Paper Citation Record · LEDGER

Positional Encoding meets Persistent Homology on Graphs

As of 21 August 2026, this Paper Citation Record lists 26 of 26 outbound references and 0 inbound Pith citation observations for arXiv:2506.05814.

A citation records a reference. It does not transfer a finding from one paper to another.

pith.paper-citation-record.v1
2506.05814 v1

Coverage vector

measured 26 of 26 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T10:29:43.515050Z

measured 26 of 26 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-21T06:32:19.484+00:00

measured 0 of 0 inbound itemization

Pith citing papers itemized under the disclosed page cap.

Source: paper_references, paper_reference_links

measured 0 of 1 external citation measurements

A source-named dated measurement, never combined with another source.

Source: cited_works

Reference resolution

26 of 26 outbound references displayed

verified exact3
verified fuzzy7
unresolved16
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation e9776a0b-5215-4ede-b6df-cf7fe6d0d67e · outbound

This paper cites Since, Q2 and Q3 are distance regular graphs (Brouwer et al., 2012), both G and G′ have the same DE-1 i.e.

Positional Encoding meets Persistent Homology on Graphs Since, Q2 and Q3 are distance regular graphs (Brouwer et al., 2012), both G and G′ have the same DE-1 i.e

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.838451Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 6ac7648c-31a2-40f5-bd76-69d2514e08a8 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.860692Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 180c26a2-123d-421a-8e08-0f3fae107c37 · outbound

This paper cites A Generalization of Transformer Networks to Graphs.

Positional Encoding meets Persistent Homology on Graphs A Generalization of Transformer Networks to Graphs

Reference 4

Resolution: unresolved; no resolver link, observed 2026-08-07T10:29:42.561242Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation fcefc3fe-f3dd-4fd7-9d4c-231968dc41b3 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 6

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.990775Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation fee927ea-4eae-4919-af4a-7659ee26b5fb · outbound

This paper cites Generalized Laplacian Positional Encoding for Graph Representation Learning.

Positional Encoding meets Persistent Homology on Graphs Generalized Laplacian Positional Encoding for Graph Representation Learning

Reference 7

Resolution: unresolved; no resolver link, observed 2026-08-07T10:29:42.792619Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation c0238e04-9da7-4fad-aae2-d14c9113adba · outbound

This paper cites What Do GNNs Actually Learn? Towards Understanding their Representations.

Positional Encoding meets Persistent Homology on Graphs What Do GNNs Actually Learn? Towards Understanding their Representations

Reference 8

Resolution: unresolved; no resolver link, observed 2026-08-07T10:29:42.845714Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 131737db-a6fc-4d3e-a21d-f60b7caab38f · outbound

This paper cites Related works Graph positional encodings.Positional encodings enhance representations in Graph Neural Networks (GNNs) (Gilmer et al., 2017; Xu et al., 2019; Velickovic et al.,.

Positional Encoding meets Persistent Homology on Graphs Related works Graph positional encodings.Positional encodings enhance representations in Graph Neural Networks (GNNs) (Gilmer et al., 2017; Xu et al., 2019; Velickovic et al.,

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.977906Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 4b954fd0-8da2-4e0c-819e-4aa6a83edfdd · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 12

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.965233Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 771bb2c1-1b82-46c4-862a-ff9233598386 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 14

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.940394Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 96278dea-5904-4424-b1a9-6617b7fe209f · outbound

This paper cites For i >0, the algorithm proceeds as, C k i+1(v) = RELABEL (C k i (v), M(v)) (6) where the multi-set M(v) ={ {C k i (ϕ1(v, w)),.

Positional Encoding meets Persistent Homology on Graphs For i >0, the algorithm proceeds as, C k i+1(v) = RELABEL (C k i (v), M(v)) (6) where the multi-set M(v) ={ {C k i (ϕ1(v, w)),

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.915338Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation ce8879a9-24ca-4714-a0db-f7cde0d18245 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 17

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.902174Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation ac0091f8-7f33-41a1-8ca0-7f4ba3023ffa · outbound

This paper cites Similarly, G′ has the same eigenvalues with identical constant eigenvectors.

Positional Encoding meets Persistent Homology on Graphs Similarly, G′ has the same eigenvalues with identical constant eigenvectors

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.889733Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation d36d1745-4d9d-4c04-a94f-264f322df4c9 · outbound

This paper cites The positional encodings for both the graphs are same i.e., Φk(G) = Φ k(G′) as shown in Equation (9).

Positional Encoding meets Persistent Homology on Graphs The positional encodings for both the graphs are same i.e., Φk(G) = Φ k(G′) as shown in Equation (9)

Reference 19

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.877605Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 751783d3-99a6-4e7b-be60-7e182182eca1 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 22

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.820557Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation b7af9daa-9910-462e-8228-02ff551793e9 · outbound

This paper cites To compute the Persistence Homology (PH) diagrams, we employed the learnable PH method proposed by Immonen et al.

Positional Encoding meets Persistent Homology on Graphs To compute the Persistence Homology (PH) diagrams, we employed the learnable PH method proposed by Immonen et al

Reference 23

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.793947Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation a3cddd8a-5613-48fb-8db0-c7295f96ab9e · outbound

This paper cites Out of distribution Prediction We adhered to the precise hyperparameters and training configuration outlined in Huang et al.

Positional Encoding meets Persistent Homology on Graphs Out of distribution Prediction We adhered to the precise hyperparameters and training configuration outlined in Huang et al

Reference 24

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T10:29:43.771856Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 932fb7e6-ca70-4c6d-a7f3-dc19a3d67893 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.743188Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation a5577d65-5e8c-43f4-b335-e1992869a239 · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 26

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.715557Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 2210a41a-d86d-46c0-b212-b38626e6d388 · outbound

This paper cites Persistent Homology and Graphs Representation Learning.

Positional Encoding meets Persistent Homology on Graphs Persistent Homology and Graphs Representation Learning

Reference 2017

Resolution: verified exact; local_arxiv, observed 2026-08-07T10:29:43.633607Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 08669a2a-0bb8-42f5-9f4a-d124370b883c · outbound

This paper cites Jet characterization in Heavy Ion Collisions by QCD-Aware Graph Neural Networks.

Positional Encoding meets Persistent Homology on Graphs Jet characterization in Heavy Ion Collisions by QCD-Aware Graph Neural Networks

Reference 2018

Resolution: verified exact; local_arxiv, observed 2026-08-07T10:29:43.580438Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation ed4ae075-dcb0-440e-bbcf-0df5f3e92cf7 · outbound

This paper cites Learning Symbolic Physics with Graph Networks.

Positional Encoding meets Persistent Homology on Graphs Learning Symbolic Physics with Graph Networks

Reference 2019

Resolution: unresolved; no resolver link, observed 2026-08-07T10:29:42.512499Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f3141a02-757b-440b-9a9c-b4adb0b70a97 · outbound

This paper cites Algebraic Topological Networks via the Persistent Local Homology Sheaf.

Positional Encoding meets Persistent Homology on Graphs Algebraic Topological Networks via the Persistent Local Homology Sheaf

Reference 2020

Resolution: verified exact; local_arxiv, observed 2026-08-07T10:29:43.690342Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 19f89397-d572-43b6-b376-d4e15dacafcc · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 2021

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.952311Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Observation 50b4feb1-ca10-4903-86db-730e266cb462 · outbound

This paper cites Alchemy: A Quantum Chemistry Dataset for Benchmarking AI Models.

Positional Encoding meets Persistent Homology on Graphs Alchemy: A Quantum Chemistry Dataset for Benchmarking AI Models

Reference 2023

Resolution: unresolved; no resolver link, observed 2026-08-07T10:29:42.467925Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation 92186848-f92c-4ef5-845e-1fca7c6aefdf · outbound

This paper cites An Empirical Study of Realized GNN Expressiveness.

Positional Encoding meets Persistent Homology on Graphs An Empirical Study of Realized GNN Expressiveness

Reference 2024

Resolution: unresolved; no resolver link, observed 2026-08-07T10:29:42.990099Z

Source-reported events for the cited work

Unavailable: canonical work link unavailable.

Observation f679c5c9-c475-4e14-ab10-e9592b88265c · outbound

This paper cites an unresolved cited work.

Positional Encoding meets Persistent Homology on Graphs Unresolved cited work

Reference 2025

Resolution: unresolved; raw_fallback, observed 2026-08-07T10:29:43.928961Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-21T06:32:19.484+00:00.

Pith citing papers

No inbound Pith citation observations are available.