Universality is a principle that fundamentally underlies many critical phenomena, ranging from epidemic spreading to the emergence or breakdown of global connectivity in networks. Percolation, the transition to global connectedness on gradual addition of links, may exhibit substantial gaps in the size of the largest connected network component. We uncover that the largest gap statistics is governed by extreme-value theory. This allows us to unify continuous and discontinuous percolation by virtue of universal critical scaling functions, obtained from normal and extreme-value statistics. Specifically, we show that the universal scaling function of the size of the largest gap is given by the extreme-value Gumbel distribution. This links extreme-value statistics to universality and criticality in percolation.

普遍性是许多关键现象的根本基础，从流行病传播到网络中全球连通性的出现或崩溃。渗透，即通过逐渐添加链接向全局连通性的过渡，可能会在最大的连接网络组件的大小上表现出巨大的差距。我们发现最大的差距统计是由极值理论支配的。这使我们能够通过从正态和极值统计中获得的通用临界尺度函数来统一连续和不连续的渗流。具体来说，我们表明最大间隙大小的通用标度函数由极值 Gumbel 分布给出。这将极值统计与渗透的普遍性和临界性联系起来。