Gradient networks

Gradient networks are defined (Toroczkai and Bassler 2004 Nature 428 716) as directed graphs formed by local gradients of a scalar field distributed on the nodes of a substrate network G. We present the derivation for some of the general properties of gradient graphs and give an exact expression for the in-degree distribution of the gradient network when the substrate is a binomial (Erds–Rényi) random graph, , and the scalars are independent identically distributed (i.i.d.) random variables. We show that in the limit for , i.e., gradient networks become scale-free graphs up to a cut-off degree. This paper presents the detailed derivation of the results announced in Toroczkai and Bassler (2004 Nature 428 716).