Selection pressure transforms the nature of social dilemmas in adaptive networks
Identifieur interne : 001720 ( Istex/Corpus ); précédent : 001719; suivant : 001721Selection pressure transforms the nature of social dilemmas in adaptive networks
Auteurs : Sven Van Segbroeck ; Francisco C. Santos ; Tom Lenaerts ; Jorge M. PachecoSource :
- New Journal of Physics [ 1367-2630 ] ; 2011.
Abstract
We have studied the evolution of cooperation in structured populations whose topology coevolves with the game strategies of the individuals. Strategy evolution proceeds according to an update rule with a free parameter, which measures the selection pressure. We explore how this parameter affects the interplay between network dynamics and strategy dynamics. A dynamical network topology can influence the strategy dynamics in two ways: (i) by modifying the expected payoff associated with each strategy and (ii) by reshaping the imitation network that underlies the evolutionary process. We show here that the selection pressure tunes the relative contribution of each of these two forces to the final outcome of strategy evolution. The dynamics of the imitation network plays only a minor role under strong selection, but becomes the dominant force under weak selection. We demonstrate how these findings constitute a mechanism supporting cooperative behavior.
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DOI: 10.1088/1367-2630/13/1/013007
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<record><TEI wicri:istexFullTextTei="biblStruct"><teiHeader><fileDesc><titleStmt><title xml:lang="en">Selection pressure transforms the nature of social dilemmas in adaptive networks</title><author><name sortKey="Segbroeck, Sven Van" sort="Segbroeck, Sven Van" uniqKey="Segbroeck S" first="Sven Van" last="Segbroeck">Sven Van Segbroeck</name><affiliation><mods:affiliation>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>COMO, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>Author to whom any correspondence should be addressed.</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: svsegbro@ulb.ac.be</mods:affiliation></affiliation></author><author><name sortKey="Santos, Francisco C" sort="Santos, Francisco C" uniqKey="Santos F" first="Francisco C" last="Santos">Francisco C. Santos</name><affiliation><mods:affiliation>CENTRIA, Departamento de Informtica, Faculdade de Cincias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal</mods:affiliation></affiliation><affiliation><mods:affiliation>ATP-Group, CMAF, Complexo Interdisciplinar, P-1649-003 Lisboa Codex, Portugal</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: fcsantos@fct.unl.pt</mods:affiliation></affiliation></author><author><name sortKey="Lenaerts, Tom" sort="Lenaerts, Tom" uniqKey="Lenaerts T" first="Tom" last="Lenaerts">Tom Lenaerts</name><affiliation><mods:affiliation>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>Department of Computer Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: tlenaert@ulb.ac.be</mods:affiliation></affiliation></author><author><name sortKey="Pacheco, Jorge M" sort="Pacheco, Jorge M" uniqKey="Pacheco J" first="Jorge M" last="Pacheco">Jorge M. 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Santos</name><affiliation><mods:affiliation>CENTRIA, Departamento de Informtica, Faculdade de Cincias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal</mods:affiliation></affiliation><affiliation><mods:affiliation>ATP-Group, CMAF, Complexo Interdisciplinar, P-1649-003 Lisboa Codex, Portugal</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: fcsantos@fct.unl.pt</mods:affiliation></affiliation></author><author><name sortKey="Lenaerts, Tom" sort="Lenaerts, Tom" uniqKey="Lenaerts T" first="Tom" last="Lenaerts">Tom Lenaerts</name><affiliation><mods:affiliation>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>Department of Computer Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</mods:affiliation></affiliation><affiliation><mods:affiliation>E-mail: tlenaert@ulb.ac.be</mods:affiliation></affiliation></author><author><name sortKey="Pacheco, Jorge M" sort="Pacheco, Jorge M" uniqKey="Pacheco J" first="Jorge M" last="Pacheco">Jorge M. 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Phys.</title><idno type="ISSN">1367-2630</idno><idno type="eISSN">1367-2630</idno><imprint><publisher>IOP Publishing</publisher><date type="published" when="2011">2011</date><biblScope unit="volume">13</biblScope><biblScope unit="issue">1</biblScope><biblScope unit="page" from="1">1</biblScope><biblScope unit="page" to="13">13</biblScope></imprint><idno type="ISSN">1367-2630</idno></series><idno type="istex">643BCB406A6541BD7C697A56A11A105AB6542273</idno><idno type="DOI">10.1088/1367-2630/13/1/013007</idno><idno type="articleID">368509</idno><idno type="articleNumber">013007</idno></biblStruct></sourceDesc><seriesStmt><idno type="ISSN">1367-2630</idno></seriesStmt></fileDesc><profileDesc><textClass></textClass><langUsage><language ident="en">en</language></langUsage></profileDesc></teiHeader><front><div type="abstract">We have studied the evolution of cooperation in structured populations whose topology coevolves with the game strategies of the individuals. Strategy evolution proceeds according to an update rule with a free parameter, which measures the selection pressure. We explore how this parameter affects the interplay between network dynamics and strategy dynamics. A dynamical network topology can influence the strategy dynamics in two ways: (i) by modifying the expected payoff associated with each strategy and (ii) by reshaping the imitation network that underlies the evolutionary process. We show here that the selection pressure tunes the relative contribution of each of these two forces to the final outcome of strategy evolution. The dynamics of the imitation network plays only a minor role under strong selection, but becomes the dominant force under weak selection. We demonstrate how these findings constitute a mechanism supporting cooperative behavior.</div></front></TEI><istex><corpusName>iop</corpusName><author><json:item><name>Sven Van Segbroeck</name><affiliations><json:string>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</json:string><json:string>COMO, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</json:string><json:string>Author to whom any correspondence should be addressed.</json:string><json:string>E-mail: svsegbro@ulb.ac.be</json:string></affiliations></json:item><json:item><name>Francisco C Santos</name><affiliations><json:string>CENTRIA, Departamento de Informtica, Faculdade de Cincias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal</json:string><json:string>ATP-Group, CMAF, Complexo Interdisciplinar, P-1649-003 Lisboa Codex, Portugal</json:string><json:string>E-mail: fcsantos@fct.unl.pt</json:string></affiliations></json:item><json:item><name>Tom Lenaerts</name><affiliations><json:string>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</json:string><json:string>Department of Computer Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</json:string><json:string>E-mail: tlenaert@ulb.ac.be</json:string></affiliations></json:item><json:item><name>Jorge M Pacheco</name><affiliations><json:string>ATP-Group, CMAF, Complexo Interdisciplinar, P-1649-003 Lisboa Codex, Portugal</json:string><json:string>Departamento de Matemtica e Aplicaes, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal</json:string><json:string>E-mail: jmpacheco@math.uminho.pt</json:string></affiliations></json:item></author><language><json:string>eng</json:string></language><originalGenre><json:string>paper</json:string></originalGenre><abstract>We have studied the evolution of cooperation in structured populations whose topology coevolves with the game strategies of the individuals. 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We explore how this parameter affects the interplay between network dynamics and strategy dynamics. A dynamical network topology can influence the strategy dynamics in two ways: (i) by modifying the expected payoff associated with each strategy and (ii) by reshaping the imitation network that underlies the evolutionary process. We show here that the selection pressure tunes the relative contribution of each of these two forces to the final outcome of strategy evolution. The dynamics of the imitation network plays only a minor role under strong selection, but becomes the dominant force under weak selection. 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Phys.</jnl-abbreviation><jnl-shortname>NJP</jnl-shortname><jnl-issn>1367-2630</jnl-issn><jnl-coden>NJOPFM</jnl-coden><jnl-imprint>IOP Publishing</jnl-imprint><jnl-web-address>stacks.iop.org/NJP</jnl-web-address></jnl-data><volume-data><year-publication>2011</year-publication><volume-number>13</volume-number></volume-data><issue-data><issue-number>1</issue-number><coverdate>January 2011</coverdate></issue-data><article-data><article-type type="paper" sort="regular"></article-type><type-number type="paper" numbering="article" artnum="013007">013007</type-number><article-number>368509</article-number><first-page>1</first-page><last-page>13</last-page><length>13</length><pii></pii><doi>10.1088/1367-2630/13/1/013007</doi><copyright>IOP Publishing and Deutsche Physikalische Gesellschaft</copyright><ccc>1367-2630/11/013007+13$33.00</ccc><printed></printed></article-data></article-metadata><header><title-group><title>Selection pressure transforms the nature of social dilemmas in adaptive networks</title><short-title>Selection pressure transforms the nature of social dilemmas in adaptive networks</short-title><ej-title>Selection pressure transforms the nature of social dilemmas in adaptive networks</ej-title></title-group><author-group><author address="nj368509ad1" second-address="nj368509ad2" alt-address="nj368509aad7" email="nj368509ea1"><first-names>Sven Van</first-names><second-name>Segbroeck</second-name></author><author address="nj368509ad3" second-address="nj368509ad4" email="nj368509ea2"><first-names>Francisco C</first-names><second-name>Santos</second-name></author><author address="nj368509ad1" second-address="nj368509ad5" email="nj368509ea3"><first-names>Tom</first-names><second-name>Lenaerts</second-name></author><author address="nj368509ad4" second-address="nj368509ad6" email="nj368509ea4"><first-names>Jorge M</first-names><second-name>Pacheco</second-name></author><short-author-list>Sven Van Segbroeck <italic>et al</italic></short-author-list></author-group><address-group><address id="nj368509ad1" showid="yes">MLG, <orgname>Université Libre de Bruxelles</orgname>, Boulevard du Triomphe—CP 212, 1050 Brussels, <country>Belgium</country></address><address id="nj368509ad2" showid="yes">COMO, <orgname>Vrije Universiteit Brussel</orgname>, Pleinlaan 2, 1050 Brussels, <country>Belgium</country></address><address id="nj368509ad3" showid="yes">CENTRIA, Departamento de Informática, Faculdade de Ciências e Tecnologia, <orgname>Universidade Nova de Lisboa</orgname>, 2829-516 Caparica, <country>Portugal</country></address><address id="nj368509ad4" showid="yes">ATP-Group, CMAF, <orgname>Complexo Interdisciplinar</orgname>, P-1649-003 Lisboa Codex, <country>Portugal</country></address><address id="nj368509ad5" showid="yes">Department of Computer Science, <orgname>Vrije Universiteit Brussel</orgname>, Pleinlaan 2, 1050 Brussels, <country>Belgium</country></address><address id="nj368509ad6" showid="yes">Departamento de Matemática e Aplicações, <orgname>Universidade do Minho</orgname>, Campus de Gualtar, 4710-057 Braga, <country>Portugal</country></address><address id="nj368509aad7" alt="yes" showid="yes">Author to whom any correspondence should be addressed.</address><e-address id="nj368509ea1"><email mailto="svsegbro@ulb.ac.be">svsegbro@ulb.ac.be</email></e-address><e-address id="nj368509ea2"><email mailto="fcsantos@fct.unl.pt">fcsantos@fct.unl.pt</email></e-address><e-address id="nj368509ea3"><email mailto="tlenaert@ulb.ac.be">tlenaert@ulb.ac.be</email></e-address><e-address id="nj368509ea4"><email mailto="jmpacheco@math.uminho.pt">jmpacheco@math.uminho.pt</email></e-address></address-group><history received="8 September 2010" online="11 January 2011"></history><abstract-group><abstract><heading>Abstract</heading><p indent="no">We have studied the evolution of cooperation in structured populations whose topology coevolves with the game strategies of the individuals. Strategy evolution proceeds according to an update rule with a free parameter, which measures the selection pressure. We explore how this parameter affects the interplay between network dynamics and strategy dynamics. A dynamical network topology can influence the strategy dynamics in two ways: (i) by modifying the expected payoff associated with each strategy and (ii) by reshaping the imitation network that underlies the evolutionary process. We show here that the selection pressure tunes the relative contribution of each of these two forces to the final outcome of strategy evolution. The dynamics of the imitation network plays only a minor role under strong selection, but becomes the dominant force under weak selection. We demonstrate how these findings constitute a mechanism supporting cooperative behavior.</p></abstract></abstract-group></header><body refstyle="numeric"><sec-level1 id="nj368509s1" label="1"><heading>Introduction</heading><p indent="no">Many biological systems, and especially human societies, show persistent cooperative patterns. The evolution of such patterns is often studied using evolutionary game theory (EGT) [<cite linkend="nj368509bib1">1</cite>]–[<cite linkend="nj368509bib4">4</cite>]. This framework models interactions between the individuals of a population in terms of simple games. Successful behavior—success being measured in terms of game payoff—will be imitated and spreads in the population. The prototypical game to represent dilemmas of cooperation is the conventional prisoner's dilemma [<cite linkend="nj368509bib5">5</cite>]. In this game, a cooperative act goes at a cost <inline-eqn><math-text><italic>c</italic></math-text></inline-eqn> to the cooperator while conferring a benefit <inline-eqn><math-text><italic>b</italic></math-text></inline-eqn> to another individual (assuming <inline-eqn><math-text><italic>b</italic>><italic>c</italic></math-text></inline-eqn>). Defectors receive the benefits without spending any costs and are therefore expected to have an evolutionary advantage over cooperators.</p><p>However, given the omnipresence of prosocial behavior, certain conditions exist under which cooperation becomes viable [<cite linkend="nj368509bib4">4</cite>], [<cite linkend="nj368509bib6">6</cite>]–[<cite linkend="nj368509bib12">12</cite>]. It has, for instance, been recognized that the topology of the network along which individuals interact and reproduce/imitate affects drastically the evolutionary chances of cooperators [<cite linkend="nj368509bib13">13</cite>]–[<cite linkend="nj368509bib42">42</cite>] (see Szabó and Fáth for a review [<cite linkend="nj368509bib43">43</cite>]). The individuals also shape and reshape their social environment themselves and are, at least partially, responsible for the specific features that characterize their social network [<cite linkend="nj368509bib44">44</cite>]–[<cite linkend="nj368509bib47">47</cite>]. The process of network reshaping is often coupled with the strategy dynamics: the behavior of an individual influences his social position and vice versa [<cite linkend="nj368509bib48">48</cite>]. Networks exhibiting such feedback loops provide sophisticated examples of adaptive networks [<cite linkend="nj368509bib49">49</cite>]. Several authors have studied the functioning of such networks in the context of cooperation [<cite linkend="nj368509bib50">50</cite>]–[<cite linkend="nj368509bib65">65</cite>] (see [<cite linkend="nj368509bib66">66</cite>] for a recent review). In general, cooperation has been shown to emerge more easily when interactions that benefit both partners last longer than interactions where one partner is exploited by the other [<cite linkend="nj368509bib55">55</cite>, <cite linkend="nj368509bib56">56</cite>, <cite linkend="nj368509bib60">60</cite>].</p><p>In structured populations, the outcome of the evolutionary process depends also on the update rule, which dictates how strategies evolve from one generation to the next [<cite linkend="nj368509bib67">67</cite>]. The selection pressure enters this update rule, and in EGT this is no exception, also being called intensity of selection [<cite linkend="nj368509bib68">68</cite>]. A maximal selection strength implies that individuals only imitate those with a higher game payoff. Reducing the selection strength increases the amount of noise in the imitation process, until, ultimately, evolution becomes a purely random process (neutral selection or random drift). The impact of the selection pressure on the evolutionary dynamics of a finite, well-mixed population is well understood by now [<cite linkend="nj368509bib68">68</cite>]–[<cite linkend="nj368509bib71">71</cite>]. Additional effects may, however, arise in adaptive networks, where the social environment of an individual changes according to his behavior. Such a coevolutionary interplay between strategy evolution and graph evolution may give rise to correlations between the strategies of connected individuals. The strategy of an individual will therefore influence not only the payoff he acquires, but also the probability that he will be considered as a potential role model by someone else. When some strategies are more popular than others, the rate of strategy update may effectively change, depending on the intensity of selection.</p><p>We study this issue in the framework of active linking (AL) dynamics [<cite linkend="nj368509bib55">55</cite>, <cite linkend="nj368509bib56">56</cite>, <cite linkend="nj368509bib62"> 62</cite>, <cite linkend="nj368509bib65">65</cite>, <cite linkend="nj368509bib72">72</cite>], a model that allows us to study cooperation in adaptive networks analytically in certain limits. Section <secref linkend="nj368509s2">2</secref> introduces both our model and the framework in which we will make our analysis. In section <secref linkend="nj368509s3">3</secref>, we analyze the gradient of selection as a function of the intensity of selection, assuming that the network topology evolves much faster than the individual behavior. Section <secref linkend="nj368509s4">4</secref> presents the conclusions.</p></sec-level1><sec-level1 id="nj368509s2" label="2"><heading>A minimal model</heading><p indent="no">Consider a finite population of <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn> individuals interacting in symmetric, one-shot, two-player games of cooperation defined by the payoff matrix
<display-eqn id="nj368509eqn1" textype="equation" notation="LaTeX" eqnnum="1"></display-eqn></p><p>We distinguish two possible game strategies: cooperate unconditionally (<inline-eqn><math-text><italic>C</italic></math-text></inline-eqn>) and defect unconditionally (<inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>). Payoff matrix <secref linkend="nj368509s1">1</secref> shows that individuals receive a reward <inline-eqn><math-text><italic>R</italic></math-text></inline-eqn> upon mutual cooperation and a punishment <inline-eqn><math-text><italic>P</italic></math-text></inline-eqn> upon mutual defection. When a <inline-eqn><math-text><italic>C</italic></math-text></inline-eqn> meets a <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>, the <inline-eqn><math-text><italic>C</italic></math-text></inline-eqn> receives the sucker's payoff <inline-eqn><math-text><italic>S</italic></math-text></inline-eqn>, whereas the <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn> acquires the temptation to defect <inline-eqn><math-text><italic>T</italic></math-text></inline-eqn>.</p><p>Individuals do not interact with everyone in the population. Instead, a network indicates who meets whom. The structure of this network is dynamic, in the sense that edges appear and disappear over time. Simultaneously, individuals may reconsider their game play. The network dynamics proceed on a characteristic time scale <inline-eqn><math-text>τ<sub><italic>a</italic></sub></math-text></inline-eqn>, the strategy dynamics on another time scale <inline-eqn><math-text>τ<sub><italic>s</italic></sub></math-text></inline-eqn>. Below, we define each of these two dynamical processes separately.</p><p>We use the AL model, developed earlier by Pacheco <italic>et al</italic> [<cite linkend="nj368509bib55">55</cite>, <cite linkend="nj368509bib56">56</cite>], to define the network's evolution. Each individual has a propensity <inline-eqn><math-text>α</math-text></inline-eqn> to engage in new interactions, such that new edges are formed at a rate <inline-eqn><math-text>α<sup>2</sup></math-text></inline-eqn>. The lifetime of existing edges depends on the behavior of the individuals connected by this link. Specifically, the rate at which <italic>CC</italic>-links, <italic>CD</italic>-links and <italic>DD</italic>-links disappear is given by <inline-eqn><math-text>γ<sub><italic>CC</italic></sub></math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CD</italic></sub></math-text></inline-eqn> and <inline-eqn><math-text>γ<sub><italic>DD</italic></sub></math-text></inline-eqn>, respectively.</p><p>Consider a network with <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn><italic>C</italic>s and <inline-eqn><math-text><italic>N</italic>−<italic>k</italic></math-text></inline-eqn><italic>D</italic>s. The number of <italic>CC</italic>-links, <italic>CD</italic>-links and <italic>DD</italic>-links in such a network can never exceed <inline-eqn></inline-eqn>, <inline-eqn><math-text><italic>N</italic><sub><italic>CD</italic></sub>≡<italic>k</italic>(<italic>N</italic>−<italic>k</italic>)</math-text></inline-eqn> and <inline-eqn></inline-eqn>, respectively. Under the assumption that the individuals stick to their game behavior, for a large number of links, we can describe the time evolution of the number of links of each type using the following ordinary differential equations,
<display-eqn id="nj368509eqn2" textype="equation" notation="LaTeX" eqnnum="2" lines="multiline"></display-eqn>where <inline-eqn><math-text><italic>L</italic><sub><italic>ij</italic></sub>(<italic>t</italic>)</math-text></inline-eqn> denotes the actual number of links at time <inline-eqn><math-text><italic>t</italic></math-text></inline-eqn> between individuals adopting strategy <inline-eqn><math-text><italic>i</italic></math-text></inline-eqn> and individuals adopting strategy <inline-eqn><math-text><italic>j</italic></math-text></inline-eqn> (<inline-eqn><math-text><italic>i</italic>,<italic>j</italic>∈{<italic>C</italic>,<italic>D</italic>}</math-text></inline-eqn>). In the steady state, the number of links of each type is given by
<display-eqn id="nj368509eqn3" textype="equation" notation="LaTeX" eqnnum="3" lines="multiline"></display-eqn>where <inline-eqn><math-text>&phis;<sub><italic>ij</italic></sub>=α<sup>2</sup>(α<sup>2</sup>+γ<sub><italic>ij</italic></sub>)<sup>−1</sup></math-text></inline-eqn> denotes the fraction of active <italic>ij</italic>-links (<inline-eqn><math-text><italic>i</italic>,<italic>j</italic>∈{<italic>C</italic>,<italic>D</italic>}</math-text></inline-eqn>). Note that the stationary configuration of the network depends on the actual strategy configuration of the population, illustrating the interplay between network evolution and strategy evolution.</p><p>The second dynamical process in our model, the strategy dynamics in finite populations, is defined by the pairwise-comparison rule [<cite linkend="nj368509bib18">18</cite>, <cite linkend="nj368509bib69">69</cite>]. At every strategy update event, a randomly selected individual <inline-eqn><math-text><italic>X</italic></math-text></inline-eqn> imitates a random neighbor <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn> with probability
<display-eqn id="nj368509eqn4" textype="equation" notation="LaTeX" eqnnum="4"></display-eqn>The individual <inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn> can be regarded as the role model of <inline-eqn><math-text><italic>X</italic></math-text></inline-eqn>. <inline-eqn><math-text>Π<sub><italic>X</italic></sub></math-text></inline-eqn> (<inline-eqn><math-text>Π<sub><italic>Y</italic></sub></math-text></inline-eqn>) denotes the total payoff <inline-eqn><math-text><italic>X</italic></math-text></inline-eqn> (<inline-eqn><math-text><italic>Y</italic></math-text></inline-eqn>) receives after interacting once with every neighbor. The parameter <inline-eqn><math-text>β</math-text></inline-eqn> (<inline-eqn><math-text>⩾0</math-text></inline-eqn>) controls the intensity of selection. When <inline-eqn><math-text>β</math-text></inline-eqn> is large, the imitation process is driven mainly by the payoff values that the individuals acquire. The game becomes progressively less important for decreasing <inline-eqn><math-text>β</math-text></inline-eqn>.</p><p>In the following, we assume that the network dynamics proceed much faster than the strategy dynamics (<inline-eqn><math-text>τ<sub><italic>a</italic></sub>≪τ<sub><italic>s</italic></sub></math-text></inline-eqn>). In this limit, the network always reaches a stationary configuration before a strategy update event occurs. The expected payoff of <italic>C</italic>s and <italic>D</italic>s during strategy update events is therefore given by
<display-eqn id="nj368509eqn5" textype="equation" notation="LaTeX" eqnnum="5" lines="multiline"></display-eqn>These payoff values correspond to those obtained in a well-mixed population (complete network) with the same strategy configuration, but using the following rescaled payoff matrix [<cite linkend="nj368509bib55">55</cite>, <cite linkend="nj368509bib56">56</cite>],
<display-eqn id="nj368509eqn6" textype="equation" notation="LaTeX" eqnnum="6"></display-eqn>The network dynamics not only affect the payoffs individuals acquire but also influence the imitation process. Some individuals will act more often as a role model than others, depending on their strategy. In section <secref linkend="nj368509s3">3</secref>, we show that such differences can have a profound effect on the strategy dynamics.</p><p>If <inline-eqn><math-text>τ<sub><italic>a</italic></sub>≪τ<sub><italic>s</italic></sub></math-text></inline-eqn>, a <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn> will select a <inline-eqn><math-text><italic>C</italic></math-text></inline-eqn> as his role model with probability
<display-eqn id="nj368509eqn7" textype="equation" notation="LaTeX" eqnnum="7"></display-eqn>The term in the numerator corresponds to the average number of <italic>CD</italic>-links of each <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>, whereas the term in the denominator reflects the average total number of links of each <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>. The overall probability that the number of <italic>C</italic>s will increase during a strategy update event equals
<display-eqn id="nj368509eqn8" textype="equation" notation="LaTeX" eqnnum="8"></display-eqn>Similarly, the number of <italic>C</italic>s will decrease with probability
<display-eqn id="nj368509eqn9" textype="equation" notation="LaTeX" eqnnum="9"></display-eqn>In well-mixed populations (<inline-eqn><math-text>&phis;<sub><italic>CC</italic></sub>=&phis;<sub><italic>CD</italic></sub>=&phis;<sub><italic>DD</italic></sub>=1</math-text></inline-eqn>), the difference
<display-eqn id="nj368509eqn10" textype="equation" notation="LaTeX" eqnnum="10"></display-eqn>(<inline-eqn><math-text><italic>k</italic>∈]0, <italic>N[</italic></math-text></inline-eqn>) [<cite linkend="nj368509bib69">69</cite>] can be regarded as a finite population analogue of the gradient of selection associated with the replicator equation in infinite, well-mixed populations [<cite linkend="nj368509bib2">2</cite>], which is defined as <inline-eqn></inline-eqn>, where <inline-eqn><math-text><italic>x</italic>∈[0, 1]</math-text></inline-eqn> stands for the fraction of cooperators. In both cases, <italic>C</italic>s are favored over <italic>D</italic>s when <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)>0</math-text></inline-eqn> (<inline-eqn></inline-eqn>), whereas the opposite is true whenever <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)<0</math-text></inline-eqn> (<inline-eqn></inline-eqn>).</p><p>It is noteworthy that we consider the evolutionary dynamics as a discrete stochastic system while assuming that the number of links is sufficiently large, so that the linking dynamics can be described by a set of ordinary differential equations (see equation (<eqnref linkend="nj368509eqn2">2</eqnref>)). Stochastic effects could be included at the level of the linking dynamics as well, for instance by adopting a discrete version of the AL model like the one proposed recently by Wu <italic>et al</italic> [<cite linkend="nj368509bib65">65</cite>]. Nevertheless, the continuous approximation we use here does not weaken the robustness of our conclusions, as shown in the following section by means of computer simulations (see figure <figref linkend="nj368509fig2">2</figref>(d)).</p></sec-level1><sec-level1 id="nj368509s3" label="3"><heading>Results and discussion</heading><p indent="no">We investigate how the gradient of selection depends on the selection pressure. To do so, one can study the shape of <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)</math-text></inline-eqn> or, alternatively, that of the ratio <inline-eqn><math-text><italic>h</italic>(<italic>k</italic>)=<italic>T</italic><sup>+</sup><sub><italic>k</italic></sub>/<italic>T</italic><sup>−</sup><sub><italic>k</italic></sub></math-text></inline-eqn>, which is given by
<display-eqn id="nj368509eqn11" textype="equation" notation="LaTeX" eqnnum="11"></display-eqn>As the solutions of <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)=0</math-text></inline-eqn> are sometimes more conveniently obtained solving <inline-eqn><math-text><italic>h</italic>(<italic>x</italic>)=1</math-text></inline-eqn>, we use both interchangeably. For large <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>h</italic>(<italic>x</italic>)</math-text></inline-eqn> can be approximated by
<display-eqn id="nj368509eqn12" textype="equation" notation="LaTeX" eqnnum="12"></display-eqn>where <inline-eqn></inline-eqn>, <inline-eqn></inline-eqn>, <inline-eqn></inline-eqn>, <inline-eqn><math-text><italic>u</italic>′ =<italic>R</italic>′ −<italic>S</italic>′ −<italic>T</italic>′ +<italic>P</italic>′</math-text></inline-eqn> and <inline-eqn><math-text><italic>v</italic>′ =<italic>S</italic>′ −<italic>P</italic>′</math-text></inline-eqn>.</p><p>Let us start by studying the two limiting cases: the weak selection limit (<inline-eqn><math-text>β→0</math-text></inline-eqn>) and the strong selection limit (<inline-eqn><math-text>β→∞</math-text></inline-eqn>).</p><p>In the limit of strong selection, the direction of the gradient of selection depends solely—apart from finite size effects—on the sign of <inline-eqn><math-text><italic>u</italic>′<italic>x</italic>+<italic>v</italic>′</math-text></inline-eqn> (see appendix <secref linkend="nj368509sA">A</secref>). <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> will therefore exhibit one of four possible shapes (see figure <figref linkend="nj368509fig1">1</figref>), depending on the ordering of the payoff values of the transformed game <inline-eqn><math-text><italic>M</italic>′</math-text></inline-eqn>. When <inline-eqn><math-text><italic>R</italic>′ ><italic>T</italic>′</math-text></inline-eqn> and <inline-eqn><math-text><italic>S</italic>′ ><italic>P</italic>′</math-text></inline-eqn>, <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> is positive for all <inline-eqn><math-text><italic>x</italic>∈]0, 1[</math-text></inline-eqn>. This scenario is known as <inline-eqn><math-text><italic>C</italic></math-text></inline-eqn>-dominance. Selection will always favor <italic>C</italic>s over <italic>D</italic>s, irrespective of the strategy configuration of the population. When <inline-eqn><math-text><italic>R</italic>′ <<italic>T</italic>′</math-text></inline-eqn> and <inline-eqn><math-text><italic>S</italic>′ <<italic>P</italic>′</math-text></inline-eqn>, we obtain the opposite scenario: <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance. The gradient of selection now satisfies <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)<0</math-text></inline-eqn> for all <inline-eqn><math-text><italic>x</italic>∈]0, 1[</math-text></inline-eqn>, implying that <italic>D</italic>s are always favored over <italic>C</italic>s. When <inline-eqn><math-text><italic>R</italic>′ ><italic>T</italic>′</math-text></inline-eqn> and <inline-eqn><math-text><italic>S</italic>′ <<italic>P</italic>′</math-text></inline-eqn>, <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> has a root in
<display-eqn id="nj368509eqn13" textype="equation" notation="LaTeX" eqnnum="13"></display-eqn>Furthermore, <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)<0</math-text></inline-eqn> for <inline-eqn><math-text><italic>x</italic><<italic>x</italic><sup>*</sup></math-text></inline-eqn> and <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)>0</math-text></inline-eqn> for <inline-eqn><math-text><italic>x</italic>><italic>x</italic><sup>*</sup></math-text></inline-eqn>. When the initial fraction of <italic>C</italic>s is smaller than <inline-eqn><math-text><italic>x</italic><sup>*</sup></math-text></inline-eqn>, evolution favors <italic>C</italic>s. Otherwise, <italic>D</italic>s will be favored. This is an example of coordination or bistability. Finally, when <inline-eqn><math-text><italic>R</italic>′ <<italic>T</italic>′</math-text></inline-eqn> and <inline-eqn><math-text><italic>S</italic>′ ><italic>P</italic>′</math-text></inline-eqn>, evolution favors a mixture of <italic>C</italic>s and <italic>D</italic>s. The gradient <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> has a root in <inline-eqn><math-text><italic>x</italic><sup>*</sup></math-text></inline-eqn> again, but unlike before, selection favors <italic>C</italic>s for <inline-eqn><math-text><italic>x</italic><<italic>x</italic><sup>*</sup></math-text></inline-eqn> and <italic>D</italic>s for <inline-eqn><math-text><italic>x</italic>><italic>x</italic><sup>*</sup></math-text></inline-eqn>. Note that one has to correct for self-interactions when computing the finite population analogue of the equilibrium point <inline-eqn><math-text><italic>x</italic><sup>*</sup></math-text></inline-eqn>. The resulting equilibrium <inline-eqn><math-text><italic>k</italic><sup>*</sup></math-text></inline-eqn> is given by
<display-eqn id="nj368509eqn14" textype="equation" notation="LaTeX" eqnnum="14"></display-eqn></p><figure id="nj368509fig1" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.8pc" printcolour="no" filename="images/nj368509fig1.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj368509fig1.jpg"></graphic-file></graphic><caption type="figure" id="nj368509fc1" label="Figure 1"><p indent="no">The four possible dynamical scenarios defined by two-person symmetric games in well-mixed populations. In each of the four panels, the curve represents a typical shape of the fitness difference <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> between <italic>C</italic>s and <italic>D</italic>s for a given fraction <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn> of <italic>C</italic>s. The roots of <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> are the fixed points of the evolutionary dynamics. Stable fixed points are depicted using solid circles and unstable fixed points using open circles. Arrows indicate the expected direction of evolution. <italic>C</italic>s (<italic>D</italic>s) are favored over <italic>D</italic>s (<italic>C</italic>s) when the arrow points to the right (left). The particular shape of <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> can be inferred from the sign of the derivative of <inline-eqn><math-text><italic>g</italic></math-text></inline-eqn> in 0 and in 1.</p></caption></figure><p>In the other limit, that of weak selection, we have
<display-eqn id="nj368509eqn15" textype="equation" notation="LaTeX" eqnnum="15"></display-eqn>Because this function is monotonic in <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>, <inline-eqn><math-text><italic>h</italic>(<italic>x</italic>)=1</math-text></inline-eqn> can have at most one solution and consequently the gradient <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> can have at most one root in <inline-eqn><math-text>]0, 1[</math-text></inline-eqn>. Hence, <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> exhibits one of the four shapes shown in figure <figref linkend="nj368509fig1">1</figref>. The particular type of shape that occurs depends on the parameters <inline-eqn><math-text><italic>a</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>b</italic></math-text></inline-eqn>.</p><p>In the following, we assume that links between <italic>C</italic>s satisfy both individuals and therefore last longer than other links. Two interacting <italic>D</italic>s, on the other hand, are both dissatisfied and prefer to break their connection quickly. Following the same reasoning, <italic>C</italic>s would like to break <italic>CD</italic>-links quickly, whereas <italic>D</italic>s prefer to keep such links as long as possible. Hence, on average, <italic>CD</italic>-links will have a longer time span than <italic>DD</italic>-links, but a shorter one than <italic>CC</italic>-links. Altogether, we obtain the following ordering for the fractions of active links of each type,
<display-eqn id="nj368509eqn16" textype="equation" notation="LaTeX" eqnnum="16"></display-eqn>which is equivalent to
<display-eqn id="nj368509eqn17" textype="equation" notation="LaTeX" eqnnum="17"></display-eqn>This condition ensures that <inline-eqn><math-text><italic>h</italic>(<italic>x</italic>)>1</math-text></inline-eqn> for all <inline-eqn><math-text><italic>x</italic>∈]0, 1[</math-text></inline-eqn>. Hence, <italic>C</italic>s are dominant in the limit of weak selection, irrespective of the game being played.</p><p>Having considered the two limiting cases, we now study the gradient of selection for general values of <inline-eqn><math-text>β</math-text></inline-eqn>. In appendix <secref linkend="nj368509sB">B</secref>, we show that <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> can have either zero, one or two internal equilibria, depending on the parameter settings. First we discuss examples where the specific intensity of selection leads to a scenario with at most one internal equilibrium. Next we will see that some parameter combinations lead to a scenario with two internal equilibria, which can never occur in the limits of either weak or strong selection.</p><p>We reduce the complexity of the game space by normalizing the difference between <inline-eqn><math-text><italic>R</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic></math-text></inline-eqn> to 1, taking <inline-eqn><math-text><italic>R</italic>=2</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>. The contours in figure <figref linkend="nj368509fig2">2</figref> depict the sign of <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)</math-text></inline-eqn> for three different <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance games, using <inline-eqn><math-text>α=0.8</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CC</italic></sub>=0.4</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CD</italic></sub>=0.5</math-text></inline-eqn> and <inline-eqn><math-text>γ<sub><italic>DD</italic></sub>=0.6</math-text></inline-eqn>. In the weak selection limit, these birth/death rates of links lead to dominance of <italic>C</italic>s, irrespective of the original payoff matrix. The strong selection limit leads to different scenarios, depending on the original game. For instance, <inline-eqn><math-text><italic>S</italic>=0.5</math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic>=2.1</math-text></inline-eqn> results in bistability (see figure <figref linkend="nj368509fig2">2</figref>(a), <inline-eqn><math-text><italic>S</italic>=0.9</math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic>=2.5</math-text></inline-eqn> in coexistence of <italic>C</italic>s and <italic>D</italic>s (see figure <figref linkend="nj368509fig2">2</figref>(b)), and <inline-eqn><math-text><italic>S</italic>=0.5</math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic>=2.5</math-text></inline-eqn> in <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance (see figure <figref linkend="nj368509fig2">2</figref>(c)). Hence, there exist critical values of <inline-eqn><math-text>β</math-text></inline-eqn> at which the number of internal equilibria changes. We can compute these critical values using the derivative of <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> in 0 and in 1 (see also figure <figref linkend="nj368509fig1">1</figref>). These derivatives are given by
<display-eqn id="nj368509eqn18" textype="equation" notation="LaTeX" eqnnum="18"></display-eqn><display-eqn id="nj368509eqn19" textype="equation" notation="LaTeX" eqnnum="19"></display-eqn>Assuming at most one internal equilibrium, we obtain <inline-eqn><math-text><italic>C</italic></math-text></inline-eqn>-dominance (<inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance) when both <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn> are larger (smaller) than zero. Bistability occurs when <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn>, and coexistence when <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn>. From equation (<eqnref linkend="nj368509eqn18">18</eqnref>), it follows that <inline-eqn></inline-eqn> if and only if
<display-eqn id="nj368509eqn20" textype="equation" notation="LaTeX" eqnnum="20"></display-eqn>and that <inline-eqn></inline-eqn> if and only if
<display-eqn id="nj368509eqn21" textype="equation" notation="LaTeX" eqnnum="21"></display-eqn>This means that <inline-eqn></inline-eqn> will be positive for <inline-eqn><math-text>β</math-text></inline-eqn> smaller than
<display-eqn id="nj368509eqn22" textype="equation" notation="LaTeX" eqnnum="22"></display-eqn>and negative otherwise. Similarly, <inline-eqn></inline-eqn> is positive for <inline-eqn><math-text>β</math-text></inline-eqn> smaller than
<display-eqn id="nj368509eqn23" textype="equation" notation="LaTeX" eqnnum="23"></display-eqn>and negative otherwise. Both <inline-eqn><math-text>β<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>β<sub>1</sub></math-text></inline-eqn> are indicated by vertical dotted lines in figure <figref linkend="nj368509fig2">2</figref>. They correspond clearly to the values of <inline-eqn><math-text>β</math-text></inline-eqn> at which each of the game transitions takes place.</p><figure id="nj368509fig2" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="23.2pc" printcolour="no" filename="images/nj368509fig2.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="no" filename="images/nj368509fig2.jpg"></graphic-file></graphic><caption type="figure" id="nj368509fc2" label="Figure 2"><p indent="no">The intensity of selection determines the shape of the gradient of selection. (a–c) Each of the contour plots shows the sign of the fitness difference <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)</math-text></inline-eqn>, <inline-eqn><math-text><italic>k</italic></math-text></inline-eqn> being the number of <italic>C</italic>s in a network of size <inline-eqn><math-text><italic>N</italic></math-text></inline-eqn>, as a function of <inline-eqn><math-text>β</math-text></inline-eqn>. Regions where <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)>0</math-text></inline-eqn> (<inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)<0</math-text></inline-eqn>) are indicated in gray (black). Solid white lines indicate where <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)=0</math-text></inline-eqn>. In each of the three panels, the individuals engage in a different <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance game: panel (a) shows <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)</math-text></inline-eqn> for <inline-eqn><math-text><italic>S</italic>=0.5</math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic>=2.1</math-text></inline-eqn>, panel (b) for <inline-eqn><math-text><italic>S</italic>=0.9</math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic>=2.5</math-text></inline-eqn> and, finally, panel (c) for <inline-eqn><math-text><italic>S</italic>=0.5</math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic>=2.5</math-text></inline-eqn>. The other two payoff values are normalized to <inline-eqn><math-text><italic>R</italic>=2</math-text></inline-eqn> and <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>. The network dynamics transform the game so that for <inline-eqn><math-text>β→∞</math-text></inline-eqn>, we obtain an unstable equilbrium <inline-eqn><math-text><italic>k</italic><sup>*</sup></math-text></inline-eqn> in panel (a), a stable equilibrium <inline-eqn><math-text><italic>k</italic><sup>*</sup></math-text></inline-eqn> in panel (b) and a <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance situation in panel (c). Reducing <inline-eqn><math-text>β</math-text></inline-eqn> eventually makes cooperation dominant, irrespective of the original game being played. The dotted vertical lines, indicated by <inline-eqn><math-text>β<sub>0</sub></math-text></inline-eqn> and <inline-eqn><math-text>β<sub>1</sub></math-text></inline-eqn>, show the analytical predictions for <inline-eqn><math-text>β</math-text></inline-eqn> at which the sign changes of <inline-eqn></inline-eqn> and <inline-eqn></inline-eqn>, respectively. (d) Circles (lines) show the probability, obtained in simulation (analytically), to reach <inline-eqn><math-text>100%</math-text></inline-eqn> cooperation, starting from a population with <inline-eqn><math-text>50%</math-text></inline-eqn><italic>C</italic>'s. We use <inline-eqn><math-text>τ=10<sup>−3</sup></math-text></inline-eqn> as the relative time scale for network dynamics. Individuals engage in the same game as in panel (c). In all four panels, we use <inline-eqn><math-text>γ<sub><italic>CC</italic></sub>=0.4</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CD</italic></sub>=0.5</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>DD</italic></sub>=0.6</math-text></inline-eqn>, <inline-eqn><math-text>α=0.8</math-text></inline-eqn> and <inline-eqn><math-text><italic>N</italic>=100</math-text></inline-eqn>.</p></caption></figure><p>We further confirm the validity of the analytical results by means of computer simulations. Figure <figref linkend="nj368509fig2">2</figref>(d) shows the probability, as a function of <inline-eqn><math-text>β</math-text></inline-eqn>, that a population with initially <inline-eqn><math-text>50%</math-text></inline-eqn> cooperators evolves to full cooperation. Circles, representing simulation results, fit nicely with the analytical predictions, which are indicated by a solid line. Each simulation starts with a complete network of size <inline-eqn><math-text><italic>N</italic>=100</math-text></inline-eqn>, <inline-eqn><math-text>50%</math-text></inline-eqn> of them being <italic>C</italic>s, and runs until the population fixates into either full cooperation or full defection. Strategy and network structure evolve simultaneously under asynchronous updating. Strategy update events take place with probability <inline-eqn><math-text>(1+τ)<sup>−1</sup></math-text></inline-eqn>, where <inline-eqn></inline-eqn>; network update events occur otherwise. We run <inline-eqn><math-text>10<sup>4</sup></math-text></inline-eqn> simulations for each value of <inline-eqn><math-text>β</math-text></inline-eqn> and plot the fraction of runs that end in full cooperation. The analytical predictions are calculated using the following formula [<cite linkend="nj368509bib69">69</cite>],
<display-eqn id="nj368509eqn24" textype="equation" notation="LaTeX" eqnnum="24"></display-eqn>with <inline-eqn><math-text><italic>i</italic></math-text></inline-eqn> being equal to 50. The expressions for <inline-eqn><math-text><italic>T</italic><sub><italic>j</italic></sub><sup>+</sup></math-text></inline-eqn> and <inline-eqn><math-text><italic>T</italic><sub><italic>j</italic></sub><sup>−</sup></math-text></inline-eqn> are given by equations (<eqnref linkend="nj368509eqn8">8</eqnref>) and (<eqnref linkend="nj368509eqn9">9</eqnref>).</p><p>Earlier in this section, we showed that there can be at most one internal equilibrium for the limiting cases <inline-eqn><math-text>β→0</math-text></inline-eqn> and <inline-eqn><math-text>β→∞</math-text></inline-eqn>. At intermediate intensities of selection, two internal equilibria can, however, occur simultaneously (see also appendix <secref linkend="nj368509sB">B</secref>). Figure <figref linkend="nj368509fig3">3</figref> shows two examples where this is the case. The upper two panels correspond to a scenario where <italic>C</italic>s are expected to disappear from the population in case they are rare. If a sufficiently large fraction of <italic>C</italic>s is present, however, a mixture of <italic>C</italic>s and <italic>D</italic>s is favored. The lower two panels illustrate an example of the opposite scenario, in which <italic>D</italic>s are expected to go extinct when they are rare.</p><figure id="nj368509fig3" parts="single" width="column" position="float" pageposition="top" printstyle="normal" orientation="port"><graphic position="indented"><graphic-file version="print" format="EPS" width="31.8pc" printcolour="no" filename="images/nj368509fig3.eps"></graphic-file><graphic-file version="ej" format="JPEG" printcolour="yes" filename="images/nj368509fig3.jpg"></graphic-file></graphic><caption type="figure" id="nj368509fc3" label="Figure 3"><p indent="no">Two internal equilibria at intermediate intensities of selection. The contour plots show the sign of the fitness difference <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)</math-text></inline-eqn>, like in figure <figref linkend="nj368509fig2">2</figref>. In (a), we use the parameters <inline-eqn><math-text><italic>R</italic>=2</math-text></inline-eqn>, <inline-eqn><math-text><italic>S</italic>=−9</math-text></inline-eqn>, <inline-eqn><math-text><italic>T</italic>=40</math-text></inline-eqn>, <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>, <inline-eqn><math-text>α=0.2</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CC</italic></sub>=0.04</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CD</italic></sub>=0.46</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>DD</italic></sub>=0.85</math-text></inline-eqn> and <inline-eqn><math-text><italic>N</italic>=100</math-text></inline-eqn>. Two internal equilibria take place at intermediate intensities of selection. The equilibrium closest to full defection is unstable and the other one is stable. This is emphasized in (b), where we zoom in on the actual gradient <inline-eqn><math-text><italic>g</italic>(<italic>k</italic>)</math-text></inline-eqn> for <inline-eqn><math-text>β=1.0</math-text></inline-eqn>. Panels (c) and (d) show the same plots, but using <inline-eqn><math-text><italic>R</italic>=2</math-text></inline-eqn>, <inline-eqn><math-text><italic>S</italic>=−8</math-text></inline-eqn>, <inline-eqn><math-text><italic>T</italic>=3</math-text></inline-eqn>, <inline-eqn><math-text><italic>P</italic>=1</math-text></inline-eqn>, <inline-eqn><math-text>α=0.02</math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CC</italic></sub>=10<sup>−5</sup></math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>CD</italic></sub>=6*10<sup>−4</sup></math-text></inline-eqn>, <inline-eqn><math-text>γ<sub><italic>DD</italic></sub>=0.9</math-text></inline-eqn> and <inline-eqn><math-text><italic>N</italic>=100</math-text></inline-eqn>. The equilibrium closest to full defection is now stable, while the other one is unstable.</p></caption></figure></sec-level1><sec-level1 id="nj368509s4" label="4"><heading>Conclusion</heading><p indent="no">In this paper, we have explored, both analytically and numerically, the evolution of cooperation in dynamical networks that evolve side by side with individuals' behavior, the evolution of the network structure being affected by the dynamics of the individuals and vice versa. We indicate that the adaptive nature of the network of contacts affects both the average payoff associated with each game strategy and the likelihood for each of the different strategies to serve as a role model. The intensity of selection, which controls the contribution of game payoff to fitness, regulates the importance of each of these two effects in the final outcome of strategy evolution. When the intensity of selection is strong, the payoff transformation resulting from the network dynamics provides the dominating contribution to evolution. Weakening the intensity of selection enhances the effect of the adaptive imitation network. By doing so, one is able to effectively transform a <inline-eqn><math-text><italic>D</italic></math-text></inline-eqn>-dominance dilemma into any of the conventional <inline-eqn><math-text>2×2</math-text></inline-eqn> symmetric games. We derive analytical conditions, valid under the assumption that there is only one internal equilibrium, that predict which range of intensities of selection leads to which game scenario. We show also that certain parameter combinations may lead to the occurrence of two internal equilibria at intermediate intensities of selection. Our results clearly demonstrate the relevance of the intensity of selection when evolution proceeds on an adaptive structured population. Furthermore, we indicate the conditions under which this coupled dynamics work as an efficient mechanism for the promotion of cooperative behavior.</p></sec-level1><acknowledgment><heading>Acknowledgments</heading><p indent="no">SVS and TL acknowledge financial support from F.R.S.-FNRS (Belgium). JMP and FCS acknowledge financial support from FCT (Portugal).</p></acknowledgment><appendix id="nj368509a1"><sec-level1 id="nj368509sA" type="num" appendix="yes" style="ALPHA" label="Appendix A"><heading>The strong selection limit</heading><p indent="no">The following inequality holds,
<display-eqn id="nj368509eqnA.1" textype="equation" notation="LaTeX" eqnnum="A.1"></display-eqn><display-eqn id="nj368509eqnA.2" textype="equation" notation="LaTeX" eqnnum="A.2"></display-eqn>where <inline-eqn><math-text><italic>K</italic>>0</math-text></inline-eqn> is a constant, whose specific value depends on the parameters <inline-eqn><math-text><italic>a</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>b</italic></math-text></inline-eqn>. Therefore,
<display-eqn id="nj368509eqnA.3" textype="equation" notation="LaTeX" eqnnum="A.3" lines="multiline"></display-eqn>Hence, the direction of the gradient of selection depends solely on the sign of <inline-eqn><math-text><italic>u</italic>′<italic>x</italic>+<italic>v</italic>′</math-text></inline-eqn>. <italic>C</italic>s are favored over <italic>D</italic>s if and only if <inline-eqn><math-text><italic>u</italic>′<italic>x</italic>+<italic>v</italic>′ >0</math-text></inline-eqn>.</p></sec-level1><sec-level1 id="nj368509sB" type="num" appendix="yes" style="ALPHA" label="Appendix B"><heading>General intensities of selection</heading><p indent="no">Let us designate the rational factor in equation (<eqnref linkend="nj368509eqn12">12</eqnref>) by <inline-eqn><math-text><italic>h</italic><sub>1</sub>(<italic>x</italic>)</math-text></inline-eqn> and the exponential factor by <inline-eqn><math-text><italic>h</italic><sub>2</sub>(<italic>x</italic>)</math-text></inline-eqn>. Both <inline-eqn><math-text><italic>h</italic><sub>1</sub>(<italic>x</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>h</italic><sub>2</sub>(<italic>x</italic>)</math-text></inline-eqn> are monotonic in <inline-eqn><math-text><italic>x</italic></math-text></inline-eqn>: <itemized-list id="list1" type="bullet" style="hanging"><list-item id="list1-i1"><p>The derivative <inline-eqn></inline-eqn>is given by <inline-eqn></inline-eqn>. Hence, <inline-eqn><math-text><italic>h</italic><sub>1</sub>(<italic>x</italic>)</math-text></inline-eqn> increases monotonically if <inline-eqn><math-text><italic>b</italic>><italic>a</italic><sup>2</sup></math-text></inline-eqn> and decreases monotonically if <inline-eqn><math-text><italic>b</italic><<italic>a</italic><sup>2</sup></math-text></inline-eqn>.</p></list-item><list-item id="list1-i2"><p>The derivative <inline-eqn></inline-eqn>is given by <inline-eqn><math-text><italic>u</italic>′ <upright>e</upright><sup><italic>u</italic>′<italic>x</italic>+<italic>v</italic>′</sup></math-text></inline-eqn>. Hence, <inline-eqn><math-text><italic>h</italic><sub>2</sub>(<italic>x</italic>)</math-text></inline-eqn> increases monotonically if <inline-eqn><math-text><italic>u</italic>′ >0</math-text></inline-eqn> and decreases monotonically if <inline-eqn><math-text><italic>u</italic>′ <0</math-text></inline-eqn>.</p></list-item></itemized-list><inline-eqn><math-text><italic>h</italic>(<italic>x</italic>)</math-text></inline-eqn> increases (decreases) monotonically in case both <inline-eqn><math-text><italic>h</italic><sub>1</sub>(<italic>x</italic>)</math-text></inline-eqn> and <inline-eqn><math-text><italic>h</italic><sub>2</sub>(<italic>x</italic>)</math-text></inline-eqn> increase (decrease) monotonically. Hence, if <inline-eqn><math-text><italic>a</italic><sup>2</sup><<italic>b</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>u</italic>′ >0</math-text></inline-eqn> (or <inline-eqn><math-text><italic>a</italic><sup>2</sup>><italic>b</italic></math-text></inline-eqn> and <inline-eqn><math-text><italic>u</italic>′ <0</math-text></inline-eqn>), then <inline-eqn><math-text><italic>g</italic>(<italic>x</italic>)</math-text></inline-eqn> can have at most one root in <inline-eqn><math-text>]0, 1[</math-text></inline-eqn>. 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Biol.</jnl-title><volume>6</volume><pages>e1000895</pages></journal-ref></reference-list></references></back></article></istex:document></istex:metadataXml><mods version="3.6"><titleInfo lang="eng"><title>Selection pressure transforms the nature of social dilemmas in adaptive networks</title></titleInfo><titleInfo type="abbreviated"><title>Selection pressure transforms the nature of social dilemmas in adaptive networks</title></titleInfo><titleInfo type="alternative" lang="eng"><title>Selection pressure transforms the nature of social dilemmas in adaptive networks</title></titleInfo><name type="personal"><namePart type="given">Sven Van</namePart><namePart type="family">Segbroeck</namePart><affiliation>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</affiliation><affiliation>COMO, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</affiliation><affiliation>Author to whom any correspondence should be addressed.</affiliation><affiliation>E-mail: svsegbro@ulb.ac.be</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Francisco C</namePart><namePart type="family">Santos</namePart><affiliation>CENTRIA, Departamento de Informtica, Faculdade de Cincias e Tecnologia, Universidade Nova de Lisboa, 2829-516 Caparica, Portugal</affiliation><affiliation>ATP-Group, CMAF, Complexo Interdisciplinar, P-1649-003 Lisboa Codex, Portugal</affiliation><affiliation>E-mail: fcsantos@fct.unl.pt</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Tom</namePart><namePart type="family">Lenaerts</namePart><affiliation>MLG, Universit Libre de Bruxelles, Boulevard du TriompheCP 212, 1050 Brussels, Belgium</affiliation><affiliation>Department of Computer Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050 Brussels, Belgium</affiliation><affiliation>E-mail: tlenaert@ulb.ac.be</affiliation><role><roleTerm type="text">author</roleTerm></role></name><name type="personal"><namePart type="given">Jorge M</namePart><namePart type="family">Pacheco</namePart><affiliation>ATP-Group, CMAF, Complexo Interdisciplinar, P-1649-003 Lisboa Codex, Portugal</affiliation><affiliation>Departamento de Matemtica e Aplicaes, Universidade do Minho, Campus de Gualtar, 4710-057 Braga, Portugal</affiliation><affiliation>E-mail: jmpacheco@math.uminho.pt</affiliation><role><roleTerm type="text">author</roleTerm></role></name><typeOfResource>text</typeOfResource><genre type="research-article" displayLabel="paper">paper</genre><originInfo><publisher>IOP Publishing</publisher><dateIssued encoding="w3cdtf">2011</dateIssued><copyrightDate encoding="w3cdtf">2011</copyrightDate></originInfo><language><languageTerm type="code" authority="iso639-2b">eng</languageTerm><languageTerm type="code" authority="rfc3066">en</languageTerm></language><physicalDescription><internetMediaType>text/html</internetMediaType></physicalDescription><abstract>We have studied the evolution of cooperation in structured populations whose topology coevolves with the game strategies of the individuals. Strategy evolution proceeds according to an update rule with a free parameter, which measures the selection pressure. We explore how this parameter affects the interplay between network dynamics and strategy dynamics. A dynamical network topology can influence the strategy dynamics in two ways: (i) by modifying the expected payoff associated with each strategy and (ii) by reshaping the imitation network that underlies the evolutionary process. We show here that the selection pressure tunes the relative contribution of each of these two forces to the final outcome of strategy evolution. The dynamics of the imitation network plays only a minor role under strong selection, but becomes the dominant force under weak selection. We demonstrate how these findings constitute a mechanism supporting cooperative behavior.</abstract><relatedItem type="host"><titleInfo><title>New Journal of Physics</title></titleInfo><titleInfo type="abbreviated"><title>New J. Phys.</title></titleInfo><genre type="journal">journal</genre><identifier type="ISSN">1367-2630</identifier><identifier type="eISSN">1367-2630</identifier><identifier type="PublisherID">nj</identifier><identifier type="CODEN">NJOPFM</identifier><identifier type="URL">stacks.iop.org/NJP</identifier><part><date>2011</date><detail type="volume"><caption>vol.</caption><number>13</number></detail><detail type="issue"><caption>no.</caption><number>1</number></detail><extent unit="pages"><start>1</start><end>13</end><total>13</total></extent></part></relatedItem><identifier type="istex">643BCB406A6541BD7C697A56A11A105AB6542273</identifier><identifier type="DOI">10.1088/1367-2630/13/1/013007</identifier><identifier type="articleID">368509</identifier><identifier type="articleNumber">013007</identifier><accessCondition type="use and reproduction" contentType="copyright">IOP Publishing and Deutsche Physikalische Gesellschaft</accessCondition><recordInfo><recordContentSource>IOP</recordContentSource><recordOrigin>IOP Publishing and Deutsche Physikalische Gesellschaft</recordOrigin></recordInfo></mods></metadata><enrichments><json:item><type>multicat</type><uri>https://api.istex.fr/document/643BCB406A6541BD7C697A56A11A105AB6542273/enrichments/multicat</uri></json:item></enrichments><serie></serie></istex></record>Pour manipuler ce document sous Unix (Dilib)
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