Expected degree of finite preferential attachment networks

TL;DR

This paper derives an exact implicit formula for the expected degree distribution exponent in finite preferential attachment networks, validated through numerical simulations and curve fitting.

Contribution

It provides an analytic expression for the degree distribution exponent in finite scale-free networks generated by preferential attachment, with validation against simulations.

Findings

Exact implicit relationship for degree exponent $\

Good agreement between theory, numerical calculations, and simulations.

Curve fitting estimates parameters accurately.

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.