WIAS Preprint No. 2898, (2021)

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


  • Andreis, Luisa
  • König, Wolfgang
    ORCID: 0000-0002-7673-4364
  • Langhammer, Heide
    ORCID: 0009-0005-0353-0343
  • Patterson, Robert I. A.
    ORCID: 0000-0002-3583-2857

2020 Mathematics Subject Classification

  • 05C30 05C80 60F10 60G57 60J80


  • Sparse random graph, empirical measures of components, large deviations, projective limits, giant cluster phase transition, asymptotics for connection probabilities, spatial coagulation model, Flory equation, stochastic block model




We study an inhomogeneous sparse random graph, GN, on [N] = { 1,...,N } as introduced in a seminal paper [BJR07] by Bollobás, Janson and Riordan (2007): vertices have a type (here in a compact metric space S), and edges between different vertices occur randomly and independently over all vertex pairs, with a probability depending on the two vertex types. In the limit N → ∞ , we consider the sparse regime, where the average degree is O(1). We prove a large-deviations principle with explicit rate function for the statistics of the collection of all the connected components, registered according to their vertex type sets, and distinguished according to being microscopic (of finite size) or macroscopic (of size ≈ N). In doing so, we derive explicit logarithmic asymptotics for the probability that GN is connected. We present a full analysis of the rate function including its minimizers. From this analysis we deduce a number of limit laws, conditional and unconditional, which provide comprehensive information about all the microscopic and macroscopic components of GN. In particular, we recover the criterion for the existence of the phase transition given in [BJR07].

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