# Gradient networks

Citation:

Toroczkai, Zoltan, et al. "Gradient networks." Journal Of Physics A: Mathematical And Theoretical 41.15 (2008): 155103.

Publication Date:

April, 2008

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 os–Renyi) random graph, ´ GN ,p, and the scalars are independent identically distributed (i.i.d.) random variables. We show that in the limit N → ∞, p → 0, z = pN = const 1, R(l) ∝ l−1 for l