A transition from river networks to scale-free networks

Abstract

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: $π_i(t) \sim k_i(t)\ell_{ti}^α$ 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 $α$ is a continuously tunable parameter and while for $α=0$ one gets the simple Barabási-Albert network, the case for $α\to -\infty$ corresponds to the spatially continuous version of the well known Scheidegger's river network problem. The modulating parameter $α$ is tuned to study the transition between the two different critical behaviors at a specific value $α_c$ which we numerically estimate to be -2.