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Zhao X, Hu K, Tao Y, Jin L, Shi L. The impact of dynamic linking on cooperation on complex networks. CHAOS (WOODBURY, N.Y.) 2024; 34:073130. [PMID: 38995990 DOI: 10.1063/5.0221942] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/04/2024] [Accepted: 06/27/2024] [Indexed: 07/14/2024]
Abstract
In complex social systems, individual relationships and the surrounding environment are constantly changing, allowing individuals to interact on dynamic networks. This study aims to investigate how individuals in a dynamic network engaged in a prisoner's dilemma game adapt their competitive environment through random edge breaks and reconnections when faced with incomplete information and adverse local conditions, thereby influencing the evolution of cooperative behavior. We find that random edge breaks and reconnections in dynamic networks can disrupt cooperative clusters, significantly hindering the development of cooperation. This negative impact becomes more pronounced over larger time scales. However, we also observe that nodes with higher degrees of connectivity exhibit greater resilience to this cooperation disruption. Our research reveals the profound impact of dynamic network structures on the evolution of cooperation and provides new insights into the mechanisms of cooperation in complex systems.
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Affiliation(s)
- Xiaoqian Zhao
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Kaipeng Hu
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Yewei Tao
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
| | - Libin Jin
- Research Institute of Data Science, Shanghai Lixin University of Accounting and Finance, 201209 Shanghai, China
| | - Lei Shi
- School of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming 650221, China
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Wang C, Perc M, Szolnoki A. Evolutionary dynamics of any multiplayer game on regular graphs. Nat Commun 2024; 15:5349. [PMID: 38914550 PMCID: PMC11196707 DOI: 10.1038/s41467-024-49505-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/23/2023] [Accepted: 06/05/2024] [Indexed: 06/26/2024] Open
Abstract
Multiplayer games on graphs are at the heart of theoretical descriptions of key evolutionary processes that govern vital social and natural systems. However, a comprehensive theoretical framework for solving multiplayer games with an arbitrary number of strategies on graphs is still missing. Here, we solve this by drawing an analogy with the Balls-and-Boxes problem, based on which we show that the local configuration of multiplayer games on graphs is equivalent to distributing k identical co-players among n distinct strategies. We use this to derive the replicator equation for any n-strategy multiplayer game under weak selection, which can be solved in polynomial time. As an example, we revisit the second-order free-riding problem, where costly punishment cannot truly resolve social dilemmas in a well-mixed population. Yet, in structured populations, we derive an accurate threshold for the punishment strength, beyond which punishment can either lead to the extinction of defection or transform the system into a rock-paper-scissors-like cycle. The analytical solution also qualitatively agrees with the phase diagrams that were previously obtained for non-marginal selection strengths. Our framework thus allows an exploration of any multi-strategy multiplayer game on regular graphs.
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Affiliation(s)
- Chaoqian Wang
- Department of Computational and Data Sciences, George Mason University, Fairfax, VA, 22030, USA.
| | - Matjaž Perc
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000, Maribor, Slovenia
- Community Healthcare Center Dr. Adolf Drolc Maribor, Vošnjakova ulica 2, 2000, Maribor, Slovenia
- Complexity Science Hub Vienna, Josefstädterstraße 39, 1080, Vienna, Austria
- Department of Physics, Kyung Hee University, 26 Kyungheedae-ro, Dongdaemun-gu, Seoul, Republic of Korea
| | - Attila Szolnoki
- Institute of Technical Physics and Materials Science, Centre for Energy Research, P.O. Box 49, H-1525, Budapest, Hungary
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Allen B, McAvoy A. The coalescent in finite populations with arbitrary, fixed structure. Theor Popul Biol 2024; 158:150-169. [PMID: 38880430 DOI: 10.1016/j.tpb.2024.06.004] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/02/2023] [Revised: 06/03/2024] [Accepted: 06/12/2024] [Indexed: 06/18/2024]
Abstract
The coalescent is a stochastic process representing ancestral lineages in a population undergoing neutral genetic drift. Originally defined for a well-mixed population, the coalescent has been adapted in various ways to accommodate spatial, age, and class structure, along with other features of real-world populations. To further extend the range of population structures to which coalescent theory applies, we formulate a coalescent process for a broad class of neutral drift models with arbitrary - but fixed - spatial, age, sex, and class structure, haploid or diploid genetics, and any fixed mating pattern. Here, the coalescent is represented as a random sequence of mappings [Formula: see text] from a finite set G to itself. The set G represents the "sites" (in individuals, in particular locations and/or classes) at which these alleles can live. The state of the coalescent, Ct:G→G, maps each site g∈G to the site containing g's ancestor, t time-steps into the past. Using this representation, we define and analyze coalescence time, coalescence branch length, mutations prior to coalescence, and stationary probabilities of identity-by-descent and identity-by-state. For low mutation, we provide a recipe for computing identity-by-descent and identity-by-state probabilities via the coalescent. Applying our results to a diploid population with arbitrary sex ratio r, we find that measures of genetic dissimilarity, among any set of sites, are scaled by 4r(1-r) relative to the even sex ratio case.
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Affiliation(s)
- Benjamin Allen
- Department of Mathematics, Emmanuel College, 400 The Fenway, Boston, MA, 02115, USA.
| | - Alex McAvoy
- School of Data Science and Society, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA; Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC, 27599, USA
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Wang X, Fu F, Wang L. Deterministic theory of evolutionary games on temporal networks. J R Soc Interface 2024; 21:20240055. [PMID: 38807526 DOI: 10.1098/rsif.2024.0055] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/24/2024] [Accepted: 03/28/2024] [Indexed: 05/30/2024] Open
Abstract
Recent empirical studies have revealed that social interactions among agents in realistic networks merely exist intermittently and occur in a particular sequential order. However, it remains unexplored how to theoretically describe evolutionary dynamics of multiple strategies on temporal networks. Herein, we develop a deterministic theory for studying evolutionary dynamics of any [Formula: see text] pairwise games in structured populations where individuals are connected and organized by temporally activated edges. In the limit of weak selection, we derive replicator-like equations with a transformed payoff matrix characterizing how the mean frequency of each strategy varies over time, and then obtain critical conditions for any strategy to be evolutionarily stable on temporal networks. Interestingly, the re-scaled payoff matrix is a linear combination of the original payoff matrix with an additional one describing local competitions between any pair of different strategies, whose weights are solely determined by network topology and selection intensity. As a particular example, we apply the deterministic theory to analysing the impacts of temporal networks in the mini-ultimatum game, and find that temporally networked population structures result in the emergence of fairness. Our work offers theoretical insights into the subtle effects of network temporality on evolutionary game dynamics.
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Affiliation(s)
- Xiaofeng Wang
- Department of Automation, School of Information Science and Technology, Donghua University , Shanghai 201620, People's Republic of China
- Engineering Research Center of Digitized Textile and Apparel Technology (Ministry of Education), Donghua University , Shanghai 201620, People's Republic of China
| | - Feng Fu
- Department of Mathematics, Dartmouth College , Hanover, NH 03755, USA
- Department of Biomedical Data Science, Geisel School of Medicine at Dartmouth , Lebanon, NH 03756, USA
| | - Long Wang
- Center for Systems and Control, College of Engineering, Peking University , Beijing 100871, People's Republic of China
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Allen B, Khwaja AR, Donahue JL, Kelly TJ, Hyacinthe SR, Proulx J, Lattanzio C, Dementieva YA, Sample C. Nonlinear social evolution and the emergence of collective action. PNAS NEXUS 2024; 3:pgae131. [PMID: 38595801 PMCID: PMC11002786 DOI: 10.1093/pnasnexus/pgae131] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 03/13/2024] [Accepted: 03/21/2024] [Indexed: 04/11/2024]
Abstract
Organisms from microbes to humans engage in a variety of social behaviors, which affect fitness in complex, often nonlinear ways. The question of how these behaviors evolve has consequences ranging from antibiotic resistance to human origins. However, evolution with nonlinear social interactions is challenging to model mathematically, especially in combination with spatial, group, and/or kin assortment. We derive a mathematical condition for natural selection with synergistic interactions among any number of individuals. This result applies to populations with arbitrary (but fixed) spatial or network structure, group subdivision, and/or mating patterns. In this condition, nonlinear fitness effects are ascribed to collectives, and weighted by a new measure of collective relatedness. For weak selection, this condition can be systematically evaluated by computing branch lengths of ancestral trees. We apply this condition to pairwise games between diploid relatives, and to dilemmas of collective help or harm among siblings and on spatial networks. Our work provides a rigorous basis for extending the notion of "actor", in the study of social evolution, from individuals to collectives.
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Affiliation(s)
- Benjamin Allen
- Department of Mathematics, Emmanuel College, Boston, MA 02115, USA
| | | | - James L Donahue
- Department of Mathematics, Emmanuel College, Boston, MA 02115, USA
| | - Theodore J Kelly
- Department of Mathematics, Emmanuel College, Boston, MA 02115, USA
| | | | - Jacob Proulx
- Department of Mathematics, Emmanuel College, Boston, MA 02115, USA
| | | | | | - Christine Sample
- Department of Mathematics, Emmanuel College, Boston, MA 02115, USA
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Zarei F, Gandica Y, Rocha LEC. Bursts of communication increase opinion diversity in the temporal Deffuant model. Sci Rep 2024; 14:2222. [PMID: 38278824 PMCID: PMC10817933 DOI: 10.1038/s41598-024-52458-w] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/16/2023] [Accepted: 01/18/2024] [Indexed: 01/28/2024] Open
Abstract
Human interactions create social networks forming the backbone of societies. Individuals adjust their opinions by exchanging information through social interactions. Two recurrent questions are whether social structures promote opinion polarisation or consensus and whether polarisation can be avoided, particularly on social media. In this paper, we hypothesise that not only network structure but also the timings of social interactions regulate the emergence of opinion clusters. We devise a temporal version of the Deffuant opinion model where pairwise social interactions follow temporal patterns. Individuals may self-organise into a multi-partisan society due to network clustering promoting the reinforcement of local opinions. Burstiness has a similar effect and is alone sufficient to refrain the population from consensus and polarisation by also promoting the reinforcement of local opinions. The diversity of opinions in socially clustered networks thus increases with burstiness, particularly, and counter-intuitively, when individuals have low tolerance and prefer to adjust to similar peers. The emergent opinion landscape is well-balanced regarding groups' size, with relatively short differences between groups, and a small fraction of extremists. We argue that polarisation is more likely to emerge in social media than offline social networks because of the relatively low social clustering observed online, despite the observed online burstiness being sufficient to promote more diversity than would be expected offline. Increasing the variance of burst activation times, e.g. by being less active on social media, could be a venue to reduce polarisation. Furthermore, strengthening online social networks by increasing social redundancy, i.e. triangles, may also promote diversity.
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Affiliation(s)
- Fatemeh Zarei
- Department of Economics, Ghent University, Ghent, Belgium
- Department of Physics and Astronomy, Ghent University, Ghent, Belgium
| | - Yerali Gandica
- Department of Mathematics, Valencian International University, Valencia, Spain
| | - Luis E C Rocha
- Department of Economics, Ghent University, Ghent, Belgium.
- Department of Physics and Astronomy, Ghent University, Ghent, Belgium.
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Allen B. Symmetry in models of natural selection. J R Soc Interface 2023; 20:20230306. [PMID: 37963562 PMCID: PMC10645516 DOI: 10.1098/rsif.2023.0306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/27/2023] [Accepted: 10/20/2023] [Indexed: 11/16/2023] Open
Abstract
Symmetry arguments are frequently used-often implicitly-in mathematical modelling of natural selection. Symmetry simplifies the analysis of models and reduces the number of distinct population states to be considered. Here, I introduce a formal definition of symmetry in mathematical models of natural selection. This definition applies to a broad class of models that satisfy a minimal set of assumptions, using a framework developed in previous works. In this framework, population structure is represented by a set of sites at which alleles can live, and transitions occur via replacement of some alleles by copies of others. A symmetry is defined as a permutation of sites that preserves probabilities of replacement and mutation. The symmetries of a given selection process form a group, which acts on population states in a way that preserves the Markov chain representing selection. Applying classical results on group actions, I formally characterize the use of symmetry to reduce the states of this Markov chain, and obtain bounds on the number of states in the reduced chain.
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Affiliation(s)
- Benjamin Allen
- Department of Mathematics, Emmanuel College, Boston, MA, USA
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Bhaumik J, Masuda N. Fixation probability in evolutionary dynamics on switching temporal networks. J Math Biol 2023; 87:64. [PMID: 37768362 PMCID: PMC10539469 DOI: 10.1007/s00285-023-01987-5] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Grants] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/03/2023] [Revised: 08/03/2023] [Accepted: 08/13/2023] [Indexed: 09/29/2023]
Abstract
Population structure has been known to substantially affect evolutionary dynamics. Networks that promote the spreading of fitter mutants are called amplifiers of selection, and those that suppress the spreading of fitter mutants are called suppressors of selection. Research in the past two decades has found various families of amplifiers while suppressors still remain somewhat elusive. It has also been discovered that most networks are amplifiers of selection under the birth-death updating combined with uniform initialization, which is a standard condition assumed widely in the literature. In the present study, we extend the birth-death processes to temporal (i.e., time-varying) networks. For the sake of tractability, we restrict ourselves to switching temporal networks, in which the network structure deterministically alternates between two static networks at constant time intervals or stochastically in a Markovian manner. We show that, in a majority of cases, switching networks are less amplifying than both of the two static networks constituting the switching networks. Furthermore, most small switching networks, i.e., networks on six nodes or less, are suppressors, which contrasts to the case of static networks.
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Affiliation(s)
- Jnanajyoti Bhaumik
- Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, 14260-2900, USA
| | - Naoki Masuda
- Department of Mathematics, State University of New York at Buffalo, Buffalo, NY, 14260-2900, USA.
- Computational and Data-Enabled Science and Engineering Program, State University of New York at Buffalo, Buffalo, NY, 14260-5030, USA.
- Center for Computational Social Science, Kobe University, Kobe, 657-8501, Japan.
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Allen B. Flipping the intuition for games on dynamic networks. NATURE COMPUTATIONAL SCIENCE 2023; 3:737-738. [PMID: 38177776 DOI: 10.1038/s43588-023-00513-3] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/06/2024]
Affiliation(s)
- Benjamin Allen
- Department of Mathematics, Emmanuel College, Boston, MA, USA.
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