Geometric fractal growth model for scalefree networks
Abstract
We introduce a deterministic model for scalefree networks, whose degree distribution follows a power law with the exponent γ. At each time step, each vertex generates its offspring, whose number is proportional to the degree of that vertex with proportionality constant m1 (m>1). 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 powerlaw behaviors in their degree distributions with the exponent γ=1+ln(2m1)/ln m. Thus, by tuning m, the degree exponent can be adjusted in the range, 2<γ<3. We also solve analytically a mean shortestpath distance d between two vertices for the tree structure, showing the smallworld behavior, that is, d~ln N/ln k&;, where N is system size, and k&; 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 exponentialdecay behavior.
 Publication:

Physical Review E
 Pub Date:
 May 2002
 DOI:
 10.1103/PhysRevE.65.056101
 arXiv:
 arXiv:condmat/0112361
 Bibcode:
 2002PhRvE..65e6101J
 Keywords:

 89.70.+c;
 89.75.k;
 05.10.a;
 Information theory and communication theory;
 Complex systems;
 Computational methods in statistical physics and nonlinear dynamics;
 Condensed Matter  Statistical Mechanics;
 Condensed Matter  Disordered Systems and Neural Networks
 EPrint:
 doi:10.1103/PhysRevE.65.056101