 Barabási–Albert model

The Barabási-Albert model constructs a growing random graph by preferential attachment. Starting from a suitable seed, each new vertex brings $m$ edges and selects existing endpoints with probability proportional to their current degrees. A convention for sampling distinct endpoints must be specified.^[ref]

Older high-degree vertices tend to attract more edges. In the standard large-graph limit, the degree distribution has a tail proportional to $k^{-3}$, unlike the sharply concentrated degrees of a basic Erdős-Rényi model. Both growth and the attachment rule matter. The model supplies a mechanism for hubs; observing a heavy degree tail alone does not establish that this mechanism generated the data.

The Barabási–Albert (BA) model is a preferential attachment model for generating scale-free networks, which are networks characterized by a degree distribution that follows a power law. This model was proposed by Albert-László Barabási and Réka Albert in their seminal 1999 paper. ### Key Features of the Barabási–Albert Model: 1. **Network Growth**: The BA model creates networks by starting with a small number of connected nodes and adding new nodes over time.