A large-deviations principle for all the components in a sparse inhomogeneous random graph

Abstract We study an inhomogeneous sparse random graph, $${\mathcal G }_N$$ G N , on $$[N]=\{1,\dots ,N\}$$ [ ] = { 1 , ⋯ } as introduced in a seminal paper by Bollobás et al. (Random Struct Algorithms 31(1):3–122, 2007): vertices have type (here compact metric space S }$$ S ), and edges between different occur randomly independently over all vertex pairs, with probability depending the two types. In limit $$N\rightarrow \infty $$ → ∞ we consider regime, where average degree is O (1). prove large-deviations principle explicit rate function for statistics of collection connected components, registered according to their sets, distinguished being microscopic (of finite size) or macroscopic size $$\asymp N$$ ≍ ). doing so, derive logarithmic asymptotics that connected. present full analysis including its minimizers. From this deduce number laws, conditional unconditional, which provide comprehensive information about components . particular, recover criterion existence phase transition given (2007).