Giant component in a configuration-model power-law graph with a variable number of links

Abstract

We generalize an algorithm used widely in the configuration model such that power-law degree sequences with the degree exponent λ and the number of links per node K controllable independently may be generated. It yields the degree distribution in a different form from that of the static model or under random removal of links while sharing the same λ and K. With this generalized power-law degree distribution, the critical point K c for the appearance of the giant component remains zero not only for λ ≤ 3 but also for 3 < λ < λ l ≃ 3.81. This is contrasted with K c = 0 only for λ ≤ 3 in the static model and under random link removal. The critical exponents and the cluster-size distribution for λ < λ l are also different from known results. By analyzing the moments and the generating function of the degree distribution and comparison with those of other models, we show that the asymptotic behavior and the degree exponent may not be the only properties of the degree distribution relevant to the critical phenomena but that its whole functional form can be relevant. These results can be useful in designing and assessing the structure and robustness of networked systems.

Degree distributions under general node removal: Power-law or Poisson?

“…A natural extension can be the link removal process introduced in Ref. [56], where the non-random case 𝜃 ≠ 0 can be realized in various ways to reflect different possible scenarios. In addition, one can try different power-law degree distributions from those considered in this work.…”

Giant graviton expansion of Schur index and quasimodular forms

“…In the simplest example of four dimensional N = 4 U(N ) super Yang-Mills (SYM) theory dual to type IIB superstring in AdS 5 × S 5 , finite N effects originate from wrapped D3 brane states maximally stretched in S 5 called giant gravitons [13] and having charge of order JHEP05(2024)282 N . 1 From the corresponding giant-graviton expansion of the index, it is possible to reproduce the entropy of dual black holes in AdS 5 × S 5 as first shown in [15] for small black holes with charge Q ≪ N 2 and later generalized to black holes with charge Q ∼ N 2 , first in [16], using a large-charge expansion, and later in [17], using a large N expansion. Recently, the giant graviton expansion has also been described by enumerating BPS geometries studying bubbling solutions in supergravity [18,19].…”