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Gradient networks. (English) Zbl 1135.90007

Summary: Gradient networks are defined [B. Toroczkai and K. E. Bassler, Nature 428, 716 (2004)] 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 (Erdős-Rényi) random graph, \(G_{N,p}\), and the scalars are independent identically distributed (i.i.d.) random variables. We show that in the limit \(N\to\infty\), \(p\to 0\), \(z=pN= \text{const.}\gg1\), \(R(l)\propto l^{-1}\) 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 by Toroczkai and Bassler (loc. cit.).

MSC:

90B15 Stochastic network models in operations research
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