Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach Article Swipe
Minyi Huang
,
Mengjie Zhou
·
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1903.08776
YOU?
·
· 2019
· Open Access
·
· DOI: https://doi.org/10.48550/arxiv.1903.08776
Mean field game theory has been developed largely following two routes. One of them, called the direct approach, starts by solving a large-scale game and next derives a set of limiting equations as the population size tends to infinity. The second route is to apply mean field approximations and formalize a fixed point problem by analyzing the best response of a representative player. This paper addresses the connection and difference of the two approaches in a linear quadratic (LQ) setting. We first introduce an asymptotic solvability notion for the direct approach, which means for all sufficiently large population sizes, the corresponding game has a set of feedback Nash strategies in addition to a mild regularity requirement. We provide a necessary and sufficient condition for asymptotic solvability and show that in this case the solution converges to a mean field limit. This is accomplished by developing a re-scaling method to derive a low dimensional ordinary differential equation (ODE) system, where a non-symmetric Riccati ODE has a central role. We next compare with the fixed point approach which determines a two point boundary value (TPBV) problem, and show that asymptotic solvability implies feasibility of the fixed point approach, but the converse is not true. We further address non-uniqueness in the fixed point approach and examine the long time behavior of the non-symmetric Riccati ODE in the asymptotic solvability problem.
Related Topics
Concepts
Metadata
- Type
- preprint
- Language
- en
- Landing Page
- http://arxiv.org/abs/1903.08776
- https://arxiv.org/pdf/1903.08776
- OA Status
- green
- Cited By
- 3
- References
- 50
- Related Works
- 10
- OpenAlex ID
- https://openalex.org/W2923270835
All OpenAlex metadata
Raw OpenAlex JSON
- OpenAlex ID
-
https://openalex.org/W2923270835Canonical identifier for this work in OpenAlex
- DOI
-
https://doi.org/10.48550/arxiv.1903.08776Digital Object Identifier
- Title
-
Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approachWork title
- Type
-
preprintOpenAlex work type
- Language
-
enPrimary language
- Publication year
-
2019Year of publication
- Publication date
-
2019-03-20Full publication date if available
- Authors
-
Minyi Huang, Mengjie ZhouList of authors in order
- Landing page
-
https://arxiv.org/abs/1903.08776Publisher landing page
- PDF URL
-
https://arxiv.org/pdf/1903.08776Direct link to full text PDF
- Open access
-
YesWhether a free full text is available
- OA status
-
greenOpen access status per OpenAlex
- OA URL
-
https://arxiv.org/pdf/1903.08776Direct OA link when available
- Concepts
-
Mathematics, Ode, Fixed point, Uniqueness, Applied mathematics, Ordinary differential equation, Mathematical analysis, Differential equationTop concepts (fields/topics) attached by OpenAlex
- Cited by
-
3Total citation count in OpenAlex
- Citations by year (recent)
-
2025: 1, 2021: 1, 2019: 1Per-year citation counts (last 5 years)
- References (count)
-
50Number of works referenced by this work
- Related works (count)
-
10Other works algorithmically related by OpenAlex
Full payload
| id | https://openalex.org/W2923270835 |
|---|---|
| doi | https://doi.org/10.48550/arxiv.1903.08776 |
| ids.doi | https://doi.org/10.48550/arxiv.1903.08776 |
| ids.mag | 2923270835 |
| ids.openalex | https://openalex.org/W2923270835 |
| fwci | |
| type | preprint |
| title | Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach |
| biblio.issue | |
| biblio.volume | |
| biblio.last_page | |
| biblio.first_page | |
| 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 | 0.9933000206947327 |
| 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/T11764 |
| topics[1].field.id | https://openalex.org/fields/13 |
| topics[1].field.display_name | Biochemistry, Genetics and Molecular Biology |
| topics[1].score | 0.9829999804496765 |
| topics[1].domain.id | https://openalex.org/domains/1 |
| topics[1].domain.display_name | Life Sciences |
| topics[1].subfield.id | https://openalex.org/subfields/1311 |
| topics[1].subfield.display_name | Genetics |
| topics[1].display_name | Evolution and Genetic Dynamics |
| topics[2].id | https://openalex.org/T11520 |
| topics[2].field.id | https://openalex.org/fields/31 |
| topics[2].field.display_name | Physics and Astronomy |
| topics[2].score | 0.9821000099182129 |
| topics[2].domain.id | https://openalex.org/domains/3 |
| topics[2].domain.display_name | Physical Sciences |
| topics[2].subfield.id | https://openalex.org/subfields/3109 |
| topics[2].subfield.display_name | Statistical and Nonlinear Physics |
| topics[2].display_name | Advanced Thermodynamics and Statistical Mechanics |
| is_xpac | False |
| apc_list | |
| apc_paid | |
| concepts[0].id | https://openalex.org/C33923547 |
| concepts[0].level | 0 |
| concepts[0].score | 0.810347318649292 |
| concepts[0].wikidata | https://www.wikidata.org/wiki/Q395 |
| concepts[0].display_name | Mathematics |
| concepts[1].id | https://openalex.org/C34862557 |
| concepts[1].level | 2 |
| concepts[1].score | 0.6648975610733032 |
| concepts[1].wikidata | https://www.wikidata.org/wiki/Q178985 |
| concepts[1].display_name | Ode |
| concepts[2].id | https://openalex.org/C61445026 |
| concepts[2].level | 2 |
| concepts[2].score | 0.6612837910652161 |
| concepts[2].wikidata | https://www.wikidata.org/wiki/Q217608 |
| concepts[2].display_name | Fixed point |
| concepts[3].id | https://openalex.org/C2777021972 |
| concepts[3].level | 2 |
| concepts[3].score | 0.5850106477737427 |
| concepts[3].wikidata | https://www.wikidata.org/wiki/Q22976830 |
| concepts[3].display_name | Uniqueness |
| concepts[4].id | https://openalex.org/C28826006 |
| concepts[4].level | 1 |
| concepts[4].score | 0.49706724286079407 |
| concepts[4].wikidata | https://www.wikidata.org/wiki/Q33521 |
| concepts[4].display_name | Applied mathematics |
| concepts[5].id | https://openalex.org/C51544822 |
| concepts[5].level | 3 |
| concepts[5].score | 0.494804322719574 |
| concepts[5].wikidata | https://www.wikidata.org/wiki/Q465274 |
| concepts[5].display_name | Ordinary differential equation |
| concepts[6].id | https://openalex.org/C134306372 |
| concepts[6].level | 1 |
| concepts[6].score | 0.29310721158981323 |
| concepts[6].wikidata | https://www.wikidata.org/wiki/Q7754 |
| concepts[6].display_name | Mathematical analysis |
| concepts[7].id | https://openalex.org/C78045399 |
| concepts[7].level | 2 |
| concepts[7].score | 0.2339804470539093 |
| concepts[7].wikidata | https://www.wikidata.org/wiki/Q11214 |
| concepts[7].display_name | Differential equation |
| keywords[0].id | https://openalex.org/keywords/mathematics |
| keywords[0].score | 0.810347318649292 |
| keywords[0].display_name | Mathematics |
| keywords[1].id | https://openalex.org/keywords/ode |
| keywords[1].score | 0.6648975610733032 |
| keywords[1].display_name | Ode |
| keywords[2].id | https://openalex.org/keywords/fixed-point |
| keywords[2].score | 0.6612837910652161 |
| keywords[2].display_name | Fixed point |
| keywords[3].id | https://openalex.org/keywords/uniqueness |
| keywords[3].score | 0.5850106477737427 |
| keywords[3].display_name | Uniqueness |
| keywords[4].id | https://openalex.org/keywords/applied-mathematics |
| keywords[4].score | 0.49706724286079407 |
| keywords[4].display_name | Applied mathematics |
| keywords[5].id | https://openalex.org/keywords/ordinary-differential-equation |
| keywords[5].score | 0.494804322719574 |
| keywords[5].display_name | Ordinary differential equation |
| keywords[6].id | https://openalex.org/keywords/mathematical-analysis |
| keywords[6].score | 0.29310721158981323 |
| keywords[6].display_name | Mathematical analysis |
| keywords[7].id | https://openalex.org/keywords/differential-equation |
| keywords[7].score | 0.2339804470539093 |
| keywords[7].display_name | Differential equation |
| language | en |
| locations[0].id | pmh:oai:arXiv.org:1903.08776 |
| locations[0].is_oa | True |
| locations[0].source.id | https://openalex.org/S4306400194 |
| locations[0].source.issn | |
| locations[0].source.type | repository |
| locations[0].source.is_oa | True |
| locations[0].source.issn_l | |
| locations[0].source.is_core | False |
| locations[0].source.is_in_doaj | False |
| locations[0].source.display_name | arXiv (Cornell University) |
| locations[0].source.host_organization | https://openalex.org/I205783295 |
| locations[0].source.host_organization_name | Cornell University |
| locations[0].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[0].license | |
| locations[0].pdf_url | https://arxiv.org/pdf/1903.08776 |
| locations[0].version | submittedVersion |
| locations[0].raw_type | text |
| locations[0].license_id | |
| locations[0].is_accepted | False |
| locations[0].is_published | False |
| locations[0].raw_source_name | |
| locations[0].landing_page_url | http://arxiv.org/abs/1903.08776 |
| locations[1].id | doi:10.48550/arxiv.1903.08776 |
| locations[1].is_oa | True |
| locations[1].source.id | https://openalex.org/S4306400194 |
| locations[1].source.issn | |
| locations[1].source.type | repository |
| locations[1].source.is_oa | True |
| locations[1].source.issn_l | |
| locations[1].source.is_core | False |
| locations[1].source.is_in_doaj | False |
| locations[1].source.display_name | arXiv (Cornell University) |
| locations[1].source.host_organization | https://openalex.org/I205783295 |
| locations[1].source.host_organization_name | Cornell University |
| locations[1].source.host_organization_lineage | https://openalex.org/I205783295 |
| locations[1].license | |
| locations[1].pdf_url | |
| locations[1].version | |
| locations[1].raw_type | article |
| locations[1].license_id | |
| locations[1].is_accepted | False |
| locations[1].is_published | |
| locations[1].raw_source_name | |
| locations[1].landing_page_url | https://doi.org/10.48550/arxiv.1903.08776 |
| indexed_in | arxiv, datacite |
| authorships[0].author.id | https://openalex.org/A5082825311 |
| authorships[0].author.orcid | https://orcid.org/0000-0003-1145-5281 |
| authorships[0].author.display_name | Minyi Huang |
| authorships[0].countries | CA |
| authorships[0].affiliations[0].institution_ids | https://openalex.org/I67031392 |
| authorships[0].affiliations[0].raw_affiliation_string | Carleton University |
| authorships[0].institutions[0].id | https://openalex.org/I67031392 |
| authorships[0].institutions[0].ror | https://ror.org/02qtvee93 |
| authorships[0].institutions[0].type | education |
| authorships[0].institutions[0].lineage | https://openalex.org/I67031392 |
| authorships[0].institutions[0].country_code | CA |
| authorships[0].institutions[0].display_name | Carleton University |
| authorships[0].author_position | first |
| authorships[0].raw_author_name | Minyi Huang |
| authorships[0].is_corresponding | False |
| authorships[0].raw_affiliation_strings | Carleton University |
| authorships[1].author.id | https://openalex.org/A5103216060 |
| authorships[1].author.orcid | https://orcid.org/0000-0003-2603-2943 |
| authorships[1].author.display_name | Mengjie Zhou |
| authorships[1].countries | CA |
| authorships[1].affiliations[0].institution_ids | https://openalex.org/I67031392 |
| authorships[1].affiliations[0].raw_affiliation_string | Carleton University |
| authorships[1].institutions[0].id | https://openalex.org/I67031392 |
| authorships[1].institutions[0].ror | https://ror.org/02qtvee93 |
| authorships[1].institutions[0].type | education |
| authorships[1].institutions[0].lineage | https://openalex.org/I67031392 |
| authorships[1].institutions[0].country_code | CA |
| authorships[1].institutions[0].display_name | Carleton University |
| authorships[1].author_position | last |
| authorships[1].raw_author_name | Mengjie Zhou |
| authorships[1].is_corresponding | False |
| authorships[1].raw_affiliation_strings | Carleton University |
| has_content.pdf | True |
| has_content.grobid_xml | True |
| is_paratext | False |
| open_access.is_oa | True |
| open_access.oa_url | https://arxiv.org/pdf/1903.08776 |
| open_access.oa_status | green |
| open_access.any_repository_has_fulltext | False |
| created_date | 2019-04-01T00:00:00 |
| display_name | Linear quadratic mean field games: Asymptotic solvability and relation to the fixed point approach |
| has_fulltext | True |
| is_retracted | False |
| updated_date | 2025-11-06T06:51:31.235846 |
| 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 | 0.9933000206947327 |
| 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/W1866998390, https://openalex.org/W3082668976, https://openalex.org/W2517973022, https://openalex.org/W1543121148, https://openalex.org/W1580703421, https://openalex.org/W2996841952, https://openalex.org/W2618518959, https://openalex.org/W2231364979, https://openalex.org/W4295745414, https://openalex.org/W4382202915 |
| cited_by_count | 3 |
| counts_by_year[0].year | 2025 |
| counts_by_year[0].cited_by_count | 1 |
| counts_by_year[1].year | 2021 |
| counts_by_year[1].cited_by_count | 1 |
| counts_by_year[2].year | 2019 |
| counts_by_year[2].cited_by_count | 1 |
| locations_count | 2 |
| best_oa_location.id | pmh:oai:arXiv.org:1903.08776 |
| best_oa_location.is_oa | True |
| best_oa_location.source.id | https://openalex.org/S4306400194 |
| best_oa_location.source.issn | |
| best_oa_location.source.type | repository |
| best_oa_location.source.is_oa | True |
| best_oa_location.source.issn_l | |
| best_oa_location.source.is_core | False |
| best_oa_location.source.is_in_doaj | False |
| best_oa_location.source.display_name | arXiv (Cornell University) |
| best_oa_location.source.host_organization | https://openalex.org/I205783295 |
| best_oa_location.source.host_organization_name | Cornell University |
| best_oa_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| best_oa_location.license | |
| best_oa_location.pdf_url | https://arxiv.org/pdf/1903.08776 |
| best_oa_location.version | submittedVersion |
| best_oa_location.raw_type | text |
| best_oa_location.license_id | |
| best_oa_location.is_accepted | False |
| best_oa_location.is_published | False |
| best_oa_location.raw_source_name | |
| best_oa_location.landing_page_url | http://arxiv.org/abs/1903.08776 |
| primary_location.id | pmh:oai:arXiv.org:1903.08776 |
| primary_location.is_oa | True |
| primary_location.source.id | https://openalex.org/S4306400194 |
| primary_location.source.issn | |
| primary_location.source.type | repository |
| primary_location.source.is_oa | True |
| primary_location.source.issn_l | |
| primary_location.source.is_core | False |
| primary_location.source.is_in_doaj | False |
| primary_location.source.display_name | arXiv (Cornell University) |
| primary_location.source.host_organization | https://openalex.org/I205783295 |
| primary_location.source.host_organization_name | Cornell University |
| primary_location.source.host_organization_lineage | https://openalex.org/I205783295 |
| primary_location.license | |
| primary_location.pdf_url | https://arxiv.org/pdf/1903.08776 |
| primary_location.version | submittedVersion |
| primary_location.raw_type | text |
| primary_location.license_id | |
| primary_location.is_accepted | False |
| primary_location.is_published | False |
| primary_location.raw_source_name | |
| primary_location.landing_page_url | http://arxiv.org/abs/1903.08776 |
| publication_date | 2019-03-20 |
| publication_year | 2019 |
| referenced_works | https://openalex.org/W3102551889, https://openalex.org/W2129951390, https://openalex.org/W1927857046, https://openalex.org/W2149953340, https://openalex.org/W1983321045, https://openalex.org/W847771221, https://openalex.org/W1986431940, https://openalex.org/W2791784110, https://openalex.org/W1981479913, https://openalex.org/W3202580040, https://openalex.org/W2077233700, https://openalex.org/W2136776938, https://openalex.org/W2021590899, https://openalex.org/W1992169727, https://openalex.org/W2011000015, https://openalex.org/W3141151088, https://openalex.org/W2128008927, https://openalex.org/W2606543100, https://openalex.org/W2026814250, https://openalex.org/W1485289509, https://openalex.org/W2898765808, https://openalex.org/W2477284523, https://openalex.org/W2120459890, https://openalex.org/W574171861, https://openalex.org/W1714215715, https://openalex.org/W3102711288, https://openalex.org/W2055058434, https://openalex.org/W2154720456, https://openalex.org/W2038686546, https://openalex.org/W1997001104, https://openalex.org/W2289473542, https://openalex.org/W2111512254, https://openalex.org/W2964226889, https://openalex.org/W2557843815, https://openalex.org/W2036475626, https://openalex.org/W2963980460, https://openalex.org/W2345177722, https://openalex.org/W586490843, https://openalex.org/W1600349401, https://openalex.org/W1983997656, https://openalex.org/W2397671592, https://openalex.org/W2560316583, https://openalex.org/W1578630563, https://openalex.org/W3111908840, https://openalex.org/W2062240654, https://openalex.org/W1990967271, https://openalex.org/W2152872399, https://openalex.org/W2078532197, https://openalex.org/W3102247888, https://openalex.org/W1484739396 |
| referenced_works_count | 50 |
| abstract_inverted_index.a | 21, 27, 50, 60, 75, 103, 112, 118, 136, 145, 150, 159, 164, 177 |
| abstract_inverted_index.We | 80, 116, 167, 202 |
| abstract_inverted_index.an | 83 |
| abstract_inverted_index.as | 32 |
| abstract_inverted_index.by | 19, 54, 143 |
| abstract_inverted_index.in | 74, 109, 129, 206, 222 |
| abstract_inverted_index.is | 42, 141, 199 |
| abstract_inverted_index.of | 12, 29, 59, 70, 105, 191, 217 |
| abstract_inverted_index.to | 37, 43, 111, 135, 148 |
| abstract_inverted_index.ODE | 162, 221 |
| abstract_inverted_index.One | 11 |
| abstract_inverted_index.The | 39 |
| abstract_inverted_index.all | 94 |
| abstract_inverted_index.and | 24, 48, 68, 120, 126, 184, 211 |
| abstract_inverted_index.but | 196 |
| abstract_inverted_index.for | 87, 93, 123 |
| abstract_inverted_index.has | 4, 102, 163 |
| abstract_inverted_index.low | 151 |
| abstract_inverted_index.not | 200 |
| abstract_inverted_index.set | 28, 104 |
| abstract_inverted_index.the | 15, 33, 56, 66, 71, 88, 99, 132, 171, 192, 197, 207, 213, 218, 223 |
| abstract_inverted_index.two | 9, 72, 178 |
| abstract_inverted_index.(LQ) | 78 |
| abstract_inverted_index.Mean | 0 |
| abstract_inverted_index.Nash | 107 |
| abstract_inverted_index.This | 63, 140 |
| abstract_inverted_index.been | 5 |
| abstract_inverted_index.best | 57 |
| abstract_inverted_index.case | 131 |
| abstract_inverted_index.game | 2, 23, 101 |
| abstract_inverted_index.long | 214 |
| abstract_inverted_index.mean | 45, 137 |
| abstract_inverted_index.mild | 113 |
| abstract_inverted_index.next | 25, 168 |
| abstract_inverted_index.show | 127, 185 |
| abstract_inverted_index.size | 35 |
| abstract_inverted_index.that | 128, 186 |
| abstract_inverted_index.this | 130 |
| abstract_inverted_index.time | 215 |
| abstract_inverted_index.with | 170 |
| abstract_inverted_index.(ODE) | 156 |
| abstract_inverted_index.apply | 44 |
| abstract_inverted_index.field | 1, 46, 138 |
| abstract_inverted_index.first | 81 |
| abstract_inverted_index.fixed | 51, 172, 193, 208 |
| abstract_inverted_index.large | 96 |
| abstract_inverted_index.means | 92 |
| abstract_inverted_index.paper | 64 |
| abstract_inverted_index.point | 52, 173, 179, 194, 209 |
| abstract_inverted_index.role. | 166 |
| abstract_inverted_index.route | 41 |
| abstract_inverted_index.tends | 36 |
| abstract_inverted_index.them, | 13 |
| abstract_inverted_index.true. | 201 |
| abstract_inverted_index.value | 181 |
| abstract_inverted_index.where | 158 |
| abstract_inverted_index.which | 91, 175 |
| abstract_inverted_index.(TPBV) | 182 |
| abstract_inverted_index.called | 14 |
| abstract_inverted_index.derive | 149 |
| abstract_inverted_index.direct | 16, 89 |
| abstract_inverted_index.limit. | 139 |
| abstract_inverted_index.linear | 76 |
| abstract_inverted_index.method | 147 |
| abstract_inverted_index.notion | 86 |
| abstract_inverted_index.second | 40 |
| abstract_inverted_index.sizes, | 98 |
| abstract_inverted_index.starts | 18 |
| abstract_inverted_index.theory | 3 |
| abstract_inverted_index.Riccati | 161, 220 |
| abstract_inverted_index.address | 204 |
| abstract_inverted_index.central | 165 |
| abstract_inverted_index.compare | 169 |
| abstract_inverted_index.derives | 26 |
| abstract_inverted_index.examine | 212 |
| abstract_inverted_index.further | 203 |
| abstract_inverted_index.implies | 189 |
| abstract_inverted_index.largely | 7 |
| abstract_inverted_index.player. | 62 |
| abstract_inverted_index.problem | 53 |
| abstract_inverted_index.provide | 117 |
| abstract_inverted_index.routes. | 10 |
| abstract_inverted_index.solving | 20 |
| abstract_inverted_index.system, | 157 |
| abstract_inverted_index.addition | 110 |
| abstract_inverted_index.approach | 174, 210 |
| abstract_inverted_index.behavior | 216 |
| abstract_inverted_index.boundary | 180 |
| abstract_inverted_index.converse | 198 |
| abstract_inverted_index.equation | 155 |
| abstract_inverted_index.feedback | 106 |
| abstract_inverted_index.limiting | 30 |
| abstract_inverted_index.ordinary | 153 |
| abstract_inverted_index.problem, | 183 |
| abstract_inverted_index.problem. | 226 |
| abstract_inverted_index.response | 58 |
| abstract_inverted_index.setting. | 79 |
| abstract_inverted_index.solution | 133 |
| abstract_inverted_index.addresses | 65 |
| abstract_inverted_index.analyzing | 55 |
| abstract_inverted_index.approach, | 17, 90, 195 |
| abstract_inverted_index.condition | 122 |
| abstract_inverted_index.converges | 134 |
| abstract_inverted_index.developed | 6 |
| abstract_inverted_index.equations | 31 |
| abstract_inverted_index.following | 8 |
| abstract_inverted_index.formalize | 49 |
| abstract_inverted_index.infinity. | 38 |
| abstract_inverted_index.introduce | 82 |
| abstract_inverted_index.necessary | 119 |
| abstract_inverted_index.quadratic | 77 |
| abstract_inverted_index.approaches | 73 |
| abstract_inverted_index.asymptotic | 84, 124, 187, 224 |
| abstract_inverted_index.connection | 67 |
| abstract_inverted_index.determines | 176 |
| abstract_inverted_index.developing | 144 |
| abstract_inverted_index.difference | 69 |
| abstract_inverted_index.population | 34, 97 |
| abstract_inverted_index.re-scaling | 146 |
| abstract_inverted_index.regularity | 114 |
| abstract_inverted_index.strategies | 108 |
| abstract_inverted_index.sufficient | 121 |
| abstract_inverted_index.dimensional | 152 |
| abstract_inverted_index.feasibility | 190 |
| abstract_inverted_index.large-scale | 22 |
| abstract_inverted_index.solvability | 85, 125, 188, 225 |
| abstract_inverted_index.accomplished | 142 |
| abstract_inverted_index.differential | 154 |
| abstract_inverted_index.requirement. | 115 |
| abstract_inverted_index.sufficiently | 95 |
| abstract_inverted_index.corresponding | 100 |
| abstract_inverted_index.non-symmetric | 160, 219 |
| abstract_inverted_index.approximations | 47 |
| abstract_inverted_index.non-uniqueness | 205 |
| abstract_inverted_index.representative | 61 |
| cited_by_percentile_year | |
| countries_distinct_count | 1 |
| institutions_distinct_count | 2 |
| sustainable_development_goals[0].id | https://metadata.un.org/sdg/10 |
| sustainable_development_goals[0].score | 0.4699999988079071 |
| sustainable_development_goals[0].display_name | Reduced inequalities |
| citation_normalized_percentile |