  We define and study two new kinds of "effective resistances" based on
hubs-biased -- hubs-repelling and hubs-attracting -- models of navigating a
graph/network. We prove that these effective resistances are squared Euclidean
distances between the vertices of a graph. They can be expressed in terms of
the Moore-Penrose pseudoinverse of the hubs-biased Laplacian matrices of the
graph. We define the analogous of the Kirchhoff indices of the graph based of
these resistance distances. We prove several results for the new resistance
distances and the Kirchhoff indices based on spectral properties of the
corresponding Laplacians. After an intensive computational search we conjecture
that the Kirchhoff index based on the hubs-repelling resistance distance is not
smaller than that based on the standard resistance distance, and that the last
is not smaller than the one based on the hubs-attracting resistance distance.
We also observe that in real-world brain and neural systems the efficiency of
standard random walk processes is as high as that of hubs-attracting schemes.
On the contrary, infrastructures and modular software networks seem to be
designed to be navigated by using their hubs.
