Dynamics Analysis and Optimal Control for a Delayed Rumor-Spreading Model Article Swipe
In this work, we analyze a delayed rumor-propagation model. First, we analyze the existence and boundedness of the solution of the model. Then, we give the conditions for the existence of the rumor-endemic equilibrium. Regrading the delay as a bifurcating parameter, we explore the local asymptotic stability and Hopf bifurcation of the rumor-endemic equilibrium. By a Lyapunov functional technique, we examine the global asymptotically stability of the rumor-free and the rumor-endemic equilibria. We provide two control variables in the rumor-spreading model with time delay, and get the optimal solution via the optimal procedures. Finally, we present some numerical simulations to verify our theoretical predictions. They illustrate that the delay is a crucial issue for system, and it can lead to not just Hopf bifurcation but also chaos.
Related Topics
Concepts
Rumor
Hopf bifurcation
Stability (learning theory)
Applied mathematics
Control theory (sociology)
Bifurcation
Mathematics
Mathematical economics
Computer science
Control (management)
Physics
Nonlinear system
Political science
Machine learning
Artificial intelligence
Public relations
Quantum mechanics
Metadata
- Type
- article
- Language
- en
- Landing Page
- https://doi.org/10.3390/math10193455
- https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676
- OA Status
- gold
- Cited By
- 6
- References
- 23
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W4296848642
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W4296848642Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.3390/math10193455Digital Object Identifier
- Title
-
Dynamics Analysis and Optimal Control for a Delayed Rumor-Spreading ModelWork title
- Type
-
articleOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2022Year of publication
- Publication date
-
2022-09-22Full publication date if available
- Authors
-
Chunru Li, Zujun MaList of authors in order
- Landing page
-
https://doi.org/10.3390/math10193455Publisher landing page
- PDF URL
-
https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
goldOpen access status per OpenAlex
- OA URL
-
https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676Direct OA link when available
- Concepts
-
Rumor, Hopf bifurcation, Stability (learning theory), Applied mathematics, Control theory (sociology), Bifurcation, Mathematics, Mathematical economics, Computer science, Control (management), Physics, Nonlinear system, Political science, Machine learning, Artificial intelligence, Public relations, Quantum mechanicsTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
6Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 2, 2024: 1, 2023: 3Per-year citation counts (last 5 years)
- References (count)
-
23Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W4296848642 |
|---|---|
| doi | https://doi.org/10.3390/math10193455 |
| ids.doi | https://doi.org/10.3390/math10193455 |
| ids.openalex | https://openalex.org/W4296848642 |
| fwci | 1.71856013 |
| type | article |
| title | Dynamics Analysis and Optimal Control for a Delayed Rumor-Spreading Model |
| biblio.issue | 19 |
| biblio.volume | 10 |
| biblio.last_page | 3455 |
| biblio.first_page | 3455 |
| topics[0].id | https://openalex.org/T10482 |
| topics[0].field.id | https://openalex.org/fields/27 |
| topics[0].field.display_name | Medicine |
| topics[0].score | 0.9994000196456909 |
| topics[0].domain.id | https://openalex.org/domains/4 |
| topics[0].domain.display_name | Health Sciences |
| topics[0].subfield.id | https://openalex.org/subfields/2739 |
| topics[0].subfield.display_name | Public Health, Environmental and Occupational Health |
| topics[0].display_name | Mathematical and Theoretical Epidemiology and Ecology Models |
| topics[1].id | https://openalex.org/T12592 |
| topics[1].field.id | https://openalex.org/fields/31 |
| topics[1].field.display_name | Physics and Astronomy |
| topics[1].score | 0.9976000189781189 |
| topics[1].domain.id | https://openalex.org/domains/3 |
| topics[1].domain.display_name | Physical Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/3109 |
| topics[1].subfield.display_name | Statistical and Nonlinear Physics |
| topics[1].display_name | Opinion Dynamics and Social Influence |
| topics[2].id | https://openalex.org/T11764 |
| topics[2].field.id | https://openalex.org/fields/13 |
| topics[2].field.display_name | Biochemistry, Genetics and Molecular Biology |
| topics[2].score | 0.9869999885559082 |
| topics[2].domain.id | https://openalex.org/domains/1 |
| topics[2].domain.display_name | Life Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/1311 |
| topics[2].subfield.display_name | Genetics |
| topics[2].display_name | Evolution and Genetic Dynamics |
| is_xpac | False |
| apc_list.value | 1800 |
| apc_list.currency | CHF |
| apc_list.value_usd | 1949 |
| apc_paid.value | 1800 |
| apc_paid.currency | CHF |
| apc_paid.value_usd | 1949 |
| concepts[0].id | https://openalex.org/C2780469804 |
| concepts[0].level | 2 |
| concepts[0].score | 0.9521327018737793 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q878352 |
| concepts[0].display_name | Rumor |
| concepts[1].id | https://openalex.org/C172435161 |
| concepts[1].level | 4 |
| concepts[1].score | 0.7015539407730103 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q531126 |
| concepts[1].display_name | Hopf bifurcation |
| concepts[2].id | https://openalex.org/C112972136 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6295408606529236 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q7595718 |
| concepts[2].display_name | Stability (learning theory) |
| concepts[3].id | https://openalex.org/C28826006 |
| concepts[3].level | 1 |
| concepts[3].score | 0.5122258067131042 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q33521 |
| concepts[3].display_name | Applied mathematics |
| concepts[4].id | https://openalex.org/C47446073 |
| concepts[4].level | 3 |
| concepts[4].score | 0.500232458114624 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q5165890 |
| concepts[4].display_name | Control theory (sociology) |
| concepts[5].id | https://openalex.org/C2781349735 |
| concepts[5].level | 3 |
| concepts[5].score | 0.4700920581817627 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q4904824 |
| concepts[5].display_name | Bifurcation |
| concepts[6].id | https://openalex.org/C33923547 |
| concepts[6].level | 0 |
| concepts[6].score | 0.46709010004997253 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[6].display_name | Mathematics |
| concepts[7].id | https://openalex.org/C144237770 |
| concepts[7].level | 1 |
| concepts[7].score | 0.39309877157211304 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q747534 |
| concepts[7].display_name | Mathematical economics |
| concepts[8].id | https://openalex.org/C41008148 |
| concepts[8].level | 0 |
| concepts[8].score | 0.36441877484321594 |
| concepts[8].wikidata | https://www.wikidata.org/wiki/Q21198 |
| concepts[8].display_name | Computer science |
| concepts[9].id | https://openalex.org/C2775924081 |
| concepts[9].level | 2 |
| concepts[9].score | 0.3299925923347473 |
| concepts[9].wikidata | https://www.wikidata.org/wiki/Q55608371 |
| concepts[9].display_name | Control (management) |
| concepts[10].id | https://openalex.org/C121332964 |
| concepts[10].level | 0 |
| concepts[10].score | 0.17877477407455444 |
| concepts[10].wikidata | https://www.wikidata.org/wiki/Q413 |
| concepts[10].display_name | Physics |
| concepts[11].id | https://openalex.org/C158622935 |
| concepts[11].level | 2 |
| concepts[11].score | 0.07997533679008484 |
| concepts[11].wikidata | https://www.wikidata.org/wiki/Q660848 |
| concepts[11].display_name | Nonlinear system |
| concepts[12].id | https://openalex.org/C17744445 |
| concepts[12].level | 0 |
| concepts[12].score | 0.0 |
| concepts[12].wikidata | https://www.wikidata.org/wiki/Q36442 |
| concepts[12].display_name | Political science |
| concepts[13].id | https://openalex.org/C119857082 |
| concepts[13].level | 1 |
| concepts[13].score | 0.0 |
| concepts[13].wikidata | https://www.wikidata.org/wiki/Q2539 |
| concepts[13].display_name | Machine learning |
| concepts[14].id | https://openalex.org/C154945302 |
| concepts[14].level | 1 |
| concepts[14].score | 0.0 |
| concepts[14].wikidata | https://www.wikidata.org/wiki/Q11660 |
| concepts[14].display_name | Artificial intelligence |
| concepts[15].id | https://openalex.org/C39549134 |
| concepts[15].level | 1 |
| concepts[15].score | 0.0 |
| concepts[15].wikidata | https://www.wikidata.org/wiki/Q133080 |
| concepts[15].display_name | Public relations |
| concepts[16].id | https://openalex.org/C62520636 |
| concepts[16].level | 1 |
| concepts[16].score | 0.0 |
| concepts[16].wikidata | https://www.wikidata.org/wiki/Q944 |
| concepts[16].display_name | Quantum mechanics |
| keywords[0].id | https://openalex.org/keywords/rumor |
| keywords[0].score | 0.9521327018737793 |
| keywords[0].display_name | Rumor |
| keywords[1].id | https://openalex.org/keywords/hopf-bifurcation |
| keywords[1].score | 0.7015539407730103 |
| keywords[1].display_name | Hopf bifurcation |
| keywords[2].id | https://openalex.org/keywords/stability |
| keywords[2].score | 0.6295408606529236 |
| keywords[2].display_name | Stability (learning theory) |
| keywords[3].id | https://openalex.org/keywords/applied-mathematics |
| keywords[3].score | 0.5122258067131042 |
| keywords[3].display_name | Applied mathematics |
| keywords[4].id | https://openalex.org/keywords/control-theory |
| keywords[4].score | 0.500232458114624 |
| keywords[4].display_name | Control theory (sociology) |
| keywords[5].id | https://openalex.org/keywords/bifurcation |
| keywords[5].score | 0.4700920581817627 |
| keywords[5].display_name | Bifurcation |
| keywords[6].id | https://openalex.org/keywords/mathematics |
| keywords[6].score | 0.46709010004997253 |
| keywords[6].display_name | Mathematics |
| keywords[7].id | https://openalex.org/keywords/mathematical-economics |
| keywords[7].score | 0.39309877157211304 |
| keywords[7].display_name | Mathematical economics |
| keywords[8].id | https://openalex.org/keywords/computer-science |
| keywords[8].score | 0.36441877484321594 |
| keywords[8].display_name | Computer science |
| keywords[9].id | https://openalex.org/keywords/control |
| keywords[9].score | 0.3299925923347473 |
| keywords[9].display_name | Control (management) |
| keywords[10].id | https://openalex.org/keywords/physics |
| keywords[10].score | 0.17877477407455444 |
| keywords[10].display_name | Physics |
| keywords[11].id | https://openalex.org/keywords/nonlinear-system |
| keywords[11].score | 0.07997533679008484 |
| keywords[11].display_name | Nonlinear system |
| language | en |
| locations[0].id | doi:10.3390/math10193455 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4210192031 |
| locations[0].source.issn | 2227-7390 |
| locations[0].source.type | journal |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | 2227-7390 |
| locations[0].source.is_core | True |
| locations[0].source.is_in_doaj | True |
| locations[0].source.display_name | Mathematics |
| locations[0].source.host_organization | https://openalex.org/P4310310987 |
| locations[0].source.host_organization_name | Multidisciplinary Digital Publishing Institute |
| locations[0].source.host_organization_lineage | https://openalex.org/P4310310987 |
| locations[0].source.host_organization_lineage_names | Multidisciplinary Digital Publishing Institute |
| locations[0].license | cc-by |
| locations[0].pdf_url | https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676 |
| locations[0].version | publishedVersion |
| locations[0].raw_type | journal-article |
| locations[0].license_id | https://openalex.org/licenses/cc-by |
| locations[0].is_accepted | True |
| locations[0].is_published | True |
| locations[0].raw_source_name | Mathematics |
| locations[0].landing_page_url | https://doi.org/10.3390/math10193455 |
| locations[1].id | pmh:oai:doaj.org/article:cb059762991a48cb946ed0369f2fb5f6 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306401280 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | False |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | DOAJ (DOAJ: Directory of Open Access Journals) |
| locations[1].source.host_organization | |
| locations[1].source.host_organization_name | |
| locations[1].license | cc-by-sa |
| locations[1].pdf_url | |
| locations[1].version | submittedVersion |
| locations[1].raw_type | article |
| locations[1].license_id | https://openalex.org/licenses/cc-by-sa |
| locations[1].is_accepted | False |
| locations[1].is_published | False |
| locations[1].raw_source_name | Mathematics, Vol 10, Iss 19, p 3455 (2022) |
| locations[1].landing_page_url | https://doaj.org/article/cb059762991a48cb946ed0369f2fb5f6 |
| locations[2].id | pmh:oai:mdpi.com:/2227-7390/10/19/3455/ |
| locations[2].is_oa | True |
| locations[2].source.id | https://openalex.org/S4306400947 |
| locations[2].source.issn | |
| locations[2].source.type | repository |
| locations[2].source.is_oa | True |
| locations[2].source.issn_l | |
| locations[2].source.is_core | False |
| locations[2].source.is_in_doaj | False |
| locations[2].source.display_name | MDPI (MDPI AG) |
| locations[2].source.host_organization | https://openalex.org/I4210097602 |
| locations[2].source.host_organization_name | Multidisciplinary Digital Publishing Institute (Switzerland) |
| locations[2].source.host_organization_lineage | https://openalex.org/I4210097602 |
| locations[2].license | cc-by |
| locations[2].pdf_url | |
| locations[2].version | submittedVersion |
| locations[2].raw_type | Text |
| locations[2].license_id | https://openalex.org/licenses/cc-by |
| locations[2].is_accepted | False |
| locations[2].is_published | False |
| locations[2].raw_source_name | Mathematics; Volume 10; Issue 19; Pages: 3455 |
| locations[2].landing_page_url | https://dx.doi.org/10.3390/math10193455 |
| indexed_in | crossref, doaj |
| authorships[0].author.id | https://openalex.org/A5012436609 |
| authorships[0].author.orcid | https://orcid.org/0000-0003-0381-7262 |
| authorships[0].author.display_name | Chunru Li |
| authorships[0].countries | CN |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I21741975 |
| authorships[0].affiliations[0].raw_affiliation_string | Business School, Changshu Institute of Technology, Changshu 215500, China |
| authorships[0].institutions[0].id | https://openalex.org/I21741975 |
| authorships[0].institutions[0].ror | https://ror.org/05g6ben79 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I21741975 |
| authorships[0].institutions[0].country_code | CN |
| authorships[0].institutions[0].display_name | Changshu Institute of Technology |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Chunru Li |
| authorships[0].is_corresponding | True |
| authorships[0].raw_affiliation_strings | Business School, Changshu Institute of Technology, Changshu 215500, China |
| authorships[1].author.id | https://openalex.org/A5103087520 |
| authorships[1].author.orcid | https://orcid.org/0000-0002-7120-9106 |
| authorships[1].author.display_name | Zujun Ma |
| authorships[1].countries | CN |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I4800084 |
| authorships[1].affiliations[0].raw_affiliation_string | School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China |
| authorships[1].institutions[0].id | https://openalex.org/I4800084 |
| authorships[1].institutions[0].ror | https://ror.org/00hn7w693 |
| authorships[1].institutions[0].type | education |
| authorships[1].institutions[0].lineage | https://openalex.org/I4800084 |
| authorships[1].institutions[0].country_code | CN |
| authorships[1].institutions[0].display_name | Southwest Jiaotong University |
| authorships[1].author_position | last |
| authorships[1].raw_author_name | Zujun Ma |
| authorships[1].is_corresponding | False |
| authorships[1].raw_affiliation_strings | School of Economics and Management, Southwest Jiaotong University, Chengdu 610031, China |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676 |
| open_access.oa_status | gold |
| open_access.any_repository_has_fulltext | False |
| created_date | 2025-10-10T00:00:00 |
| display_name | Dynamics Analysis and Optimal Control for a Delayed Rumor-Spreading Model |
| has_fulltext | False |
| is_retracted | False |
| updated_date | 2025-11-06T03:46:38.306776 |
| primary_topic.id | https://openalex.org/T10482 |
| primary_topic.field.id | https://openalex.org/fields/27 |
| primary_topic.field.display_name | Medicine |
| primary_topic.score | 0.9994000196456909 |
| primary_topic.domain.id | https://openalex.org/domains/4 |
| primary_topic.domain.display_name | Health Sciences |
| primary_topic.subfield.id | https://openalex.org/subfields/2739 |
| primary_topic.subfield.display_name | Public Health, Environmental and Occupational Health |
| primary_topic.display_name | Mathematical and Theoretical Epidemiology and Ecology Models |
| related_works | https://openalex.org/W2372853429, https://openalex.org/W2793319716, https://openalex.org/W3110636174, https://openalex.org/W4210503589, https://openalex.org/W2908713064, https://openalex.org/W2810386322, https://openalex.org/W2801267666, https://openalex.org/W2188429085, https://openalex.org/W2886888575, https://openalex.org/W3140274682 |
| cited_by_count | 6 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 2 |
| counts_by_year[1].year | 2024 |
| counts_by_year[1].cited_by_count | 1 |
| counts_by_year[2].year | 2023 |
| counts_by_year[2].cited_by_count | 3 |
| locations_count | 3 |
| best_oa_location.id | doi:10.3390/math10193455 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4210192031 |
| best_oa_location.source.issn | 2227-7390 |
| best_oa_location.source.type | journal |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | 2227-7390 |
| best_oa_location.source.is_core | True |
| best_oa_location.source.is_in_doaj | True |
| best_oa_location.source.display_name | Mathematics |
| best_oa_location.source.host_organization | https://openalex.org/P4310310987 |
| best_oa_location.source.host_organization_name | Multidisciplinary Digital Publishing Institute |
| best_oa_location.source.host_organization_lineage | https://openalex.org/P4310310987 |
| best_oa_location.source.host_organization_lineage_names | Multidisciplinary Digital Publishing Institute |
| best_oa_location.license | cc-by |
| best_oa_location.pdf_url | https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676 |
| best_oa_location.version | publishedVersion |
| best_oa_location.raw_type | journal-article |
| best_oa_location.license_id | https://openalex.org/licenses/cc-by |
| best_oa_location.is_accepted | True |
| best_oa_location.is_published | True |
| best_oa_location.raw_source_name | Mathematics |
| best_oa_location.landing_page_url | https://doi.org/10.3390/math10193455 |
| primary_location.id | doi:10.3390/math10193455 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4210192031 |
| primary_location.source.issn | 2227-7390 |
| primary_location.source.type | journal |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | 2227-7390 |
| primary_location.source.is_core | True |
| primary_location.source.is_in_doaj | True |
| primary_location.source.display_name | Mathematics |
| primary_location.source.host_organization | https://openalex.org/P4310310987 |
| primary_location.source.host_organization_name | Multidisciplinary Digital Publishing Institute |
| primary_location.source.host_organization_lineage | https://openalex.org/P4310310987 |
| primary_location.source.host_organization_lineage_names | Multidisciplinary Digital Publishing Institute |
| primary_location.license | cc-by |
| primary_location.pdf_url | https://www.mdpi.com/2227-7390/10/19/3455/pdf?version=1663847676 |
| primary_location.version | publishedVersion |
| primary_location.raw_type | journal-article |
| primary_location.license_id | https://openalex.org/licenses/cc-by |
| primary_location.is_accepted | True |
| primary_location.is_published | True |
| primary_location.raw_source_name | Mathematics |
| primary_location.landing_page_url | https://doi.org/10.3390/math10193455 |
| publication_date | 2022-09-22 |
| publication_year | 2022 |
| referenced_works | https://openalex.org/W4232523802, https://openalex.org/W1984977694, https://openalex.org/W1976232622, https://openalex.org/W3013846983, https://openalex.org/W2593305494, https://openalex.org/W2904225341, https://openalex.org/W3002392201, https://openalex.org/W2907364371, https://openalex.org/W2997843974, https://openalex.org/W2058904112, https://openalex.org/W2075941199, https://openalex.org/W2134435043, https://openalex.org/W2565418550, https://openalex.org/W3092091480, https://openalex.org/W2031277215, https://openalex.org/W3133518996, https://openalex.org/W2870926212, https://openalex.org/W3125666695, https://openalex.org/W2902771221, https://openalex.org/W2463473857, https://openalex.org/W2278142357, https://openalex.org/W6766169582, https://openalex.org/W2965611836 |
| referenced_works_count | 23 |
| abstract_inverted_index.a | 5, 38, 55, 110 |
| abstract_inverted_index.By | 54 |
| abstract_inverted_index.In | 0 |
| abstract_inverted_index.We | 72 |
| abstract_inverted_index.as | 37 |
| abstract_inverted_index.in | 77 |
| abstract_inverted_index.is | 109 |
| abstract_inverted_index.it | 116 |
| abstract_inverted_index.of | 16, 19, 30, 50, 65 |
| abstract_inverted_index.to | 99, 119 |
| abstract_inverted_index.we | 3, 10, 23, 41, 59, 94 |
| abstract_inverted_index.and | 14, 47, 68, 84, 115 |
| abstract_inverted_index.but | 124 |
| abstract_inverted_index.can | 117 |
| abstract_inverted_index.for | 27, 113 |
| abstract_inverted_index.get | 85 |
| abstract_inverted_index.not | 120 |
| abstract_inverted_index.our | 101 |
| abstract_inverted_index.the | 12, 17, 20, 25, 28, 31, 35, 43, 51, 61, 66, 69, 78, 86, 90, 107 |
| abstract_inverted_index.two | 74 |
| abstract_inverted_index.via | 89 |
| abstract_inverted_index.Hopf | 48, 122 |
| abstract_inverted_index.They | 104 |
| abstract_inverted_index.also | 125 |
| abstract_inverted_index.give | 24 |
| abstract_inverted_index.just | 121 |
| abstract_inverted_index.lead | 118 |
| abstract_inverted_index.some | 96 |
| abstract_inverted_index.that | 106 |
| abstract_inverted_index.this | 1 |
| abstract_inverted_index.time | 82 |
| abstract_inverted_index.with | 81 |
| abstract_inverted_index.Then, | 22 |
| abstract_inverted_index.delay | 36, 108 |
| abstract_inverted_index.issue | 112 |
| abstract_inverted_index.local | 44 |
| abstract_inverted_index.model | 80 |
| abstract_inverted_index.work, | 2 |
| abstract_inverted_index.First, | 9 |
| abstract_inverted_index.chaos. | 126 |
| abstract_inverted_index.delay, | 83 |
| abstract_inverted_index.global | 62 |
| abstract_inverted_index.model. | 8, 21 |
| abstract_inverted_index.verify | 100 |
| abstract_inverted_index.analyze | 4, 11 |
| abstract_inverted_index.control | 75 |
| abstract_inverted_index.crucial | 111 |
| abstract_inverted_index.delayed | 6 |
| abstract_inverted_index.examine | 60 |
| abstract_inverted_index.explore | 42 |
| abstract_inverted_index.optimal | 87, 91 |
| abstract_inverted_index.present | 95 |
| abstract_inverted_index.provide | 73 |
| abstract_inverted_index.system, | 114 |
| abstract_inverted_index.Finally, | 93 |
| abstract_inverted_index.Lyapunov | 56 |
| abstract_inverted_index.solution | 18, 88 |
| abstract_inverted_index.Regrading | 34 |
| abstract_inverted_index.existence | 13, 29 |
| abstract_inverted_index.numerical | 97 |
| abstract_inverted_index.stability | 46, 64 |
| abstract_inverted_index.variables | 76 |
| abstract_inverted_index.asymptotic | 45 |
| abstract_inverted_index.conditions | 26 |
| abstract_inverted_index.functional | 57 |
| abstract_inverted_index.illustrate | 105 |
| abstract_inverted_index.parameter, | 40 |
| abstract_inverted_index.rumor-free | 67 |
| abstract_inverted_index.technique, | 58 |
| abstract_inverted_index.bifurcating | 39 |
| abstract_inverted_index.bifurcation | 49, 123 |
| abstract_inverted_index.boundedness | 15 |
| abstract_inverted_index.equilibria. | 71 |
| abstract_inverted_index.procedures. | 92 |
| abstract_inverted_index.simulations | 98 |
| abstract_inverted_index.theoretical | 102 |
| abstract_inverted_index.equilibrium. | 33, 53 |
| abstract_inverted_index.predictions. | 103 |
| abstract_inverted_index.rumor-endemic | 32, 52, 70 |
| abstract_inverted_index.asymptotically | 63 |
| abstract_inverted_index.rumor-spreading | 79 |
| abstract_inverted_index.rumor-propagation | 7 |
| cited_by_percentile_year.max | 97 |
| cited_by_percentile_year.min | 90 |
| corresponding_author_ids | https://openalex.org/A5012436609 |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| corresponding_institution_ids | https://openalex.org/I21741975 |
| citation_normalized_percentile.value | 0.7929142 |
| citation_normalized_percentile.is_in_top_1_percent | False |
| citation_normalized_percentile.is_in_top_10_percent | False |