Scaling Limits for Critical Inhomogeneous Random Graphs with Finite Third Moments

Abstract

We find scaling limits for the sizes of the largest components at criticality for the rank-1 inhomogeneous random graphs with power-law degrees with exponent τ . We investigate the case where τ ∈ (3, 4), so that the degrees have finite variance but infinite third moment. The sizes of the largest clusters, rescaled by n −(τ −2)/(τ −1) , converge to hitting times of a 'thinned' Lévy process. This process is intimately connected to the general multiplicative coalescents studied in [1] and [3]. In particular, we use the results in [3] to show that, when interpreting the location λ inside the critical window as time, the limiting process is a multiplicative process with diffusion constant 0 and the entrance boundary describing the size of relative components in the λ → −∞ regime proportional to i. A crucial ingredient is the identification of the scaling of the largest connected components in the barely subcritical regime.Our results should be contrasted to the case where the degree exponent τ satisfies τ > 4, so that the third moment is finite. There, instead, we see that the sizes of the components rescaled by n −2/3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in [1] for the Erdős-Rényi random graph and extended to the present setting in [6,26]. The limit again is a multiplicative coalescent, the only difference with the limit for τ ∈ (3, 4) being the initial state, corresponding to the barely subcritical regime.

Critical epidemics, random graphs, and Brownian motion with a parabolic drift

“…The hitting time T a σ,κ (b), for specific choices of σ , a, and κ, occurs in the study of SIR models with general sampling procedures [14], SIR models with vaccination [7] and inhomogeneous random graphs [3], [20]. Our results for the simple SIR model and the Erdős-Rényi (homogeneous) random graph thus extend to these more general settings via the scaling relation (4.1).…”

Component structure of the configuration model: Barely supercritical case

“…This proof is standard, and can, for example, be found in or [, Section 7.7]. The critical case of these models was studied in .…”

Project management educational software (ProMES)

“…Their proofs use related means as in [1], and apply under the slightly weaker condition that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb, dsfont}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{E}}\lbrack W^3\rbrack < \infty\end{align*} \end{document}. The results in [2, 34] for τ > 4 strengthen Theorem 1.1 considerably. We have decided to include Theorem 1.1 and its proof, as this proof follows the same lines as the proof of the novel result in Theorem 1.2, and the proof nicely elucidates the place where the restriction τ > 4 is used.…”

“…Since the completion of the first version of this paper, the weak convergence of the rescaled ordered cluster sizes has been proved independently and almost at the same time in [2, 34]. Their proofs use related means as in [1], and apply under the slightly weaker condition that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb, dsfont}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{E}}\lbrack W^3\rbrack < \infty\end{align*} \end{document}.…”

“…This also sheds light on the critical cases τ = 4 and τ = 3. Indeed, when τ = 4 and u ↦ ℓ ( u ) is such that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb, dsfont}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{E}}\lbrack W^3\rbrack < \infty\end{align*} \end{document}, then the results in [2, 34] prove that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb, dsfont}\pagestyle{empty}\begin{document}\begin{align*}|{\mathcal C}_{\max}|\end{align*} \end{document} is of order n 2/3 as in Theorem 1.1 with a critical window of size n ‐1/3 . When τ = 4 and u ↦ ℓ ( u ) is such that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb, dsfont}\pagestyle{empty}\begin{document}\begin{align*}{\mathbb{E}}\lbrack W^3\rbrack =\infty\end{align*} \end{document}, then we predict that \documentclass{article}\usepackage{mathrsfs, amsmath, amssymb, dsfont}\pagestyle{empty}\begin{document}\begin{align*}|{\mathcal C}_{\max}|\end{align*} \end{document} is of order n 2/3 / ℓ (1/ n ) with critical window of width n ‐1/3 ℓ (1/ n ) 2 , instead.…”