Critical percolation on preferential attachment graphs with infinite variance

Abstract

We study the inhomogeneous random graph with preferential attachment kernel and degree distribution with power-law exponent $\tau\in(2,3)$ as a representative of the class of graphs of preferential attachment type with infinite variance degrees. Under bond percolation with a positive retention probability independent of the size $n$ of the graph there is a unique macroscopic component with high probability. We therefore investigate percolation probabilities $p_n\downarrow0$. We identify a moving critical window at $p_c \sim \beta n^{(\tau-3)/(2\tau-2)}$. Above this window, when $p_n \gg p_c$, the maximal component has size of order $n p^{_{(\tau-1)/(3-\tau)}}_{_n}$ and it is unique. Below this window, when $n^{1/(1-\tau)} \ll p_n \ll p_c$, it is non-unique, star-shaped and has size of order $n^{1/(\tau-1)} p_n$. In the critical window itself, the largest component scaled by $\sqrt{n}$ converges in distribution to a positive random variable with a law given in terms of a subcritical Norros-Reittu graph. This behaviour is markedly different from that seen for other classes of scale-free graphs and is conjectured to persist throughout the broad class of growing graphs with infinite variance.