress-test point about skip-gram/negative-sampling dot products approximating PMI is a legitimate theoretical worry: cosine normalization could introduce a nonlinear dependence on node strengths and degrees, so a method that encodes weights well in the inner product might score poorly on cosine, and vice versa. This does not sink the paper—the rankings are consistent and the null model helps—but it means the central claim rests on a single unvalidated metric. The authors should at least test an alternative similarity (e.g., inner product or Euclidean distance) and show the rankings are stable.

The second issue is statistical: they generate ten instances per synthetic configuration but report no error bars or significance tests, and the real-network results appear to be single realizations. That makes it hard to know how much of the observed differences are noise. A few confidence intervals would fix this.

Minor but worth mentioning: the methodology section says both skip-gram and CBOW were used, but the experimental section only describes skip-gram; and there is a garbled string of gid tokens in the manuscript between Section III and IV. These are fixable but suggest a hasty final pass.

Bottom line: this is a useful empirical reference for practitioners choosing walk strategies for weighted graphs, and it deserves a serious peer review. I would accept it with the expectation of a revision that validates or at least discusses the metric, adds error bars, and cleans up the text.

Referee Report

3 major / 6 minor

Summary. The paper asks whether, and under what conditions, random-walk-based node embeddings preserve edge-weight information. It compares three transition strategies—unweighted random walk (RW), strength-biased random walk (SRW), and weight-proportional random walk (WRW)—on weighted Erdős–Rényi, stochastic block, Waxman, and Barabási–Albert models, on 11 real-world networks, and on complete graphs subjected to progressive low-weight edge removal. The evaluation metric is the Pearson correlation between original edge weights and the cosine similarity of connected node-pair embeddings produced by skip-gram/negative sampling. The main reported findings are that WRW achieves the highest correlations in most synthetic and real-world settings, with synthetic correlations above 0.9 in some models, real-world correlations peaking at 0.446, and a three-phase threshold response where moderate pruning improves correlation but excessive pruning degrades it. The paper concludes that weight-aware walks are generally the best strategy for preserving edge-weight relationships, while acknowledging that the result is not universal.

Significance. If the central result is accepted, the paper provides a useful systematic benchmark: it is a straightforward, parameter-free comparison of three walk strategies across diverse network structures, with a shuffled-weight null model and a robustness analysis that are both methodologically valuable. The connection between walk design and interpretable weight preservation in embeddings is of practical interest for transportation, finance, and recommendation applications. The paper does not introduce a new model or fit any free parameters, which makes the qualitative comparison easy to reproduce. However, the entire conclusion rests on a single unvalidated similarity metric, and the reported comparisons lack uncertainty quantification; these issues currently limit the strength of the claims.

major comments (3)

[Section IV.D, Eq. (3), Table II] The Pearson correlation between original edge weights and cosine similarity of connected-pair embeddings is the sole evaluation metric, and it is never validated. For skip-gram with negative sampling, the embedding dot product approximates the PMI of co-occurrence; under the WRW transition P(u→v)=w_uv/Σ_w w_uw, this PMI is approximately log(w_uv) − log(s_u) − log(s_v) plus constants, so dot-product similarity confounds edge weight with node strength. Cosine similarity adds a norm normalization that depends on strength and degree. Thus a walk can encode weights well in the dot product while showing modest cosine correlation, and vice versa. This concern is visible in Table II: for Sp High School Diaries and Faculty Hiring US Academia, RW/SRW correlations on shuffled weights are nearly identical to or higher than on original weights (e.g., Faculty Hiring SRW 0.311 shuffled vs 0.245 origina
[Section V.A, Figures 4–7, Table II] The text states that 10 random instances of each synthetic configuration were generated, but no error bars, standard deviations, confidence intervals, or significance tests are reported; the word 'significantly' is used qualitatively. Point estimates alone are insufficient to support the claim that WRW 'consistently' outperforms the alternatives, especially for real-world networks where differences are modest (e.g., Budapest Connectome: WRW 0.129 vs SRW 0.067) and for the null-model comparisons. Please report variability across instances and embedding initializations and, where appropriate, paired tests or bootstrap confidence intervals for the walk-strategy contrasts.
[Section VI and Table II] The concluding claim that 'weight-aware random walks are essential' is stronger than the evidence. In Table II, Faculty Hiring US Academia has its highest observed correlation with SRW on shuffled weights (0.311), exceeding WRW on original weights (0.267); New Zealand Collab has a higher original-weight correlation with SRW (0.023) than WRW (0.003); and three networks have negative WRW correlations. The abstract's 'not a universal solution' qualifier is appropriate, but Section VI should state the recommendation in a more conditional form or specify the network characteristics for which WRW is expected to help.

minor comments (6)

[Section IV, step 3 vs. Section IV.C] The methodology says 'we employ both Skip-Gram and Continuous Bag-of-Words (CBOW) architectures,' but Section IV.C and all reported results describe only skip-gram; no CBOW results appear. Please clarify or provide CBOW results.
[Section IV.B vs. Section V.A.1] The graph-size parameter range is inconsistent: Section IV.B says 'the number of nodes from 16 to 1024,' while Section V.A.1 and Figure 4 say 128 to 4096 nodes.
[After Section III] The manuscript contains a stray block of 'gid' tokens between Section III and Figure 1. This appears to be a corruption artifact and should be removed before publication.
[Table I] The graph-selection criterion 'bidirectional' is unclear; please specify whether all networks are undirected/symmetric and how the adjacency matrix is symmetrized if needed.
[Section V.B] The null model is described only as 'randomly shuffling the weights while preserving the overall distribution.' Please state the exact shuffling procedure (e.g., permutation of weights over edges) and the number of shuffles used.
[Abstract and Section V.A.2] Minor language issues: 'Using edges weights,' 'random walk strategies ... keeps,' and 'a plato' in Section V.A.2 should be corrected.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reports direct empirical correlations with external null-model controls; self-citations are contextual, not load-bearing.

full rationale

This paper does not derive a prediction from an input in a way that reduces to its own assumptions. It reports direct measurements of the Pearson correlation between original edge weights and the cosine similarity of node-pair embeddings produced by three random-walk strategies. The weighted random walk (Eq. 3) is defined using edge weights, so its better weight-correlation is an expected empirical consequence of sampling bias, not a fitted parameter renamed as a prediction. No parameters are fitted to the target correlations, no uniqueness theorem is imported, and no central claim is justified solely by self-citation. The shuffled-weight null model provides an external benchmark against the same topology, and the across-model synthetic experiments provide independent control conditions. Author self-citations (e.g., [17,18,41]) appear only in background and motivation, not as the evidential basis for the measured correlations. The complete-graph threshold experiment uses cosine-similarity-derived weights as ground truth and then evaluates cosine similarity of embeddings; although the metric is aligned, the embeddings come from random walks on the weighted graph only, so the result is not equivalent to its input by construction. A skeptical concern about whether cosine similarity is a faithful, topology-independent measure of weight preservation is a validity question, not a circularity, and does not raise the circularity score.

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

The paper introduces no new fitted parameters or postulated entities. It relies on standard empirical assumptions about embedding quality, the validity of the correlation metric, the shuffled-weight null model, and the representativeness of synthetic graph models. The lack of free parameters is appropriate for a measurement study, but the domain assumptions carry the weight of generalization.

assumptions (4)

domain assumption Skip-gram/Word2Vec training converts random walk co-occurrences into embeddings that capture structural and relational information.
Section IV C relies on this to interpret cosine similarities as meaningful reflections of network relationships.
domain assumption Pearson correlation between cosine similarity of connected node pairs and original edge weights is an appropriate measure of weight preservation.
Section IV D; all experimental conclusions are evaluated through this single metric.
domain assumption The shuffled-weight null model isolates the contribution of actual weights from topology when comparing original versus shuffled correlations.
Section V B uses the null model to attribute correlation to real weight structure, but the shuffling procedure may not preserve all dependencies between weights and topology.
domain assumption The synthetic graph models and their weight assignment functions represent common characteristics of real-world weighted networks.
Section IV A1; generalization from synthetic results to real networks depends on this representativeness.

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Cite this review

Pith. "Pith review of Recovering link-weight structure in complex networks with weight-aware random walks." pith.science (2026). https://pith.science/paper/PUGG6RDX

@misc{pith2026250807489,
  author       = {Pith},
  title        = {Pith review of: Recovering link-weight structure in complex networks with weight-aware random walks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PUGG6RDX}},
  note         = {Machine review of arXiv:2508.07489}
}

read the original abstract

Using edge weights is essential for modeling real-world systems where links possess relevant information, and preserving this information in low-dimensional representations is relevant for classification and prediction tasks. This paper systematically investigates how different random walk strategies - traditional unweighted, strength-based, and fully weight-aware - keeps edge weight information when generating node embeddings. Using network models, real-world graphs, and networks subjected to low-weight edge removal, we measured the correlation between original edge weights and the similarity of node pairs in the embedding space generated by random walk strategies. Our results consistently showed that weight-aware random walks significantly outperform other strategies, achieving correlations above 0.90 in network models. However, performance in real-world networks was more heterogeneous, influenced by factors like topology and weight distribution. Our analysis also revealed that removing weak edges via thresholding can initially improve correlation by reducing noise, but excessive pruning degrades representation quality. Our findings suggest that simply using a weight-aware random walk is generally the best approach for preserving node weight information in embeddings, but it is not a universal solution.

Figures

Figures reproduced from arXiv: 2508.07489 by the authors.

**Figure 2.** FIG. 2. Visualization of the four different graph models with weighted edges, each containing 128 [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗

**Figure 3.** FIG. 3. Representation of the three different biased random walk models. The first model, Random [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

Figures from the paper (5 more)

**Figure 4.** Figure 4: FIG. 4. Pearson correlation between cosine similarity of node embeddings and original edge weights [PITH_FULL_IMAGE:figures/full_fig_p017_4.png]

**Figure 5.** Figure 5: FIG. 5. Pearson correlation between cosine similarity of node embeddings and original edge weights [PITH_FULL_IMAGE:figures/full_fig_p020_5.png]

**Figure 6.** Figure 6: FIG. 6. Pearson correlation between cosine similarity of node embeddings and original edge weights [PITH_FULL_IMAGE:figures/full_fig_p021_6.png]

**Figure 7.** Figure 7: FIG. 7. Pearson correlation between cosine similarity of node embeddings and original edge weights [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]

**Figure 8.** Figure 8: FIG. 8. Pearson correlation between cosine similarity of node embeddings and original edge weights [PITH_FULL_IMAGE:figures/full_fig_p026_8.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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These networks primarily come from three categories: synthetic graph models, real- world networks, and complete graphs

Graph Selection: our study on biased random walks in networks is conducted on a diverse set of networks with varying structures, sources, and weight distributions. These networks primarily come from three categories: synthetic graph models, real- world networks, and complete graphs. For graph models, we opted for those that different represent common char...

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Two of these strategies incorporated edge weights as a bias for transition probabilities, thereby increasing the likelihood of traversing higher-weight connections

Random Walk: we employed three distinct random walk strategies on each graph. Two of these strategies incorporated edge weights as a bias for transition probabilities, thereby increasing the likelihood of traversing higher-weight connections. The third strategy, a pure random walk, disregarded edge weights entirely, functioning as a control to evaluate th...

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Then we apply the same logic behind Node2Vec, in which such sequences are fed into the Word2Vec, traditionally used in language processing

Embedding: by performing several realizations of the random walks, we obtain a collec- tion of sequences of nodes. Then we apply the same logic behind Node2Vec, in which such sequences are fed into the Word2Vec, traditionally used in language processing. To convert thenode sequences into embedding vectors, we employboth Skip-Gramand Continuous Bag-of-Word...

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Each graph varied in size, following powers of 2, ranging from 128 to 4096 nodes, and the average node degree between 2 and 32, using the same logic

Weighted Graph Models We used four well-known graph models, modified to include weighted edges: the Erdős–Rényi model, the stochastic block model (SBM), the Waxman model, and the Barabási–Albert model. Each graph varied in size, following powers of 2, ranging from 128 to 4096 nodes, and the average node degree between 2 and 32, using the same logic. To en...

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We also restricted the selection to networks with fewer than 5,000 nodes

Real Graphs For the real-world networks, we used the Netzschleuder network catalog [34], selecting 11 weighted graphs that are bidirectional – meaning the origin and destination nodes are interchangeable – and unipartite, indicating that all nodes are of the same type. We also restricted the selection to networks with fewer than 5,000 nodes. Additionally,...

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We began by generating a random feature matrix, where each row represents a node and each column corresponds to a feature dimension

Complete Graphs To evaluate how the removal of low-weight edges impacts the ability of random walks to capture weight information, we conducted a controlled experiment using completed graphs – i.e., fully connected networks where every node is linked to every other node. We began by generating a random feature matrix, where each row represents a node and ...

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Graph Size The first parameter analyzed in our study is the graph size. Figure 4 presents the results of fixing the average node degree to 16, and doing 16 walks per node of length 128, while increasing the graph size in powers of two, ranging from 128 to 4096 nodes, with the rows representing the walk types, and the columns the graph models. The first ro...

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[1] [1]

These networks primarily come from three categories: synthetic graph models, real- world networks, and complete graphs

Graph Selection: our study on biased random walks in networks is conducted on a diverse set of networks with varying structures, sources, and weight distributions. These networks primarily come from three categories: synthetic graph models, real- world networks, and complete graphs. For graph models, we opted for those that different represent common char...

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[2] [2]

Two of these strategies incorporated edge weights as a bias for transition probabilities, thereby increasing the likelihood of traversing higher-weight connections

Random Walk: we employed three distinct random walk strategies on each graph. Two of these strategies incorporated edge weights as a bias for transition probabilities, thereby increasing the likelihood of traversing higher-weight connections. The third strategy, a pure random walk, disregarded edge weights entirely, functioning as a control to evaluate th...

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[3] [3]

Then we apply the same logic behind Node2Vec, in which such sequences are fed into the Word2Vec, traditionally used in language processing

Embedding: by performing several realizations of the random walks, we obtain a collec- tion of sequences of nodes. Then we apply the same logic behind Node2Vec, in which such sequences are fed into the Word2Vec, traditionally used in language processing. To convert thenode sequences into embedding vectors, we employboth Skip-Gramand Continuous Bag-of-Word...

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[4] [4]

To assess this, we compute the cosine similarity between all connected node pairs in the em- beddings generated for each combination of graph and random walk technique

Node Similarity: the primary objective of this study is to evaluate whether random walks can effectively encode edge weight information into node embeddings. To assess this, we compute the cosine similarity between all connected node pairs in the em- beddings generated for each combination of graph and random walk technique. We then measure the Pearson co...

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[5] [5]

Each graph varied in size, following powers of 2, ranging from 128 to 4096 nodes, and the average node degree between 2 and 32, using the same logic

Weighted Graph Models We used four well-known graph models, modified to include weighted edges: the Erdős–Rényi model, the stochastic block model (SBM), the Waxman model, and the Barabási–Albert model. Each graph varied in size, following powers of 2, ranging from 128 to 4096 nodes, and the average node degree between 2 and 32, using the same logic. To en...

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[6] [6]

We also restricted the selection to networks with fewer than 5,000 nodes

Real Graphs For the real-world networks, we used the Netzschleuder network catalog [34], selecting 11 weighted graphs that are bidirectional – meaning the origin and destination nodes are interchangeable – and unipartite, indicating that all nodes are of the same type. We also restricted the selection to networks with fewer than 5,000 nodes. Additionally,...

work page 2019

[7] [7]

We began by generating a random feature matrix, where each row represents a node and each column corresponds to a feature dimension

Complete Graphs To evaluate how the removal of low-weight edges impacts the ability of random walks to capture weight information, we conducted a controlled experiment using completed graphs – i.e., fully connected networks where every node is linked to every other node. We began by generating a random feature matrix, where each row represents a node and ...

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[8] [8]

Graph Size The first parameter analyzed in our study is the graph size. Figure 4 presents the results of fixing the average node degree to 16, and doing 16 walks per node of length 128, while increasing the graph size in powers of two, ranging from 128 to 4096 nodes, with the rows representing the walk types, and the columns the graph models. The first ro...

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