Let's analyze each statement about the link$-$copy model:

When $$ approaches $1,$ the network has a degree distribution similar to that of G$($n$,$p$)$ networks:

$[$False$]$

When p approaches $1$ in the link$-$copy model, almost all links are copied from the target node. This leads to a highly clustered network with a power$-$law degree distribution, which is very different from the Poisson degree distribution of G$($n$,$p$)$ random graphs.

When $=1/2,$ the link$-$copy network is identical to the network generated by a PA model:

$[$False$]$

While p$=1/2$ in the link$-$copy model does create preferential attachment$-$like behavior, it's not identical to the PA model. The link$-$copy model still has a copying mechanism that creates more clustering than a pure PA model.

When $$ approaches $0,$ the network has a power$-$law distribution with exponent $2$:

$[$True$]$

As q approaches $0,$ almost all new edges are formed by copying, which leads to a power$-$law degree distribution with exponent $2.$ This is a well$-$known result for the link$-$copy model.

When $=1/2,$ the network has a power$-$law distribution with exponent $3$:

$[$False$]$

While the link$-$copy model does produce power$-$law degree distributions, the exponent is not always $3$ when q$=1/2.$ The exact exponent depends on both p and q$,$ and q$=1/2$ alone doesn't guarantee an exponent of $3.$

Therefore, the correct answers are:

False

False

True

False