Figure 6
Optimal pinning fraction.
(a) Intersections of the curves 
and 
denote nonzero optimal pinning fraction 
given by Eq. (18b). The scale-free networks have the degree exponents 
, 2.5, 2.7 and 3.0, respectively. The response function is for 
(corresponding to 
. (b) Contour map of 
in the parameter space of 
and 
for scale-free networks with 
. In the lower-left region below the boundary 
(white dashed line), nonzero solution of 
cannot be obtained. (c) Optimal pinning fraction 
as a function of 
for scale-free networks. The analytical results from Eq. (18b) (red solid curve) and the simulation results (black open squares) agree well with each other. The red arrow marks the theoretical prediction of the boundary, where nonzero 
solutions exist on the left side. (d) For ER random networks, 
as a function of 
. Theoretical results from Eq. (18b) (red open circle) and simulation results (black open squares) are shown. The boundaries 1 and 2 obtained theoretically (pointed to by solid arrows), respectively, stand for the constraint in Eqs. (19) and (22). In (c,d), the value of 
varies but 
is set to 0.9. The scale-free and random networks used in the simulations have 
and 
.
