Geometric inhomogeneous random graphs (GIRGs) are a model for scale-free networks with underlying geometry. We study bootstrap percolation on these graphs, which is a process modelling the spread of an infection of vertices starting within a (small) local region. We show that the process exhibits a phase transition in terms of the initial infection rate in this region. We determine the speed of the process in the supercritical case, up to lower order terms, and show that its evolution is fundamentally influenced by the underlying geometry. For vertices with given position and expected degree, we determine the infection time up to lower order terms. Finally, we show how this knowledge can be used to contain the infection locally by removing relatively few edges from the graph. This is the first time that the role of geometry on bootstrap percolation is analysed mathematically for geometric scale-free networks.

We determine the asymptotic size of the largest component in the $2$-type binomial random graph $G(\mathbf{n},P)$ near criticality using a refined branching process approach. In $G(\mathbf{n},P)$ every vertex has one of two types, the vector $\mathbf{n}$ describes the number of vertices of each type, and any edge $\{u,v\}$ is present independently with a probability that is given by an entry of the probability matrix $P$ according to the types of $u$ and $v.$ We prove that in the weakly supercritical regime, i.e., if the “distance” to the critical point of the phase transition is given by $\varepsilon=\varepsilon(\mathbf{n})\to0,$ with probability $1-o(1),$ the largest component in $G(\mathbf{n},P)$ contains asymptotically $2\varepsilon \|\mathbf{n}\|_1$ vertices and all other components are of size $o(\varepsilon \|\mathbf{n}\|_1).$