On the Unreasonable Effectiveness of Federated Averaging with Heterogeneous Data Article Swipe

PDF

Jianyu Wang , Rudrajit Das , Gauri Joshi , Satyen Kale , Zheng Xu , Tong Zhang ·

YOU? · · 2022 · Open Access · · DOI: https://doi.org/10.48550/arxiv.2206.04723

Existing theory predicts that data heterogeneity will degrade the performance of the Federated Averaging (FedAvg) algorithm in federated learning. However, in practice, the simple FedAvg algorithm converges very well. This paper explains the seemingly unreasonable effectiveness of FedAvg that contradicts the previous theoretical predictions. We find that the key assumption of bounded gradient dissimilarity in previous theoretical analyses is too pessimistic to characterize data heterogeneity in practical applications. For a simple quadratic problem, we demonstrate there exist regimes where large gradient dissimilarity does not have any negative impact on the convergence of FedAvg. Motivated by this observation, we propose a new quantity, average drift at optimum, to measure the effects of data heterogeneity, and explicitly use it to present a new theoretical analysis of FedAvg. We show that the average drift at optimum is nearly zero across many real-world federated training tasks, whereas the gradient dissimilarity can be large. And our new analysis suggests FedAvg can have identical convergence rates in homogeneous and heterogeneous data settings, and hence, leads to better understanding of its empirical success.

Related Topics

Mathematical Analysis

Concepts

Simple (philosophy) Computer science Convergence (economics) Bounded function Measure (data warehouse) Homogeneous Pessimism Quadratic equation Rate of convergence Concept drift Key (lock) Mathematical optimization Data mining Mathematics Economics Data stream mining Computer security Combinatorics Philosophy Geometry Economic growth Epistemology Mathematical analysis

Metadata

Type: preprint
Language: en
Landing Page: http://arxiv.org/abs/2206.04723
PDF: https://arxiv.org/pdf/2206.04723
OA Status: green
Cited By: 19
Related Works: 10
OpenAlex ID: https://openalex.org/W4282813193

All OpenAlex metadata

Raw OpenAlex JSON

OpenAlex ID: https://openalex.org/W4282813193

Canonical identifier for this work in OpenAlex
DOI: https://doi.org/10.48550/arxiv.2206.04723

Digital Object Identifier
Title: On the Unreasonable Effectiveness of Federated Averaging with Heterogeneous Data

Work title
Type: preprint

OpenAlex work type
Language: en

Primary language
Publication year: 2022

Year of publication
Publication date: 2022-06-09

Full publication date if available
Authors: Jianyu Wang, Rudrajit Das, Gauri Joshi, Satyen Kale, Zheng Xu, Tong Zhang

List of authors in order
Landing page: https://arxiv.org/abs/2206.04723

Publisher landing page
PDF URL: https://arxiv.org/pdf/2206.04723

Direct link to full text PDF
Open access: Yes

Whether a free full text is available
OA status: green

Open access status per OpenAlex
OA URL: https://arxiv.org/pdf/2206.04723

Direct OA link when available
Concepts: Simple (philosophy), Computer science, Convergence (economics), Bounded function, Measure (data warehouse), Homogeneous, Pessimism, Quadratic equation, Rate of convergence, Concept drift, Key (lock), Mathematical optimization, Data mining, Mathematics, Economics, Data stream mining, Computer security, Combinatorics, Philosophy, Geometry, Economic growth, Epistemology, Mathematical analysis

Top concepts (fields/topics) attached by OpenAlex
Cited by: 19

Total citation count in OpenAlex
Citations by year (recent): 2025: 1, 2024: 6, 2023: 12

Per-year citation counts (last 5 years)
Related works (count): 10

Other works algorithmically related by OpenAlex

Full payload

id	https://openalex.org/W4282813193
doi	https://doi.org/10.48550/arxiv.2206.04723
ids.doi	https://doi.org/10.48550/arxiv.2206.04723
ids.openalex	https://openalex.org/W4282813193
fwci
type	preprint
title	On the Unreasonable Effectiveness of Federated Averaging with Heterogeneous Data
biblio.issue
biblio.volume
biblio.last_page
biblio.first_page
topics[0].id	https://openalex.org/T10764
topics[0].field.id	https://openalex.org/fields/17
topics[0].field.display_name	Computer Science
topics[0].score	0.9991000294685364
topics[0].domain.id	https://openalex.org/domains/3
topics[0].domain.display_name	Physical Sciences
topics[0].subfield.id	https://openalex.org/subfields/1702
topics[0].subfield.display_name	Artificial Intelligence
topics[0].display_name	Privacy-Preserving Technologies in Data
topics[1].id	https://openalex.org/T11344
topics[1].field.id	https://openalex.org/fields/22
topics[1].field.display_name	Engineering
topics[1].score	0.995199978351593
topics[1].domain.id	https://openalex.org/domains/3
topics[1].domain.display_name	Physical Sciences
topics[1].subfield.id	https://openalex.org/subfields/2215
topics[1].subfield.display_name	Building and Construction
topics[1].display_name	Traffic Prediction and Management Techniques
topics[2].id	https://openalex.org/T11612
topics[2].field.id	https://openalex.org/fields/17
topics[2].field.display_name	Computer Science
topics[2].score	0.9825999736785889
topics[2].domain.id	https://openalex.org/domains/3
topics[2].domain.display_name	Physical Sciences
topics[2].subfield.id	https://openalex.org/subfields/1702
topics[2].subfield.display_name	Artificial Intelligence
topics[2].display_name	Stochastic Gradient Optimization Techniques
is_xpac	False
apc_list
apc_paid
concepts[0].id	https://openalex.org/C2780586882
concepts[0].level	2
concepts[0].score	0.6888573169708252
concepts[0].wikidata	https://www.wikidata.org/wiki/Q7520643
concepts[0].display_name	Simple (philosophy)
concepts[1].id	https://openalex.org/C41008148
concepts[1].level	0
concepts[1].score	0.6708443760871887
concepts[1].wikidata	https://www.wikidata.org/wiki/Q21198
concepts[1].display_name	Computer science
concepts[2].id	https://openalex.org/C2777303404
concepts[2].level	2
concepts[2].score	0.6454726457595825
concepts[2].wikidata	https://www.wikidata.org/wiki/Q759757
concepts[2].display_name	Convergence (economics)
concepts[3].id	https://openalex.org/C34388435
concepts[3].level	2
concepts[3].score	0.6022014021873474
concepts[3].wikidata	https://www.wikidata.org/wiki/Q2267362
concepts[3].display_name	Bounded function
concepts[4].id	https://openalex.org/C2780009758
concepts[4].level	2
concepts[4].score	0.5570225715637207
concepts[4].wikidata	https://www.wikidata.org/wiki/Q6804172
concepts[4].display_name	Measure (data warehouse)
concepts[5].id	https://openalex.org/C66882249
concepts[5].level	2
concepts[5].score	0.5444768667221069
concepts[5].wikidata	https://www.wikidata.org/wiki/Q169336
concepts[5].display_name	Homogeneous
concepts[6].id	https://openalex.org/C9992130
concepts[6].level	2
concepts[6].score	0.5259276032447815
concepts[6].wikidata	https://www.wikidata.org/wiki/Q484954
concepts[6].display_name	Pessimism
concepts[7].id	https://openalex.org/C129844170
concepts[7].level	2
concepts[7].score	0.48079290986061096
concepts[7].wikidata	https://www.wikidata.org/wiki/Q41299
concepts[7].display_name	Quadratic equation
concepts[8].id	https://openalex.org/C57869625
concepts[8].level	3
concepts[8].score	0.4497992694377899
concepts[8].wikidata	https://www.wikidata.org/wiki/Q1783502
concepts[8].display_name	Rate of convergence
concepts[9].id	https://openalex.org/C60777511
concepts[9].level	3
concepts[9].score	0.42315420508384705
concepts[9].wikidata	https://www.wikidata.org/wiki/Q3045002
concepts[9].display_name	Concept drift
concepts[10].id	https://openalex.org/C26517878
concepts[10].level	2
concepts[10].score	0.38268449902534485
concepts[10].wikidata	https://www.wikidata.org/wiki/Q228039
concepts[10].display_name	Key (lock)
concepts[11].id	https://openalex.org/C126255220
concepts[11].level	1
concepts[11].score	0.3300474286079407
concepts[11].wikidata	https://www.wikidata.org/wiki/Q141495
concepts[11].display_name	Mathematical optimization
concepts[12].id	https://openalex.org/C124101348
concepts[12].level	1
concepts[12].score	0.32390618324279785
concepts[12].wikidata	https://www.wikidata.org/wiki/Q172491
concepts[12].display_name	Data mining
concepts[13].id	https://openalex.org/C33923547
concepts[13].level	0
concepts[13].score	0.25244343280792236
concepts[13].wikidata	https://www.wikidata.org/wiki/Q395
concepts[13].display_name	Mathematics
concepts[14].id	https://openalex.org/C162324750
concepts[14].level	0
concepts[14].score	0.07063648104667664
concepts[14].wikidata	https://www.wikidata.org/wiki/Q8134
concepts[14].display_name	Economics
concepts[15].id	https://openalex.org/C89198739
concepts[15].level	2
concepts[15].score	0.06631284952163696
concepts[15].wikidata	https://www.wikidata.org/wiki/Q3079880
concepts[15].display_name	Data stream mining
concepts[16].id	https://openalex.org/C38652104
concepts[16].level	1
concepts[16].score	0.0
concepts[16].wikidata	https://www.wikidata.org/wiki/Q3510521
concepts[16].display_name	Computer security
concepts[17].id	https://openalex.org/C114614502
concepts[17].level	1
concepts[17].score	0.0
concepts[17].wikidata	https://www.wikidata.org/wiki/Q76592
concepts[17].display_name	Combinatorics
concepts[18].id	https://openalex.org/C138885662
concepts[18].level	0
concepts[18].score	0.0
concepts[18].wikidata	https://www.wikidata.org/wiki/Q5891
concepts[18].display_name	Philosophy
concepts[19].id	https://openalex.org/C2524010
concepts[19].level	1
concepts[19].score	0.0
concepts[19].wikidata	https://www.wikidata.org/wiki/Q8087
concepts[19].display_name	Geometry
concepts[20].id	https://openalex.org/C50522688
concepts[20].level	1
concepts[20].score	0.0
concepts[20].wikidata	https://www.wikidata.org/wiki/Q189833
concepts[20].display_name	Economic growth
concepts[21].id	https://openalex.org/C111472728
concepts[21].level	1
concepts[21].score	0.0
concepts[21].wikidata	https://www.wikidata.org/wiki/Q9471
concepts[21].display_name	Epistemology
concepts[22].id	https://openalex.org/C134306372
concepts[22].level	1
concepts[22].score	0.0
concepts[22].wikidata	https://www.wikidata.org/wiki/Q7754
concepts[22].display_name	Mathematical analysis
keywords[0].id	https://openalex.org/keywords/simple
keywords[0].score	0.6888573169708252
keywords[0].display_name	Simple (philosophy)
keywords[1].id	https://openalex.org/keywords/computer-science
keywords[1].score	0.6708443760871887
keywords[1].display_name	Computer science
keywords[2].id	https://openalex.org/keywords/convergence
keywords[2].score	0.6454726457595825
keywords[2].display_name	Convergence (economics)
keywords[3].id	https://openalex.org/keywords/bounded-function
keywords[3].score	0.6022014021873474
keywords[3].display_name	Bounded function
keywords[4].id	https://openalex.org/keywords/measure
keywords[4].score	0.5570225715637207
keywords[4].display_name	Measure (data warehouse)
keywords[5].id	https://openalex.org/keywords/homogeneous
keywords[5].score	0.5444768667221069
keywords[5].display_name	Homogeneous
keywords[6].id	https://openalex.org/keywords/pessimism
keywords[6].score	0.5259276032447815
keywords[6].display_name	Pessimism
keywords[7].id	https://openalex.org/keywords/quadratic-equation
keywords[7].score	0.48079290986061096
keywords[7].display_name	Quadratic equation
keywords[8].id	https://openalex.org/keywords/rate-of-convergence
keywords[8].score	0.4497992694377899
keywords[8].display_name	Rate of convergence
keywords[9].id	https://openalex.org/keywords/concept-drift
keywords[9].score	0.42315420508384705
keywords[9].display_name	Concept drift
keywords[10].id	https://openalex.org/keywords/key
keywords[10].score	0.38268449902534485
keywords[10].display_name	Key (lock)
keywords[11].id	https://openalex.org/keywords/mathematical-optimization
keywords[11].score	0.3300474286079407
keywords[11].display_name	Mathematical optimization
keywords[12].id	https://openalex.org/keywords/data-mining
keywords[12].score	0.32390618324279785
keywords[12].display_name	Data mining
keywords[13].id	https://openalex.org/keywords/mathematics
keywords[13].score	0.25244343280792236
keywords[13].display_name	Mathematics
keywords[14].id	https://openalex.org/keywords/economics
keywords[14].score	0.07063648104667664
keywords[14].display_name	Economics
keywords[15].id	https://openalex.org/keywords/data-stream-mining
keywords[15].score	0.06631284952163696
keywords[15].display_name	Data stream mining
language	en
locations[0].id	pmh:oai:arXiv.org:2206.04723
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/2206.04723
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/2206.04723
locations[1].id	doi:10.48550/arxiv.2206.04723
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	cc-by
locations[1].pdf_url
locations[1].version
locations[1].raw_type	article
locations[1].license_id	https://openalex.org/licenses/cc-by
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.2206.04723
indexed_in	arxiv, datacite
authorships[0].author.id	https://openalex.org/A5100398977
authorships[0].author.orcid	https://orcid.org/0000-0002-7075-9333
authorships[0].author.display_name	Jianyu Wang
authorships[0].author_position	first
authorships[0].raw_author_name	Wang, Jianyu
authorships[0].is_corresponding	False
authorships[1].author.id	https://openalex.org/A5037384125
authorships[1].author.orcid
authorships[1].author.display_name	Rudrajit Das
authorships[1].author_position	middle
authorships[1].raw_author_name	Das, Rudrajit
authorships[1].is_corresponding	False
authorships[2].author.id	https://openalex.org/A5067441201
authorships[2].author.orcid	https://orcid.org/0000-0002-6372-9697
authorships[2].author.display_name	Gauri Joshi
authorships[2].author_position	middle
authorships[2].raw_author_name	Joshi, Gauri
authorships[2].is_corresponding	False
authorships[3].author.id	https://openalex.org/A5114021051
authorships[3].author.orcid
authorships[3].author.display_name	Satyen Kale
authorships[3].author_position	middle
authorships[3].raw_author_name	Kale, Satyen
authorships[3].is_corresponding	False
authorships[4].author.id	https://openalex.org/A5100413782
authorships[4].author.orcid	https://orcid.org/0000-0003-0648-0327
authorships[4].author.display_name	Zheng Xu
authorships[4].author_position	middle
authorships[4].raw_author_name	Xu, Zheng
authorships[4].is_corresponding	False
authorships[5].author.id	https://openalex.org/A5100378750
authorships[5].author.orcid	https://orcid.org/0000-0001-5818-4285
authorships[5].author.display_name	Tong Zhang
authorships[5].author_position	last
authorships[5].raw_author_name	Zhang, Tong
authorships[5].is_corresponding	False
has_content.pdf	False
has_content.grobid_xml	False
is_paratext	False
open_access.is_oa	True
open_access.oa_url	https://arxiv.org/pdf/2206.04723
open_access.oa_status	green
open_access.any_repository_has_fulltext	False
created_date	2025-10-10T00:00:00
display_name	On the Unreasonable Effectiveness of Federated Averaging with Heterogeneous Data
has_fulltext	False
is_retracted	False
updated_date	2025-11-06T06:51:31.235846
primary_topic.id	https://openalex.org/T10764
primary_topic.field.id	https://openalex.org/fields/17
primary_topic.field.display_name	Computer Science
primary_topic.score	0.9991000294685364
primary_topic.domain.id	https://openalex.org/domains/3
primary_topic.domain.display_name	Physical Sciences
primary_topic.subfield.id	https://openalex.org/subfields/1702
primary_topic.subfield.display_name	Artificial Intelligence
primary_topic.display_name	Privacy-Preserving Technologies in Data
related_works	https://openalex.org/W4380987628, https://openalex.org/W2418537576, https://openalex.org/W2557514562, https://openalex.org/W214945085, https://openalex.org/W1987935396, https://openalex.org/W3126025002, https://openalex.org/W2990081132, https://openalex.org/W2354456418, https://openalex.org/W2080168062, https://openalex.org/W4287117439
cited_by_count	19
counts_by_year[0].year	2025
counts_by_year[0].cited_by_count	1
counts_by_year[1].year	2024
counts_by_year[1].cited_by_count	6
counts_by_year[2].year	2023
counts_by_year[2].cited_by_count	12
locations_count	2
best_oa_location.id	pmh:oai:arXiv.org:2206.04723
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/2206.04723
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/2206.04723
primary_location.id	pmh:oai:arXiv.org:2206.04723
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/2206.04723
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/2206.04723
publication_date	2022-06-09
publication_year	2022
referenced_works_count	0
abstract_inverted_index.a	69, 99, 119
abstract_inverted_index.We	44, 125
abstract_inverted_index.at	104, 131
abstract_inverted_index.be	147
abstract_inverted_index.by	94
abstract_inverted_index.in	16, 20, 54, 65, 160
abstract_inverted_index.is	58, 133
abstract_inverted_index.it	116
abstract_inverted_index.of	10, 36, 50, 91, 110, 123, 172
abstract_inverted_index.on	88
abstract_inverted_index.to	61, 106, 117, 169
abstract_inverted_index.we	73, 97
abstract_inverted_index.And	149
abstract_inverted_index.For	68
abstract_inverted_index.and	113, 162, 166
abstract_inverted_index.any	85
abstract_inverted_index.can	146, 155
abstract_inverted_index.its	173
abstract_inverted_index.key	48
abstract_inverted_index.new	100, 120, 151
abstract_inverted_index.not	83
abstract_inverted_index.our	150
abstract_inverted_index.the	8, 11, 22, 32, 40, 47, 89, 108, 128, 143
abstract_inverted_index.too	59
abstract_inverted_index.use	115
abstract_inverted_index.This	29
abstract_inverted_index.data	4, 63, 111, 164
abstract_inverted_index.does	82
abstract_inverted_index.find	45
abstract_inverted_index.have	84, 156
abstract_inverted_index.many	137
abstract_inverted_index.show	126
abstract_inverted_index.that	3, 38, 46, 127
abstract_inverted_index.this	95
abstract_inverted_index.very	27
abstract_inverted_index.will	6
abstract_inverted_index.zero	135
abstract_inverted_index.drift	103, 130
abstract_inverted_index.exist	76
abstract_inverted_index.large	79
abstract_inverted_index.leads	168
abstract_inverted_index.paper	30
abstract_inverted_index.rates	159
abstract_inverted_index.there	75
abstract_inverted_index.well.	28
abstract_inverted_index.where	78
abstract_inverted_index.FedAvg	24, 37, 154
abstract_inverted_index.across	136
abstract_inverted_index.better	170
abstract_inverted_index.hence,	167
abstract_inverted_index.impact	87
abstract_inverted_index.large.	148
abstract_inverted_index.nearly	134
abstract_inverted_index.simple	23, 70
abstract_inverted_index.tasks,	141
abstract_inverted_index.theory	1
abstract_inverted_index.FedAvg.	92, 124
abstract_inverted_index.average	102, 129
abstract_inverted_index.bounded	51
abstract_inverted_index.degrade	7
abstract_inverted_index.effects	109
abstract_inverted_index.measure	107
abstract_inverted_index.optimum	132
abstract_inverted_index.present	118
abstract_inverted_index.propose	98
abstract_inverted_index.regimes	77
abstract_inverted_index.whereas	142
abstract_inverted_index.(FedAvg)	14
abstract_inverted_index.Existing	0
abstract_inverted_index.However,	19
abstract_inverted_index.analyses	57
abstract_inverted_index.analysis	122, 152
abstract_inverted_index.explains	31
abstract_inverted_index.gradient	52, 80, 144
abstract_inverted_index.negative	86
abstract_inverted_index.optimum,	105
abstract_inverted_index.predicts	2
abstract_inverted_index.previous	41, 55
abstract_inverted_index.problem,	72
abstract_inverted_index.success.	175
abstract_inverted_index.suggests	153
abstract_inverted_index.training	140
abstract_inverted_index.Averaging	13
abstract_inverted_index.Federated	12
abstract_inverted_index.Motivated	93
abstract_inverted_index.algorithm	15, 25
abstract_inverted_index.converges	26
abstract_inverted_index.empirical	174
abstract_inverted_index.federated	17, 139
abstract_inverted_index.identical	157
abstract_inverted_index.learning.	18
abstract_inverted_index.practical	66
abstract_inverted_index.practice,	21
abstract_inverted_index.quadratic	71
abstract_inverted_index.quantity,	101
abstract_inverted_index.seemingly	33
abstract_inverted_index.settings,	165
abstract_inverted_index.assumption	49
abstract_inverted_index.explicitly	114
abstract_inverted_index.real-world	138
abstract_inverted_index.contradicts	39
abstract_inverted_index.convergence	90, 158
abstract_inverted_index.demonstrate	74
abstract_inverted_index.homogeneous	161
abstract_inverted_index.performance	9
abstract_inverted_index.pessimistic	60
abstract_inverted_index.theoretical	42, 56, 121
abstract_inverted_index.characterize	62
abstract_inverted_index.observation,	96
abstract_inverted_index.predictions.	43
abstract_inverted_index.unreasonable	34
abstract_inverted_index.applications.	67
abstract_inverted_index.dissimilarity	53, 81, 145
abstract_inverted_index.effectiveness	35
abstract_inverted_index.heterogeneity	5, 64
abstract_inverted_index.heterogeneous	163
abstract_inverted_index.understanding	171
abstract_inverted_index.heterogeneity,	112
cited_by_percentile_year
countries_distinct_count	0
institutions_distinct_count	6
citation_normalized_percentile