Average distance in a hierarchical scale-free network: an exact solution
arXiv:0910.5349 · doi:10.1088/1742-5468/2009/10/P10022
Abstract
Various real systems simultaneously exhibit scale-free and hierarchical structure. In this paper, we study analytically average distance in a deterministic scale-free network with hierarchical organization. Using a recursive method based on the network construction, we determine explicitly the average distance, obtaining an exact expression for it, which is confirmed by extensive numerical calculations. The obtained rigorous solution shows that the average distance grows logarithmically with the network order (number of nodes in the network). We exhibit the similarity and dissimilarity in average distance between the network under consideration and some previously studied networks, including random networks and other deterministic networks. On the basis of the comparison, we argue that the logarithmic scaling of average distance with network order could be a generic feature of deterministic scale-free networks.
Definitive version published in Journal of Statistical Mechanics
References in corpus (10)
- Hierarchical structure and the prediction of missing links in networks
- Maximal planar scale-free Sierpinski networks with small-world effect and power-law strength-degree correlation
- Self-similarity, small-world, scale-free scaling, disassortativity, and robustness in hierarchical lattices
- Exact solution of mean geodesic distance for Vicsek fractals
- Evolving small-world scale-free networks consist of cliques
- Griffiths singularities and algebraic order in the exact solution of an Ising model on a fractal modular network
- Average distance in a hierarchical scale-free network: an exact solution
- Constrained spin dynamics description of random walks on hierarchical scale-free networks
- Transition from fractal to non-fractal scalings in growing scale-free networks
- Contact graphs of disk packings as a model of spatial planar networks