Expected degree of finite preferential attachment networks

Abstract We provide an analytic expression for the quantity described in the title. Namely, we perform a preferential attachment growth process to generate a scale-free network. At each stage we add a new node with m new links. Let k denote the degree of a node, and N the number of nodes in the network. The degree distribution is assumed to converge to a power-law (for k ≥ m) of the form k and we obtain an exact implicit relationship for γ, m and N . We verify this with numerical calculations over several orders of magnitude. Although this expression is exact, it provides only an implicit expression for γ(m). Nonetheless, we provide a reasonable guess as to the form of this curve and perform curve fitting to estimate the parameters of that curve — demonstrating excellent agreement between numerical fit, theory, and simulation.