Limiting Degree Distribution for a Sublinear Preferential Attachment Model with Communities

par Camille Cazaux (LPSM, Sorbonne Université)

mercredi 26 nov. 2025, 11:45 → 12:45 Europe/Paris

Amphithéâtre Léon Motchane (IHES)

Seed Seminar of Mathematics and Physics

Fall' 25: Random Forests and Fermionic Field Theories 

For many real-world networks, such as the World Wide Web, the degree distribution follows a power law. It is therefore useful to have simple random graph models whose limiting degree distribution exhibits this same feature. With this motivation, physicists Albert-László Barabási and Réka Albert introduced the preferential attachment model that now bears their name. A further advantage of this model is that it incorporates temporal dynamics: starting from an initial graph $\mathcal{G}_0$, the graph at time $n+1$ is obtained from the graph at time $n$, denoted $\mathcal{G}_n$, by adding a new vertex $v_{n+1}$. This vertex then attaches to one or several vertices of $\mathcal{G}_n$ according to a preferential attachment rule, meaning that the probability of connecting to a given vertex of $\mathcal{G}_n$ is proportional to its degree.

We present an extension of this model in which each vertex of the graph is assigned a community (or type), and in which the preferential attachment is sublinear; that is, the probability of attaching to a vertex $u$ is proportional to $\deg(u)^\gamma$, where $\gamma$ is a parameter taking values in $(0,1)$.

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