Stratification in the preferential attachment network

We study structural properties of trees grown by preferential attachment. In this mechanism, nodes are added sequentially and attached to existing nodes at a rate that is strictly proportional to the degree. We classify nodes by their depth n, defined as the distance from the root of the tree, and find that the network is strongly stratified. Most notably, the distribution f(k)((n)) of nodes with degree k at depth n has a power-law tail, f(k)((n)) similar to k(-gamma(n)). The exponent grows linearly with depth, gamma(n) = 2 + n-1/< n-1 > , where the brackets denote an average over all nodes. Therefore, nodes that are closer to the root are better connected, and moreover, the degree distribution strongly varies with depth. Similarly, the in-component size distribution has a power-law tail and the characteristic exponent grows linearly with depth. Qualitatively, these behaviors extend to a class of networks that grow by redirection.