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 $k\geq m$) of the form $k^{-\gamma}$ and we obtain an exact implicit relationship for $\gamma$, $m$ 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 $\gamma(m)$. 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.