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 R(l) of the gradient network when the substrate is a binomial (Erd{\;\kern -0.10em \raise -0.35ex \{{^{^{\prime\prime}}}}\kern -0.57em \o} s-Rényi) random graph, G_{N,p} , and the scalars are independent identically distributed (i.i.d.) random variables. We show that in the limit N \to \infty, p \to 0, z = pN = \mbox{const} \gg 1, R(l)\propto l^{-1} for l < l_c = z , 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).