
Mathematics 2010
Entropy and Hausdorff Dimension in Random Growing TreesDOI: 10.1142/S0219493712500104 Abstract: We investigate the limiting behavior of random tree growth in preferential attachment models. The tree stems from a root, and we add vertices to the system onebyone at random, according to a rule which depends on the degree distribution of the already existing tree. The socalled weight function, in terms of which the rule of attachment is formulated, is such that each vertex in the tree can have at most K children. We define the concept of a certain random measure mu on the leaves of the limiting tree, which captures a global property of the tree growth in a natural way. We prove that the Hausdorff and the packing dimension of this limiting measure is equal and constant with probability one. Moreover, the local dimension of mu equals the Hausdorff dimension at mualmost every point. We give an explicit formula for the dimension, given the rule of attachment.
