2001/06/30 by S. N. Dorogovtsev, S. N. Dorogovt︠s︡ev, J. F. F. Mendes +1 · 2 citations
Mathematics · Physics and Astronomy · #Complex Network Analysis Techniques #Opinion Dynamics and Social Influence #Stochastic processes and statistical mechanics #cond-mat.stat-mech
paper · pdf · doi:10.1103/physreve.64.066110
published as Phys. Rev. E 64, 066110 (2001) · 12 pages revtex, 3 figures
arxiv created 2001/08/27 · openalex publication_date 2001/11/19 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We describe the anomalous phase transition of the emergence of the giant connected component in scale-free networks growing under mechanism of preferential linking. We obtain exact results for the size of the giant connected component and the distribution of vertices among connected components. We show that all the derivatives of the giant connected component size S over the rate b of the emergence of new edges are zero at the percolation threshold b(c), and S infinity exp[-d(gamma)(b-b(c))(-1/2)], where the coefficient d is a function of the degree distribution exponent gamma. In the entire phase without the giant component, these networks are in a "critical state." The probability P(k) that a vertex belongs to a connected component of a size k is of a power-law form. At the phase transition point, P(k) approximately 1/(k ln k)(2). In the phase with the giant component, P(k) has an exponential cutoff at k(c) approximately 1/S. In the simplest particular case, we present exact results for growing exponential networks.