On the evolution of scale-free graphs

D. -S. Lee; K. -I. Goh; B. Kahng; D. Kim

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arXiv · cond-mat/0312336

On the evolution of scale-free graphs

D. -S. Lee·K. -I. Goh·B. Kahng·D. Kim

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Abstract

We study the evolution of random graphs where edges are added one by one between pairs of weighted vertices so that resulting graphs are scale-free with the degree exponent $γ$. We use the branching process approach to obtain scaling forms for the cluster size distribution and the largest cluster size as functions of the number of edges $L$ and vertices $N$. We find that the process of forming a spanning cluster is qualitatively different between the cases of $γ>3$ and $2<γ<3$. While for the former, a spanning cluster forms abruptly at a critical number of edges $L_c$, generating a single peak in the mean cluster size $ $ as a function of $L$, for the latter, however, the formation of a spanning cluster occurs in a broad range of $L$, generating double peaks in $ $.

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D. -S. Lee, K. -I. Goh, B. Kahng, D. Kim. 2003-12-23. On the evolution of scale-free graphs. https://arxiv.org/abs/cond-mat/0312336

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