Graph Evolution: Densification and Shrinking Diameters
  How do real graphs evolve over time? What are ``normal'' growth patterns in social, technological, and information networks? Many studies have discovered patterns in static graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these include heavy tails for in- and out-degree distributions, communities, small-world phenomena, and others. However, given the lack of information about network evolution over long periods, it has been hard to convert these findings into statements about trends over time.   Here we study a wide range of real graphs, and we observe some surprising phenomena. First, most of these graphs densify over time, with the number of edges growing super-linearly in the number of nodes. Second, the average distance between nodes often shrinks over time, in contrast to the conventional wisdom that such distance parameters should increase slowly as a function of the number of nodes (like O(log n) or O(log(log n)).   Existing graph generation models do not exhibit these types of behavior, even at a qualitative level. We provide a new graph generator, based on a ``forest fire'' spreading process, that has a simple, intuitive justification, requires very few parameters (like the ``flammability'' of nodes), and produces graphs exhibiting the full range of properties observed both in prior work and in the present study.   We also notice that the ``forest fire'' model exhibits a sharp transition between sparse graphs and graphs that are densifying. Graphs with decreasing distance between the nodes are generated around this transition point. 
Jure Leskovec, Jon Kleinberg and Christos Faloutsos