Log-Networks
  We introduce a growing network model in which a new node attaches to a randomly-selected node, as well as to all ancestors of the target node. This mechanism produces a sparse, ultra-small network where the average node degree grows logarithmically with network size while the network diameter equals 2. We determine basic geometrical network properties, such as the size dependence of the number of links and the in- and out-degree distributions. We also compare our predictions with real networks where the node degree also grows slowly with time -- the Internet and the citation network of all Physical Review papers. 
P. L. Krapivsky and S. Redner