Average distance in growing trees
Two kinds of evolving trees are considered here: the exponential trees, where subsequent nodes are linked to old nodes without any preference, and the Barabisi-Albert scale-free networks, where the probability of linking to a node is proportional to the number of its pre-existing links. In both cases, new nodes are linked to m = I nodes. The average node-node distance d is calculated numerically in evolving trees as dependent on the number of nodes N. The results for N not less than a thousand are averaged over a thousand of growing trees. The results on the mean node-node distance d for large N can be approximated by d = 2 ln(N) + c(1) for the exponential trees, and d = ln(N) + c(2) for the scale-free trees, where c(i) are constant. We also derive iterative equations for d and its dispersion for the exponential trees. The simulation and the analytical approach give the same results.
exponential trees scale-free networks evolving networks small-world effect