  $\mathit{Context.}$ The thermal conductivity of highly porous dust aggregates
is a key parameter for many subjects in planetary science; however, it is not
yet fully understood. $\mathit{Aims.}$ In this study, we investigate the
thermal conductivity of fluffy dust aggregates with filling factors of less
than $10^{-1}$. $\mathit{Methods.}$ We determine the temperature structure and
heat flux of the porous dust aggregates calculated by $N$-body simulations of
static compression in the periodic boundary condition. $\mathit{Results.}$ We
derive an empirical formula for the thermal conductivity through the solid
network $k_{\rm sol}$ as a function of the filling factor of dust aggregates
$\phi$. The results reveal that $k_{\rm sol}$ is approximately proportional to
${\phi}^{2}$, and the thermal conductivity through the solid network is
significantly lower than previously assumed. In light of these findings, we
must reconsider the thermal histories of small planetary bodies.
