---
_id: '12767'
abstract:
- lang: eng
text: "Several problems in planning and reactive synthesis can be reduced to the
analysis of two-player quantitative graph games. Optimization is one form of analysis.
We argue that in many cases it may be better to replace the optimization problem
with the satisficing problem, where instead of searching for optimal solutions,
the goal is to search for solutions that adhere to a given threshold bound.\r\nThis
work defines and investigates the satisficing problem on a two-player graph game
with the discounted-sum cost model. We show that while the satisficing problem
can be solved using numerical methods just like the optimization problem, this
approach does not render compelling benefits over optimization. When the discount
factor is, however, an integer, we present another approach to satisficing, which
is purely based on automata methods. We show that this approach is algorithmically
more performant – both theoretically and empirically – and demonstrates the broader
applicability of satisficing over optimization."
acknowledgement: We thank anonymous reviewers for valuable inputs. This work is supported
in part by NSF grant 2030859 to the CRA for the CIFellows Project, NSF grants IIS-1527668,
CCF-1704883, IIS-1830549, the ERC CoG 863818 (ForM-SMArt), and an award from the
Maryland Procurement Office.
alternative_title:
- LNCS
article_processing_charge: No
author:
- first_name: Suguman
full_name: Bansal, Suguman
last_name: Bansal
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Moshe Y.
full_name: Vardi, Moshe Y.
last_name: Vardi
citation:
ama: 'Bansal S, Chatterjee K, Vardi MY. On satisficing in quantitative games. In:
27th International Conference on Tools and Algorithms for the Construction
and Analysis of Systems. Vol 12651. Springer Nature; 2021:20-37. doi:10.1007/978-3-030-72016-2'
apa: 'Bansal, S., Chatterjee, K., & Vardi, M. Y. (2021). On satisficing in quantitative
games. In 27th International Conference on Tools and Algorithms for the Construction
and Analysis of Systems (Vol. 12651, pp. 20–37). Luxembourg City, Luxembourg:
Springer Nature. https://doi.org/10.1007/978-3-030-72016-2'
chicago: Bansal, Suguman, Krishnendu Chatterjee, and Moshe Y. Vardi. “On Satisficing
in Quantitative Games.” In 27th International Conference on Tools and Algorithms
for the Construction and Analysis of Systems, 12651:20–37. Springer Nature,
2021. https://doi.org/10.1007/978-3-030-72016-2.
ieee: S. Bansal, K. Chatterjee, and M. Y. Vardi, “On satisficing in quantitative
games,” in 27th International Conference on Tools and Algorithms for the Construction
and Analysis of Systems, Luxembourg City, Luxembourg, 2021, vol. 12651, pp.
20–37.
ista: 'Bansal S, Chatterjee K, Vardi MY. 2021. On satisficing in quantitative games.
27th International Conference on Tools and Algorithms for the Construction and
Analysis of Systems. TACAS: Tools and Algorithms for the Construction and Analysis
of Systems, LNCS, vol. 12651, 20–37.'
mla: Bansal, Suguman, et al. “On Satisficing in Quantitative Games.” 27th International
Conference on Tools and Algorithms for the Construction and Analysis of Systems,
vol. 12651, Springer Nature, 2021, pp. 20–37, doi:10.1007/978-3-030-72016-2.
short: S. Bansal, K. Chatterjee, M.Y. Vardi, in:, 27th International Conference
on Tools and Algorithms for the Construction and Analysis of Systems, Springer
Nature, 2021, pp. 20–37.
conference:
end_date: 2021-04-01
location: Luxembourg City, Luxembourg
name: 'TACAS: Tools and Algorithms for the Construction and Analysis of Systems'
start_date: 2021-03-27
date_created: 2023-03-26T22:01:09Z
date_published: 2021-03-21T00:00:00Z
date_updated: 2023-03-28T11:03:11Z
day: '21'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.1007/978-3-030-72016-2
ec_funded: 1
external_id:
arxiv:
- '2101.02594'
file:
- access_level: open_access
checksum: b020b78b23587ce7610b1aafb4e63438
content_type: application/pdf
creator: dernst
date_created: 2023-03-28T11:00:33Z
date_updated: 2023-03-28T11:00:33Z
file_id: '12777'
file_name: 2021_LNCS_Bansal.pdf
file_size: 747418
relation: main_file
success: 1
file_date_updated: 2023-03-28T11:00:33Z
has_accepted_license: '1'
intvolume: ' 12651'
language:
- iso: eng
month: '03'
oa: 1
oa_version: Published Version
page: 20-37
project:
- _id: 0599E47C-7A3F-11EA-A408-12923DDC885E
call_identifier: H2020
grant_number: '863818'
name: 'Formal Methods for Stochastic Models: Algorithms and Applications'
publication: 27th International Conference on Tools and Algorithms for the Construction
and Analysis of Systems
publication_identifier:
eissn:
- 1611-3349
isbn:
- '9783030720155'
issn:
- 0302-9743
publication_status: published
publisher: Springer Nature
quality_controlled: '1'
scopus_import: '1'
status: public
title: On satisficing in quantitative games
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: conference
user_id: 2DF688A6-F248-11E8-B48F-1D18A9856A87
volume: 12651
year: '2021'
...
---
_id: '10667'
abstract:
- lang: eng
text: Bayesian neural networks (BNNs) place distributions over the weights of a
neural network to model uncertainty in the data and the network's prediction.
We consider the problem of verifying safety when running a Bayesian neural network
policy in a feedback loop with infinite time horizon systems. Compared to the
existing sampling-based approaches, which are inapplicable to the infinite time
horizon setting, we train a separate deterministic neural network that serves
as an infinite time horizon safety certificate. In particular, we show that the
certificate network guarantees the safety of the system over a subset of the BNN
weight posterior's support. Our method first computes a safe weight set and then
alters the BNN's weight posterior to reject samples outside this set. Moreover,
we show how to extend our approach to a safe-exploration reinforcement learning
setting, in order to avoid unsafe trajectories during the training of the policy.
We evaluate our approach on a series of reinforcement learning benchmarks, including
non-Lyapunovian safety specifications.
acknowledgement: This research was supported in part by the Austrian Science Fund
(FWF) under grant Z211-N23 (Wittgenstein Award), ERC CoG 863818 (FoRM-SMArt), and
the European Union’s Horizon 2020 research and innovation programme under the Marie
Skłodowska-Curie Grant Agreement No. 665385.
alternative_title:
- ' Advances in Neural Information Processing Systems'
article_processing_charge: No
author:
- first_name: Mathias
full_name: Lechner, Mathias
id: 3DC22916-F248-11E8-B48F-1D18A9856A87
last_name: Lechner
- first_name: Ðorđe
full_name: Žikelić, Ðorđe
last_name: Žikelić
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Thomas A
full_name: Henzinger, Thomas A
id: 40876CD8-F248-11E8-B48F-1D18A9856A87
last_name: Henzinger
orcid: 0000-0002-2985-7724
citation:
ama: 'Lechner M, Žikelić Ð, Chatterjee K, Henzinger TA. Infinite time horizon safety
of Bayesian neural networks. In: 35th Conference on Neural Information Processing
Systems. ; 2021. doi:10.48550/arXiv.2111.03165'
apa: Lechner, M., Žikelić, Ð., Chatterjee, K., & Henzinger, T. A. (2021). Infinite
time horizon safety of Bayesian neural networks. In 35th Conference on Neural
Information Processing Systems. Virtual. https://doi.org/10.48550/arXiv.2111.03165
chicago: Lechner, Mathias, Ðorđe Žikelić, Krishnendu Chatterjee, and Thomas A Henzinger.
“Infinite Time Horizon Safety of Bayesian Neural Networks.” In 35th Conference
on Neural Information Processing Systems, 2021. https://doi.org/10.48550/arXiv.2111.03165.
ieee: M. Lechner, Ð. Žikelić, K. Chatterjee, and T. A. Henzinger, “Infinite time
horizon safety of Bayesian neural networks,” in 35th Conference on Neural Information
Processing Systems, Virtual, 2021.
ista: 'Lechner M, Žikelić Ð, Chatterjee K, Henzinger TA. 2021. Infinite time horizon
safety of Bayesian neural networks. 35th Conference on Neural Information Processing
Systems. NeurIPS: Neural Information Processing Systems, Advances in Neural Information
Processing Systems, .'
mla: Lechner, Mathias, et al. “Infinite Time Horizon Safety of Bayesian Neural Networks.”
35th Conference on Neural Information Processing Systems, 2021, doi:10.48550/arXiv.2111.03165.
short: M. Lechner, Ð. Žikelić, K. Chatterjee, T.A. Henzinger, in:, 35th Conference
on Neural Information Processing Systems, 2021.
conference:
end_date: 2021-12-10
location: Virtual
name: 'NeurIPS: Neural Information Processing Systems'
start_date: 2021-12-06
date_created: 2022-01-25T15:45:58Z
date_published: 2021-12-01T00:00:00Z
date_updated: 2023-06-23T07:01:11Z
day: '01'
ddc:
- '000'
department:
- _id: GradSch
- _id: ToHe
- _id: KrCh
doi: 10.48550/arXiv.2111.03165
ec_funded: 1
external_id:
arxiv:
- '2111.03165'
file:
- access_level: open_access
checksum: 0fc0f852525c10dda9cc9ffea07fb4e4
content_type: application/pdf
creator: mlechner
date_created: 2022-01-26T07:39:59Z
date_updated: 2022-01-26T07:39:59Z
file_id: '10682'
file_name: infinite_time_horizon_safety_o.pdf
file_size: 452492
relation: main_file
success: 1
file_date_updated: 2022-01-26T07:39:59Z
has_accepted_license: '1'
language:
- iso: eng
license: https://creativecommons.org/licenses/by-nc-nd/3.0/
main_file_link:
- open_access: '1'
url: https://proceedings.neurips.cc/paper/2021/hash/544defa9fddff50c53b71c43e0da72be-Abstract.html
month: '12'
oa: 1
oa_version: Published Version
project:
- _id: 2564DBCA-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '665385'
name: International IST Doctoral Program
- _id: 0599E47C-7A3F-11EA-A408-12923DDC885E
call_identifier: H2020
grant_number: '863818'
name: 'Formal Methods for Stochastic Models: Algorithms and Applications'
- _id: 25F42A32-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: Z211
name: The Wittgenstein Prize
publication: 35th Conference on Neural Information Processing Systems
publication_status: published
quality_controlled: '1'
related_material:
record:
- id: '11362'
relation: dissertation_contains
status: public
status: public
title: Infinite time horizon safety of Bayesian neural networks
tmp:
image: /images/cc_by_nc_nd.png
legal_code_url: https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode
name: Creative Commons Attribution-NonCommercial-NoDerivs 3.0 Unported (CC BY-NC-ND
3.0)
short: CC BY-NC-ND (3.0)
type: conference
user_id: 2EBD1598-F248-11E8-B48F-1D18A9856A87
year: '2021'
...
---
_id: '8793'
abstract:
- lang: eng
text: We study optimal election sequences for repeatedly selecting a (very) small
group of leaders among a set of participants (players) with publicly known unique
ids. In every time slot, every player has to select exactly one player that it
considers to be the current leader, oblivious to the selection of the other players,
but with the overarching goal of maximizing a given parameterized global (“social”)
payoff function in the limit. We consider a quite generic model, where the local
payoff achieved by a given player depends, weighted by some arbitrary but fixed
real parameter, on the number of different leaders chosen in a round, the number
of players that choose the given player as the leader, and whether the chosen
leader has changed w.r.t. the previous round or not. The social payoff can be
the maximum, average or minimum local payoff of the players. Possible applications
include quite diverse examples such as rotating coordinator-based distributed
algorithms and long-haul formation flying of social birds. Depending on the weights
and the particular social payoff, optimal sequences can be very different, from
simple round-robin where all players chose the same leader alternatingly every
time slot to very exotic patterns, where a small group of leaders (at most 2)
is elected in every time slot. Moreover, we study the question if and when a single
player would not benefit w.r.t. its local payoff when deviating from the given
optimal sequence, i.e., when our optimal sequences are Nash equilibria in the
restricted strategy space of oblivious strategies. As this is the case for many
parameterizations of our model, our results reveal that no punishment is needed
to make it rational for the players to optimize the social payoff.
acknowledgement: "We are grateful to Matthias Függer and Thomas Nowak for having raised
our interest in the problem studied in this paper.\r\nThis work has been supported
the Austrian Science Fund (FWF) projects S11405, S11407 (RiSE), and P28182 (ADynNet)."
article_processing_charge: No
article_type: original
author:
- first_name: Martin
full_name: Zeiner, Martin
last_name: Zeiner
- first_name: Ulrich
full_name: Schmid, Ulrich
last_name: Schmid
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
citation:
ama: Zeiner M, Schmid U, Chatterjee K. Optimal strategies for selecting coordinators.
Discrete Applied Mathematics. 2021;289(1):392-415. doi:10.1016/j.dam.2020.10.022
apa: Zeiner, M., Schmid, U., & Chatterjee, K. (2021). Optimal strategies for
selecting coordinators. Discrete Applied Mathematics. Elsevier. https://doi.org/10.1016/j.dam.2020.10.022
chicago: Zeiner, Martin, Ulrich Schmid, and Krishnendu Chatterjee. “Optimal Strategies
for Selecting Coordinators.” Discrete Applied Mathematics. Elsevier, 2021.
https://doi.org/10.1016/j.dam.2020.10.022.
ieee: M. Zeiner, U. Schmid, and K. Chatterjee, “Optimal strategies for selecting
coordinators,” Discrete Applied Mathematics, vol. 289, no. 1. Elsevier,
pp. 392–415, 2021.
ista: Zeiner M, Schmid U, Chatterjee K. 2021. Optimal strategies for selecting coordinators.
Discrete Applied Mathematics. 289(1), 392–415.
mla: Zeiner, Martin, et al. “Optimal Strategies for Selecting Coordinators.” Discrete
Applied Mathematics, vol. 289, no. 1, Elsevier, 2021, pp. 392–415, doi:10.1016/j.dam.2020.10.022.
short: M. Zeiner, U. Schmid, K. Chatterjee, Discrete Applied Mathematics 289 (2021)
392–415.
date_created: 2020-11-22T23:01:26Z
date_published: 2021-01-31T00:00:00Z
date_updated: 2023-08-04T11:12:41Z
day: '31'
ddc:
- '510'
department:
- _id: KrCh
doi: 10.1016/j.dam.2020.10.022
external_id:
isi:
- '000596823800035'
file:
- access_level: open_access
checksum: f1039ff5a2d6ca116720efdb84ee9d5e
content_type: application/pdf
creator: dernst
date_created: 2021-02-04T11:28:42Z
date_updated: 2021-02-04T11:28:42Z
file_id: '9089'
file_name: 2021_DiscreteApplMath_Zeiner.pdf
file_size: 652739
relation: main_file
success: 1
file_date_updated: 2021-02-04T11:28:42Z
has_accepted_license: '1'
intvolume: ' 289'
isi: 1
issue: '1'
language:
- iso: eng
month: '01'
oa: 1
oa_version: Published Version
page: 392-415
project:
- _id: 25F2ACDE-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S11402-N23
name: Rigorous Systems Engineering
- _id: 25863FF4-B435-11E9-9278-68D0E5697425
call_identifier: FWF
grant_number: S11407
name: Game Theory
publication: Discrete Applied Mathematics
publication_identifier:
issn:
- 0166218X
publication_status: published
publisher: Elsevier
quality_controlled: '1'
scopus_import: '1'
status: public
title: Optimal strategies for selecting coordinators
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 289
year: '2021'
...
---
_id: '9381'
abstract:
- lang: eng
text: 'A game of rock-paper-scissors is an interesting example of an interaction
where none of the pure strategies strictly dominates all others, leading to a
cyclic pattern. In this work, we consider an unstable version of rock-paper-scissors
dynamics and allow individuals to make behavioural mistakes during the strategy
execution. We show that such an assumption can break a cyclic relationship leading
to a stable equilibrium emerging with only one strategy surviving. We consider
two cases: completely random mistakes when individuals have no bias towards any
strategy and a general form of mistakes. Then, we determine conditions for a strategy
to dominate all other strategies. However, given that individuals who adopt a
dominating strategy are still prone to behavioural mistakes in the observed behaviour,
we may still observe extinct strategies. That is, behavioural mistakes in strategy
execution stabilise evolutionary dynamics leading to an evolutionary stable and,
potentially, mixed co-existence equilibrium.'
acknowledgement: Authors would like to thank Christian Hilbe and Martin Nowak for
their inspiring and very helpful feedback on the manuscript.
article_number: e1008523
article_processing_charge: No
article_type: original
author:
- first_name: Maria
full_name: Kleshnina, Maria
id: 4E21749C-F248-11E8-B48F-1D18A9856A87
last_name: Kleshnina
- first_name: Sabrina S.
full_name: Streipert, Sabrina S.
last_name: Streipert
- first_name: Jerzy A.
full_name: Filar, Jerzy A.
last_name: Filar
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
citation:
ama: Kleshnina M, Streipert SS, Filar JA, Chatterjee K. Mistakes can stabilise the
dynamics of rock-paper-scissors games. PLoS Computational Biology. 2021;17(4).
doi:10.1371/journal.pcbi.1008523
apa: Kleshnina, M., Streipert, S. S., Filar, J. A., & Chatterjee, K. (2021).
Mistakes can stabilise the dynamics of rock-paper-scissors games. PLoS Computational
Biology. Public Library of Science. https://doi.org/10.1371/journal.pcbi.1008523
chicago: Kleshnina, Maria, Sabrina S. Streipert, Jerzy A. Filar, and Krishnendu
Chatterjee. “Mistakes Can Stabilise the Dynamics of Rock-Paper-Scissors Games.”
PLoS Computational Biology. Public Library of Science, 2021. https://doi.org/10.1371/journal.pcbi.1008523.
ieee: M. Kleshnina, S. S. Streipert, J. A. Filar, and K. Chatterjee, “Mistakes can
stabilise the dynamics of rock-paper-scissors games,” PLoS Computational Biology,
vol. 17, no. 4. Public Library of Science, 2021.
ista: Kleshnina M, Streipert SS, Filar JA, Chatterjee K. 2021. Mistakes can stabilise
the dynamics of rock-paper-scissors games. PLoS Computational Biology. 17(4),
e1008523.
mla: Kleshnina, Maria, et al. “Mistakes Can Stabilise the Dynamics of Rock-Paper-Scissors
Games.” PLoS Computational Biology, vol. 17, no. 4, e1008523, Public Library
of Science, 2021, doi:10.1371/journal.pcbi.1008523.
short: M. Kleshnina, S.S. Streipert, J.A. Filar, K. Chatterjee, PLoS Computational
Biology 17 (2021).
date_created: 2021-05-09T22:01:38Z
date_published: 2021-04-01T00:00:00Z
date_updated: 2023-08-08T13:31:08Z
day: '01'
ddc:
- '000'
department:
- _id: KrCh
doi: 10.1371/journal.pcbi.1008523
ec_funded: 1
external_id:
isi:
- '000639711200001'
file:
- access_level: open_access
checksum: a94ebe0c4116f5047eaa6029e54d2dac
content_type: application/pdf
creator: kschuh
date_created: 2021-05-11T13:50:06Z
date_updated: 2021-05-11T13:50:06Z
file_id: '9385'
file_name: 2021_pcbi_Kleshnina.pdf
file_size: 1323820
relation: main_file
success: 1
file_date_updated: 2021-05-11T13:50:06Z
has_accepted_license: '1'
intvolume: ' 17'
isi: 1
issue: '4'
language:
- iso: eng
month: '04'
oa: 1
oa_version: Published Version
project:
- _id: 260C2330-B435-11E9-9278-68D0E5697425
call_identifier: H2020
grant_number: '754411'
name: ISTplus - Postdoctoral Fellowships
- _id: 0599E47C-7A3F-11EA-A408-12923DDC885E
call_identifier: H2020
grant_number: '863818'
name: 'Formal Methods for Stochastic Models: Algorithms and Applications'
publication: PLoS Computational Biology
publication_identifier:
eissn:
- '15537358'
issn:
- 1553734X
publication_status: published
publisher: Public Library of Science
quality_controlled: '1'
scopus_import: '1'
status: public
title: Mistakes can stabilise the dynamics of rock-paper-scissors games
tmp:
image: /images/cc_by.png
legal_code_url: https://creativecommons.org/licenses/by/4.0/legalcode
name: Creative Commons Attribution 4.0 International Public License (CC-BY 4.0)
short: CC BY (4.0)
type: journal_article
user_id: 4359f0d1-fa6c-11eb-b949-802e58b17ae8
volume: 17
year: '2021'
...
---
_id: '9640'
abstract:
- lang: eng
text: 'Selection and random drift determine the probability that novel mutations
fixate in a population. Population structure is known to affect the dynamics of
the evolutionary process. Amplifiers of selection are population structures that
increase the fixation probability of beneficial mutants compared to well-mixed
populations. Over the past 15 years, extensive research has produced remarkable
structures called strong amplifiers which guarantee that every beneficial mutation
fixates with high probability. But strong amplification has come at the cost of
considerably delaying the fixation event, which can slow down the overall rate
of evolution. However, the precise relationship between fixation probability and
time has remained elusive. Here we characterize the slowdown effect of strong
amplification. First, we prove that all strong amplifiers must delay the fixation
event at least to some extent. Second, we construct strong amplifiers that delay
the fixation event only marginally as compared to the well-mixed populations.
Our results thus establish a tight relationship between fixation probability and
time: Strong amplification always comes at a cost of a slowdown, but more than
a marginal slowdown is not needed.'
acknowledgement: 'K.C. acknowledges support from ERC Start grant no. (279307: Graph
Games), ERC Consolidator grant no. (863818: ForM-SMart), Austrian Science Fund (FWF)
grant no. P23499-N23 and S11407-N23 (RiSE). M.A.N. acknowledges support from Office
of Naval Research grant N00014-16-1-2914 and from the John Templeton Foundation.'
article_number: '4009'
article_processing_charge: No
article_type: original
author:
- first_name: Josef
full_name: Tkadlec, Josef
id: 3F24CCC8-F248-11E8-B48F-1D18A9856A87
last_name: Tkadlec
orcid: 0000-0002-1097-9684
- first_name: Andreas
full_name: Pavlogiannis, Andreas
id: 49704004-F248-11E8-B48F-1D18A9856A87
last_name: Pavlogiannis
orcid: 0000-0002-8943-0722
- first_name: Krishnendu
full_name: Chatterjee, Krishnendu
id: 2E5DCA20-F248-11E8-B48F-1D18A9856A87
last_name: Chatterjee
orcid: 0000-0002-4561-241X
- first_name: Martin A.
full_name: Nowak, Martin A.
last_name: Nowak
citation:
ama: Tkadlec J, Pavlogiannis A, Chatterjee K, Nowak MA. Fast and strong amplifiers
of natural selection. Nature Communications. 2021;12(1). doi:10.1038/s41467-021-24271-w
apa: Tkadlec, J., Pavlogiannis, A., Chatterjee, K., & Nowak, M. A. (2021). Fast
and strong amplifiers of natural selection. Nature Communications. Springer
Nature. https://doi.org/10.1038/s41467-021-24271-w
chicago: Tkadlec, Josef, Andreas Pavlogiannis, Krishnendu Chatterjee, and Martin
A. Nowak. “Fast and Strong Amplifiers of Natural Selection.” Nature Communications.
Springer Nature, 2021. https://doi.org/10.1038/s41467-021-24271-w.
ieee: J. Tkadlec, A. Pavlogiannis, K. Chatterjee, and M. A. Nowak, “Fast and strong
amplifiers of natural selection,” Nature Communications, vol. 12, no. 1.
Springer Nature, 2021.
ista: Tkadlec J, Pavlogiannis A, Chatterjee K, Nowak MA. 2021. Fast and strong amplifiers
of natural selection. Nature Communications. 12(1), 4009.
mla: Tkadlec, Josef, et al. “Fast and Strong Amplifiers of Natural Selection.” Nature
Communications, vol. 12, no. 1, 4009, Springer Nature, 2021, doi:10.1038/s41467-021-24271-w.
short: J. Tkadlec, A. Pavlogiannis, K. Chatterjee, M.A. Nowak, Nature Communications
12 (2021).
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