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Albert-László Barabási is a physicist known for work on complex networks. With Réka Albert, he introduced the Barabási-Albert model, in which a growing graph attaches new vertices preferentially to vertices with larger degree.^[ref]

The central scientific idea is a mechanism, rather than merely a fitted curve: growth and degree-dependent attachment can create a broad degree distribution. This encouraged testing network formation rules against data and helped connect graph theory with statistical mechanics. The mechanism does not imply that every heavy-tailed real network follows that model.

Laboratory website: www.barabasi.com/

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.

For the Barabási-Albert model, choose a connected seed graph on $m_0$ vertices, the number $m\le m_0$ of new edges per added vertex, and a final size $n$. A complete seed with $m_0\ge 2$ suffices. Add one vertex at a time, attaching it to $m$ distinct existing vertices selected with weights proportional to their current vertex degrees. Exclude duplicate edges and self-loops.

Increasing $n$ creates a larger network and permits older hubs to grow. Increasing $m$ makes the graph denser, with asymptotic average vertex degree $2m$, and tends to shorten graph distances. The seed affects early structure and the oldest hubs; standard linear preferential attachment gives a robust large-size tail exponent.

Its asymptotic degree distribution is $P(k)\propto k^{-3}$, so it is a scale-free network. Hubs produce short average shortest path length, but elevated clustering coefficients are not automatic: $m=1$ with a tree seed gives a triangle-free tree, and clustering decreases with size in the standard model. The strict clustered small-world network property is therefore not guaranteed.^[ref]

Réka Albert is a physicist whose work connects network science with biological networks. With Albert-László Barabási, she coauthored the 1999 paper introducing a growing preferential attachment model for the emergence of broad degree distributions, now called the Barabási-Albert model.^[ref]

Her biological work uses logical models to connect the structure of a regulatory graph with its possible dynamical states. Such models represent interactions without requiring a measured kinetic constant for every reaction. Her institutional profile describes research on mechanisms controlling complex-network dynamics and on predictive modeling of biological systems.^[ref]

Homepage: www.ralbert.me/