Probability Seminar
- Event Type
- Seminar/Symposium
- Sponsor
- Department of Mathematics
- Location
- 143 Altgeld Hall
- Date
- Dec 9, 2025 2:00 - 2:50 pm
- Speaker
- Sayan Banerjee (University of North Carolina Chapel Hill)
- Contact
- Partha Dey
- Originating Calendar
- Probability Research Area Calendar
Title: Percolation on Dynamic Random Networks.
Abstract: Percolation is, by now, a classical topic, and there are many results concerning percolation on static random networks formed out of a fixed large number of vertices, where the vertex roles are exchangeable. Typically, such networks exhibit a phase transition where the largest component transforms from a size that is logarithmic to linear in $n$, the total network size, as the percolation probability crosses a certain critical threshold. However, percolation on dynamic networks, where vertex arrivals play a crucial role, is far from being well-understood. In this talk, we will discuss percolation on two such models, the uniform attachment model and the preferential attachment model. For the percolation probability p below the critical threshold (subcritical regime), we show that the maximal component size scales like $n^{a(p)}$ for an explicitly computable exponent a(p). This is in stark contrast with static random networks, where such sizes are of order log(n). Moreover, unlike most static network models, we show that the susceptibility, that is, the expected size of the component of a uniformly chosen vertex, remains bounded as the network grows and the percolation probability approaches its critical value from below. Proofs involve stochastic approximations, branching random walks, local limits, and tree-graph inequalities. Based on joint works with Shankar Bhamidi, Remco van der Hofstad, and Rounak Ray.