Abstract
We introduce a deterministic model for scale-free networks, whose degree distribution follows a power law with the exponent At each time step, each vertex generates its offspring, whose number is proportional to the degree of that vertex with proportionality constant We consider the two cases: First, each offspring is connected to its parent vertex only, forming a tree structure. Second, it is connected to both its parent and grandparent vertices, forming a loop structure. We find that both models exhibit power-law behaviors in their degree distributions with the exponent Thus, by tuning m, the degree exponent can be adjusted in the range, We also solve analytically a mean shortest-path distance d between two vertices for the tree structure, showing the small-world behavior, that is, where N is system size, and is the mean degree. Finally, we consider the case that the number of offspring is the same for all vertices, and find that the degree distribution exhibits an exponential-decay behavior.
- Received 17 November 2001
DOI:https://doi.org/10.1103/PhysRevE.65.056101
©2002 American Physical Society