MULTISCALE GENESIS OF A TINY GIANT FOR PERCOLATION ON SCALE-FREE RANDOM GRAPHS

• 成果类型: Article
• 署名作者: Bhamidi, Shankar; Dhara, Souvik; van der Hofstad, Remco
• 署名单位: University of North Carolina; University of North Carolina Chapel Hill; Purdue University System; Purdue University; Eindhoven University of Technology
• 刊物名称: ANNALS OF PROBABILITY
• ISSN/ISSBN: 0091-1798
• DOI: 10.1214/24-AOP1735
• 发表日期: 2025
• 页码: 1331-1381
• 关键词: multiplicative coalescent; PHASE-TRANSITION; LIMITS

摘要

We study the critical behavior for percolation on inhomogeneous random networks on n vertices, where the weights of the vertices follow a power-law distribution with exponent tau is an element of (2, 3). Such networks, often referred to as scale-free networks, exhibit critical behavior when the percolation probability tends to zero at an appropriate rate, as n -> infinity. We identify the critical window for several scale-free random graph models, such as the Norros-Reittu model, Chung-Lu model and generalized random graphs. Surprisingly, there exists a finite time inside the critical window, after which we see a sudden emergence of a tiny giant component. This is a novel behavior, which is in contrast with the critical behavior in other known universality classes with tau is an element of (3, 4) and tau >4. Precisely, for edge-retention probabilities pi(n) = lambda n(-(3-tau )/2), there is an explicitly computable lambda(c) > 0 such that the critical window is of the form lambda is an element of (0, lambda(c)), where the largest clusters have size of order n(beta) with beta = (tau(2)-4 tau + 5)/[2(tau - 1)] is an element of [ root 2-1, 1/2) and have nondegenerate scaling limits, while in the supercritical regime lambda> lambda(c), a unique tiny giant component of size Theta(root n) emerges, and its size concentrates. For lambda is an element of (0, lambda(c)), the scaling limit of the maximum component sizes can be described in terms of components of a one-dimensional inhomogeneous percolation model on Z(+) studied in a seminal work by Durrett and Kesten (In A Tribute to Paul Erdos (1990) 161-176 Cambridge Univ. Press). For lambda> lambda(c), we use a relation to general inhomogeneous random graphs, as studied by Bollob & aacute;s, Janson and Riordan (Random Structures Algorithms 31 (2007) 3-122), to prove that the sudden emergence of the tiny giant is caused by a phase transition inside a smaller core of vertices of weight of order at least root n.