Novel scaling limits for critical inhomogeneous random graphs
By Shankar Bhamidi, Remco van der Hofstad, Johan van Leeuwaarden. 2009
88 citations recorded by the source corpus.
We flnd scaling limits for the sizes of the largest components at criticality for the rank-1 inho- mogeneous random graphs with power-law degrees with exponent ?. We investigate the case where ? 2 (3;4), so that the degrees have flnite variance but inflnite third moment. The sizes of the largest clusters, rescaled by n i(?i2)=(?i1) , converge to hitting times of a 'thinned' Levy process. This process is intimately connected to the general multiplicative coalescents studied in (1) and (2). In particular, we use the results in (2) to show that, when interpreting the location ‚ inside the critical window as time, the limiting process is a multiplicative process with difiusion constant 0 and the entrance boundary describing the size of relative components in the ‚ ! i1 regime proportional to i i i1=(?i1) ¢ i‚1 . A crucial ingredient is the identiflcation 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 ? satisfles ? > 4, so that the third moment is flnite. There, instead, we see that the sizes of the components rescaled by n i2=3 converge to the excursion lengths of an inhomogeneous Brownian motion, as proved in (1) for the Erd}os- Renyi random graph and extended to the present setting in (3, 4). The limit again is a multiplicative coalescent, the only difierence with the limit for ? 2 (3;4) being the initial state, corresponding to the barely subcritical regime.
Subfields: Condensed Matter Physics, Mathematical Physics, Statistics and Probability
View Paper: https://www.semanticscholar.org/paper/1f02490a0cb45e674248269d20e3d856f89bc10f
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Citations in the corpus
- Scale-free percolation (2011, unknown): “In this setting, we see that the critical behavior when γ > 3 is similar to that of the Erdős-Rényi random graph as identified in [2], while, for γ ∈ (2, 3), it is rather different (see [7, 22]).”
- Random Graphs and Complex Networks (2016, unknown): “36 is proved in [39].”
- The component sizes of a critical random graph with given degree sequence (2010, background): “As for the power law distribution case, Bhamidi, van der Hofstad and van Leeuwaarden [8] proved that if there exist γ ∈ (3,4) and c > 0 such that 1 − F(x) ∼x→∞ cx1−γ , the ordered sequence of component sizes of the inhomogeneous random graph with n vertices once rescaled by n−(γ−2)/(γ−1) then conver”