  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.
