Gradient Network - In-degree Distribution of Gradient Networks

In-degree Distribution of Gradient Networks

In a gradient network, in-degree of a node i, k_i (in) is the number of gradient edges pointing into i, and the in-degree distribution {k_i (in)}

When the substrate G is random graph, and each pair of nodes is connected with probability P, the scalars h_i are i.i.d. (independent identically distributed) the exact expression for R(l) is given by

In the limit N →∞ and P → 0, the degree distribution becomes the power law

This shows in this limit, the gradient network of random network is scale-free. If the subtstrate network G is scale-free, like BA model, then the gradient network also follow the power-law with the same exponent as those of G.