We consider the density, net weight, $\pi$-weight, the Souslin number, weakly density, local weak density, &k&-network of the space of complete linked systems with compact elements. We prove that for a topological $T_{1}$-space $X$ the followings hold: 1) $\pi w(N^{*}X)= \pi w(N_{cm}X)= \pi w(N_{c}X) = \pi w(X);$ 2) $d(X)=d(N^{*}X)= d(N_{cm}X);$ 4) $n\pi w(N^{*}X) = n\pi w(N_{cm}X) = n\pi w(X)$. We also prove that for an infinite Tychonoff space the followings hold: 1) $c(N^{*}X)= c(N_{c}X)=c(N_{cm}X)=c(NX)=Sup\{c(X^{n}: n\in X)\};$ 2) $wd(N^{*}X)= wd(N_{c}X)=wd(N_{cm}X)=wd(NX) \le wd(X).$