Figure 2

Flow and dynamics. The value of the derivative \({\dot{x}}_{v}\) is plotted as a function of \({x}_{v}\) and \({\beta }_{v}\), with \({k}_{v}=10\), together with attractive (black) and repulsive (white) steady states. For a T-driven game (\(T=3\), \(S=-1\) and \(\rho =2\)), the time derivatives of \({x}_{v}\) for a generic player connected only to full defectors (\({\bar{x}}_{v}=0\)) and only to full cooperators (\({\bar{x}}_{v}=1\)) are shown in (A.1,B.1), respectively. Similarly, (C.1,D.1) show the time derivatives of \({x}_{v}\) for a S-driven game (\(T=2\), \(S=-2\) and \(\rho =2\)), assuming a neighborhood of full defectors and full cooperators, respectively. Vertical dashed lines are drawn for \({\beta }_{v}={k}_{v}/\rho \) and \({\beta }_{v}={k}_{v}\rho \), thus separating the regions \({\mathscr{D}}\), \({\mathscr{U}}\) and \({\mathscr{C}}\). Some examples of the time courses of \({x}_{v}(t)\) (red) and of \({\bar{x}}_{v}(t)\) (blue) for a player \(v\in {\mathscr{U}}\) are depicted in A.2, B.2, C.2 and D.2 for \({\beta }_{v} < {k}_{v}\), and in A.3, B.3, C.3 and D.3 for \({\beta }_{v} > {k}_{v}\).