  A spatial network is constructed on a two dimensional space where the nodes
are geometrical points located at randomly distributed positions which are
labeled sequentially in increasing order of one of their co-ordinates. Starting
with $N$ such points the network is grown by including them one by one
according to the serial number into the growing network. The $t$-th point is
attached to the $i$-th node of the network using the probability: $\pi_i(t)
\sim k_i(t)\ell_{ti}^{\alpha}$ where $k_i(t)$ is the degree of the $i$-th node
and $\ell_{ti}$ is the Euclidean distance between the points $t$ and $i$. Here
$\alpha$ is a continuously tunable parameter and while for $\alpha=0$ one gets
the simple Barab\'asi-Albert network, the case for $\alpha \to -\infty$
corresponds to the spatially continuous version of the well known Scheidegger's
river network problem. The modulating parameter $\alpha$ is tuned to study the
transition between the two different critical behaviors at a specific value
$\alpha_c$ which we numerically estimate to be -2.

Statistical Mechanics