title: Shortest paths and load scaling in scale-free trees.
abstract: The average node-to-node distance of scale-free graphs depends logarithmically on N, the number of nodes, while the probability distribution function of the distances may take various forms. Here we analyze these by considering mean-field arguments and by mapping the m=1 case of the Barabási-Albert model into a tree with a depth-dependent branching ratio. This shows the origins of the average distance scaling and allows one to demonstrate why the distribution approaches a Gaussian in the limit of N large. The load, the number of the shortest distance paths passing through any node, is discussed in the tree presentation.
authors.affiliation: Helsinki University of Technology, Laboratory of Physics, P.O. Box 1100, FIN-02015 HUT, Finland.
authors.forename: Gábor
authors.initials: G
authors.lastname: Szabó
authors.forename: Mikko
authors.initials: M
authors.lastname: Alava
authors.forename: János
authors.initials: J
authors.lastname: Kertész