Cluster Tails for Critical Power-Law Inhomogeneous Random Graphs

Abstract

Recently, the scaling limit of cluster sizes for critical inhomogeneous random graphs of rank-1 type having finite variance but infinite third moment degrees was obtained in Bhamidi et al. (Ann Probab 40:2299–2361, 2012 ). It was proved that when the degrees obey a power law with exponent , the sequence of clusters ordered in decreasing size and multiplied through by converges as to a sequence of decreasing non-degenerate random variables. Here, we study the tails of the limit of the rescaled largest cluster, i.e., the probability that the scaling limit of the largest cluster takes a large value u , as a function of u . This extends a related result of Pittel (J Combin Theory Ser B 82(2):237–269, 2001 ) for the Erdős–Rényi random graph to the setting of rank-1 inhomogeneous random graphs with infinite third moment degrees. We make use of delicate large deviations and weak convergence arguments.

The probability of unusually large components in the near-critical Erdős–Rényi graph

“…Bollobás and Riordan [9] showed that the size of the giant component in the regime p = (1 + ε n )/n with ε n → 0 and ε 3 n n → ∞ is asymptotically normally distributed when rescaled appropriately. Results similar to those of Pittel [23] have been given by van der Hofstad et al [31] for inhomogeneous random graphs whose vertex degrees have power-law tails, a model which is not in the same universality class as the near-critical Erdős-Rényi graph. The scaling limit in this case, rather than a Brownian motion with parabolic drift, involves a thinned Lévy process; further related results can be found in [3].…”

Large deviations for power-law thinned Lévy processes

“…In [14] we make formal the conjecture that P(H 1 (0) > u) ≈ P(S u > 0) for large u. We show that P(H 1 (0) > u) has the same asymptotic behavior as P(S u > 0) in (1.11), with the same constants except for the constant D. Despite the similarity of this result, the proof method in [14] is entirely different. In order to establish the asymptotics for P(H 1 (0) > u), we establish in [14] sample path large deviations, not conditioned on the event {S u > 0}, but on the event P(H 1 (0) > u).…”

“…In particular, in [14], we establish the following two results. First, we prove that there exists A ∈ (0, D) such that…”

“…The critical components are thus of the order n −(τ −2)/(τ −1) , but to obtain information beyond the order one needs to investigate H 1 (0). In the companion paper [14] we derive the precise asymptotic results for both P(H 1 (0) > u) and the tail distribution of the largest cluster, for u → ∞. A crucial ingredient of the proofs is the asymptotic behavior of P(S u > 0), the main topic of the present paper.…”

Big Jobs Arrive Early: From Critical Queues to Random Graphs

“…The sizes of the connected components are encoded as the time between successive minima of the exploration process. Because of this, the exploration process has been extensively applied to the study of undirected random graphs in the critical regime (Bollobás et al 2007;Bhamidi et al 2014Bhamidi et al , 2010Bhamidi et al , 2012Bhamidi et al , 2017van der Hofstad et al 2018). However, until recently, it was not known if this approach was useful for the study of directed critical random graphs.…”