Invasion conditions for a hub-cooperator.

<p>From the definitions of the parameters in the figure one obtains that DPD leads systematically to less stringent conditions for invasion of the <i>C</i> (squares) occupying the left hub, explaining the increased success of Cs under DPD. On general heterogeneous populations with average connectivity <i>z₀</i>, conditions a) and b), as well as c) and d), show that it is easier to invade a <i>D</i> (circles) on a leaf than in the center of another hub. This invasion creates a positive feedback resulting from cooperative “leaves” surrounding the left hub (<i>k₁ - k₂</i> increases) allowing a subsequent invasion of the right hub.</p

Fraction of Cooperators as a function of the average degree <i>z</i> of the social network.

<p>Cooperation is able to dominate on sparse networks. Yet, only under <b>DPD</b>, combined with high levels of heterogeneity of Scale-free networks, one observes the maintenance of cooperative behavior in highly connected populations. The results were obtained for networks of <i>10³</i> nodes and <i>F = 1.8</i>.</p

Fraction of Cooperators as a function of the enhancement factor <i>F</i>.

<p>Upper panel: Under CPD Cooperation is able to dominate on Scale-free networks (lines and circles), unlike what happens on regular structures (lines and filled squares). On exponential networks, intermediate levels of cooperation emerge, as a result of the heterogeneity of such topologies. Lower panel: Under DPD the advantage of Cs is dramatically enhanced when the same cost is evenly shared among each one's neighbors. The results were obtained for networks of <i>10³</i> nodes and an average degree <i>z = 4</i>. As expected, abandoning the well-mixed regime leads to a break-up of neutrality for <i>F = 2</i>.</p

Time-independent AGoS.

<p>(<b>a</b>) We plot <i>G^A(j)</i> for a population of players interacting via a <b>PD</b> in a homogeneous random network, for two values of the benefit <b>B</b>. Globally, <i>G^A(j)</i> indicates that the population evolves towards a co-existence scenario. (<b>b</b>) Stationary distributions showing the pervasiveness of each fraction <i>j/N</i> in time. In line with the <b><i>AGoS</i></b> in <b>a</b>), the population spends most of the time in the vicinity of the stable-like root <i>x_R</i> of <i>G^A(j)</i>. When <i>j/N≈0</i>, <b><i>C</i></b>s become disadvantageous, giving rise to an unstable-like root <i>x_L</i> of <i>G^A(j)</i> which, however, plays a minor role as shown (<b><i>N</i></b><i> = 10³</i>, <i> = 4</i> and β = <i>1.0</i>). Homogeneous random networks were obtained by repeatedly swapping the ends of pairs of randomly chosen links of a regular lattice <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0032114#pone.0032114-Santos5" target="_blank">[54]</a>.</p

Gradients of selection <i>G(x)</i>.

<p>a) Under the CPD paradigm, Scale-free networks lead to the appearance of an unstable equilibrium <i>x*</i> (open circles) and a scenario characteristic of a coordination game, paving the way for cooperator dominance for frequencies above <i>x*</i>. b) Under DPD, <i>G(x)</i> becomes positive for (almost) all values of <i>x</i> (<i>x*<0.004</i> for <i>F = 1.50</i> and <i>x*<0.006</i> for <i>F = 1.25</i>), leading to a scenario characteristic of a Harmony game, where cooperators dominate unconditionally. In both panels the networks employed had <i>500</i> nodes and an average degree <i>z = 4</i>, whereas <i>β</i> = 10.0.</p

Time-dependent AGoS.

<p>(<b>a</b>) We plot <i>G^A(j,t)</i> for three different instants of evolutionary time. Each line provides a snapshot for a given moment, portraying the emergence of a population-wide (time-dependent) co-existence-like dilemma stemming from an individual (time-independent) defection dominant dilemma (<b>PD</b>). (<b>b</b>) The circles show the position of the different interior roots of <i>G^A(j,t)</i>, whereas the solid (dark blue) line and (light blue) crosses show two independent evolutionary runs starting from <i>50%</i> of <b><i>C</i></b>s and <b><i>D</i></b>s randomly placed in the networked population. Open (full) circles stand for unstable, <i>x_L</i> (stable, <i>x_R</i>) roots of <i>G^A(j,t)</i> (<b><i>B</i></b><i> = 1.01</i>, <b><i>N</i></b><i> = 10³</i>, <i> = 4</i> and <i>β = 10.0</i>).</p

The overall impact on cooperation for different <i>p</i>, <i>q</i> parameters.

<p>The dashed line indicates cost-over-punishment ratio values for which overall cooperation remains unaffected by altruistic punishment (<i>Ω</i> = 0.49). The area below (above) the dashed line comprises the parameter region with enhanced (inhibited) cooperation. Black solid circles identify the parameter values used in the plots in <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002868#pcbi-1002868-g001" target="_blank">Figures 1</a> and <a href="http://www.ploscompbiol.org/article/info:doi/10.1371/journal.pcbi.1002868#pcbi-1002868-g003" target="_blank">3</a>. Altruistic punishment (<i>p>q</i>) occurs below the solid line.</p

The effect of different <i>p</i>, <i>q</i> parameters on cooperation on the <i>T - S</i> plane.

<p>Upper panel (<i>q = 0</i>, <i>p = 0.1</i>): the introduction of punishment made cooperation dominant for an additional region (red shaded area). Middle panel (<i>q = 0.1</i>, <i>p = 0.1</i>): the ‘unsuitable’ use of punishment made a big domain of the <i>T - S</i> plane inaccessible for cooperation (blue shaded area). Lower panel (<i>q = 0.04</i>, <i>p = 0.1</i>): the effects of punishment are ambiguous: there are (<i>T, S</i>) values for which punishment enhances the overall cooperation (red shaded area); however on other areas it hinders it (blue shaded area located between vertical dashed lines).</p

Stationary bit distribution as a function of <i>N</i>.

<p>Each bit (square) corresponds to the weighted sum of the fraction of time (<i>i.e.</i> the analytically computed stationary distribution) the population spends in strategy configurations in which <i>b^q</i> = 1. Blue (red) cells identify those bits that are employed at least ¾ of the time with value <i>b^q</i> = 1.0 (<i>b^q</i> = 0.0). The analysis provided extends for groups sizes (<i>N</i>) between 2 and 10 (rows). Other model parameters: <i>Z</i> = 100, <i>β</i> = 1.0, <i>F/N</i> = 0.85, <i>w</i> = 0.96, <i>ε</i> = 0.05, <i>μ</i>≪1/<i>Z</i>.</p

Slope <i>σ</i> of the tangent to the edge-curve <i>λ</i> (see Figure 1) as a function of the temptation <i>T</i> along the 50% cooperation curve (solid line, see main text for details).

<p>Blue dashed lines show the upper and lower bounds of the slope function.</p