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 labelled sequentially in increasing order of one of their coordinates. 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 tth point is attached to the ith node of the network using the probability: πi (t) - k i (t)ℓtiα where ki (t) is the degree of the ith node and ℓti is the Euclidean distance between the points t and i. Here α is a continuously tuneable parameter and while for α = 0 one gets the simple Barabási-Albert network, the case for α → - ∞ corresponds to the spatially continuous version of the well-known Scheidegger's river network problem. The modulating parameter a is tuned to study the transition between the two different critical behaviours at a specific value αc which we numerically estimate to be -2. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.