Paper Citation Record · LEDGER

Epidemic dynamics in physical-information-social multilayer networks

As of 8 August 2026, this Paper Citation Record lists 47 of 47 outbound references and 0 inbound Pith citation observations for arXiv:2506.00104.

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

pith.paper-citation-record.v1
2506.00104 v1

Coverage vector

measured 47 of 47 reference resolution

Typed states for the displayed outbound observations.

Source: paper_references, paper_reference_links, observed 2026-08-07T12:29:26.681271Z

measured 47 of 47 standing notices

One-hop event checks from named stored sources.

Source: scholarly_work_events, retraction_status_cache, observed 2026-08-08T06:32:00.761636+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

47 of 47 outbound references displayed

verified exact1
verified fuzzy26
unresolved20
parse uncertain0
malformed identifier0
metadata mismatch0

External citation measurements

No source-named external measurement is stored.

Outbound references

Observation f4d67b71-dd30-4d1f-841e-e8080e2dc6d1 · outbound

This paper cites Conversely, the SU state (a node in both S and U) retains the baseline infection probability: βU = β.

Epidemic dynamics in physical-information-social multilayer networks Conversely, the SU state (a node in both S and U) retains the baseline infection probability: βU = β

Reference 1

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:36.626933Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation fa96e5e4-9266-44b0-9f72-c68d2637b877 · outbound

This paper cites We define the observational metrics obs for the infection severity: obs = NEI N , where NEI is the total number of nodes in states E and I.

Epidemic dynamics in physical-information-social multilayer networks We define the observational metrics obs for the infection severity: obs = NEI N , where NEI is the total number of nodes in states E and I

Reference 2

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:36.363507Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 246ff364-91eb-45c6-bba5-c0740a10e0ce · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 3

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:36.199848Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation df6931f5-e61c-4272-9519-56c09fdd39b0 · outbound

This paper cites Near the epidemic threshold, the probability of infection for each individual approaches zero, so we have pSU i ≫ pEU i , pSA i ≫ pEA i + pIA i , g ≈ gl and λM ≈ km g gl.

Epidemic dynamics in physical-information-social multilayer networks Near the epidemic threshold, the probability of infection for each individual approaches zero, so we have pSU i ≫ pEU i , pSA i ≫ pEA i + pIA i , g ≈ gl and λM ≈ km g gl

Reference 4

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:36.043386Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation fb2f9fec-312d-4dac-8107-f7e03ec985e7 · outbound

This paper cites (2), we can derive the continuous microscopic mean-field equations.

Epidemic dynamics in physical-information-social multilayer networks (2), we can derive the continuous microscopic mean-field equations

Reference 5

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:35.861250Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation c8dedf78-8492-4bf0-8b78-23a9350e7dea · outbound

This paper cites Let F =   P j βGaj1pEI j pSU 1 ...P j βGajN pEI j pSU NP j βGAaj1pEI j pSA 1 ...P j βGAajN pEI j pSA N 0.

Epidemic dynamics in physical-information-social multilayer networks Let F =   P j βGaj1pEI j pSU 1 ...P j βGajN pEI j pSU NP j βGAaj1pEI j pSA 1 ...P j βGAajN pEI j pSA N 0

Reference 6

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:35.652241Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation efa63aa9-e9ea-4d6c-8d26-f3fb992048df · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 7

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:35.329291Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 124a752b-6f7d-4cca-9e75-569a5e6ead04 · outbound

This paper cites (B2), we can derive the continuous mesoscopic mean-field equations, also termed the heterogeneous mean-field equations.

Epidemic dynamics in physical-information-social multilayer networks (B2), we can derive the continuous mesoscopic mean-field equations, also termed the heterogeneous mean-field equations

Reference 8

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:35.008339Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 61554e8b-cb5b-447a-8be8-e93b8a74afd1 · outbound

This paper cites Near the epidemic threshold, the probability of infection for each individual approaches zero, so we have pSU k ≫ pEU k , pSA k ≫ pEA k + pIA k , g ≈ gl and λM ≈ km g gl.

Epidemic dynamics in physical-information-social multilayer networks Near the epidemic threshold, the probability of infection for each individual approaches zero, so we have pSU k ≫ pEU k , pSA k ≫ pEA k + pIA k , g ≈ gl and λM ≈ km g gl

Reference 9

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:34.714961Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 34434ad2-00da-445b-a95b-60fe70227e9a · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 10

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:34.424693Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 84ecd184-091d-4fdb-aaf1-f1d8d31811fa · outbound

This paper cites Malik, M.

Epidemic dynamics in physical-information-social multilayer networks Malik, M

Reference 11

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:34.112170Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3b625135-0074-47fb-b4a1-b9d524378f42 · outbound

This paper cites Platto, T.

Epidemic dynamics in physical-information-social multilayer networks Platto, T

Reference 12

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:33.802615Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation dd6aba85-13f9-41b8-b43a-26f4e955c8bf · outbound

This paper cites Dietz and J.

Epidemic dynamics in physical-information-social multilayer networks Dietz and J

Reference 13

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:33.593721Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation e10f8c38-13ed-496b-8204-229422da4648 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 14

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:33.348614Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 033e742b-1bdc-45a6-8817-63cd686c3eda · outbound

This paper cites Albert, H.

Epidemic dynamics in physical-information-social multilayer networks Albert, H

Reference 15

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:33.127971Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation b1df770d-637a-4c94-b987-e222d202904a · outbound

This paper cites Newman, Networks: An Introduction(Oxford University Press, 2010).

Epidemic dynamics in physical-information-social multilayer networks Newman, Networks: An Introduction(Oxford University Press, 2010)

Reference 16

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:32.877541Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation e0e75fcc-9495-4b5c-96a5-bfa965d487e1 · outbound

This paper cites Pastor-Satorras and A.

Epidemic dynamics in physical-information-social multilayer networks Pastor-Satorras and A

Reference 17

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:32.685126Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 02f6f7da-ca85-42a9-b267-08c35fdfee79 · outbound

This paper cites Pastor-Satorras, C.

Epidemic dynamics in physical-information-social multilayer networks Pastor-Satorras, C

Reference 18

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:32.522827Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7c27e90a-ed96-44a1-ad12-4e13a449aa06 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 19

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:32.373545Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 6f81e051-98fb-43ee-9fb1-ee63645800ff · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 20

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:32.233968Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3c8bf239-69de-496e-bb01-92b058ea58ed · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 21

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:32.045843Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 66b653c6-df80-4f1f-a825-4994e14af8a7 · outbound

This paper cites Samanta, S.

Epidemic dynamics in physical-information-social multilayer networks Samanta, S

Reference 22

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:31.811741Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation dffd3157-a181-4b75-bef1-80758a4bbd13 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 23

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:31.631772Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3b51c880-259b-4c1b-8ea5-ddbbf2de2627 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 24

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:31.483635Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7648d8b6-043e-47ce-8720-0a748e054587 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 25

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:31.328167Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 696fde8b-f216-4714-959e-e207a48155ec · outbound

This paper cites Granell, S.

Epidemic dynamics in physical-information-social multilayer networks Granell, S

Reference 26

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:31.116274Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation a7f66152-e3d3-47e6-97ea-9c5140f9c1c5 · outbound

This paper cites Granell, S.

Epidemic dynamics in physical-information-social multilayer networks Granell, S

Reference 27

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:30.939863Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 1d25c934-f577-4fa3-aed4-4db2f5446973 · outbound

This paper cites Huang, J.

Epidemic dynamics in physical-information-social multilayer networks Huang, J

Reference 28

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:30.827868Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 353ae034-2cd5-440a-a7f0-bfd875de1ec4 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 29

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:30.682391Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation cb5378a7-97d8-4716-8824-5fa0c662d4ca · outbound

This paper cites Pan and Z.

Epidemic dynamics in physical-information-social multilayer networks Pan and Z

Reference 30

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:30.364672Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 34fecc27-661c-4524-8c2b-74709254ab5d · outbound

This paper cites Zheng, C.

Epidemic dynamics in physical-information-social multilayer networks Zheng, C

Reference 31

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:30.117285Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 41ec4400-5776-48b9-be7f-556a41ee1aeb · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 32

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:29.928849Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 827ff596-797b-4ac0-904b-cd88011d2230 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 33

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:29.650146Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7c3fd2d9-eda9-4fd5-ba1e-a00dc7e45a55 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 34

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:29.371200Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 7cc48ad0-a822-492b-89bd-b437d4b9e00a · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 35

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:29.181064Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 458a55a7-119c-4548-8dfd-37f3e07cffbd · outbound

This paper cites He and L.

Epidemic dynamics in physical-information-social multilayer networks He and L

Reference 36

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:28.916880Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation e93c7e04-c07d-4013-b0e2-17ee9c955fa5 · outbound

This paper cites Yu and L.

Epidemic dynamics in physical-information-social multilayer networks Yu and L

Reference 37

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:28.755619Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 1e92a069-52aa-42ce-8b35-f0c21f298abe · outbound

This paper cites Yu and L.

Epidemic dynamics in physical-information-social multilayer networks Yu and L

Reference 38

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:28.616571Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 3300ed8e-e123-457c-872a-f76cddecad1b · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 39

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:28.398411Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 2fd4afe4-16d0-4d9c-b553-3cfaa300bbda · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 40

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:28.291608Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation d51a15c5-2085-4d58-b984-ef30dec2ed8c · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 41

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:28.149150Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 59d59175-6993-4625-b00f-4f5b08da1c40 · outbound

This paper cites Van Den Driessche and J.

Epidemic dynamics in physical-information-social multilayer networks Van Den Driessche and J

Reference 42

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:27.942825Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation ef571721-bea4-4ae0-8e47-c7c589ec2e33 · outbound

This paper cites Chang, C.-R.

Epidemic dynamics in physical-information-social multilayer networks Chang, C.-R

Reference 43

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:27.746226Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 0827eac8-8ae2-4f90-98e5-637fd537ea7d · outbound

This paper cites Pastor-Satorras and A.

Epidemic dynamics in physical-information-social multilayer networks Pastor-Satorras and A

Reference 44

Resolution: verified fuzzy; raw_fallback, observed 2026-08-07T12:29:27.486543Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation eac030b7-05d9-4dc8-9b40-1c00116d6aa5 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 45

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:27.345054Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation ddefbc0f-d9df-4a2b-bc9f-bd0519a4f8be · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 46

Resolution: unresolved; raw_fallback, observed 2026-08-07T12:29:27.149353Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Observation 9b1d26d0-11ef-48ea-924a-673e6132b749 · outbound

This paper cites an unresolved cited work.

Epidemic dynamics in physical-information-social multilayer networks Unresolved cited work

Reference 47

Resolution: verified exact; doi, observed 2026-08-07T12:29:26.970954Z

Source-reported events for the cited work

No event found in the named queried sources as of 2026-08-08T06:32:00.761636+00:00.

Pith citing papers

No inbound Pith citation observations are available.