How do real graphs evolve over time? What are ``normal'' growth patterns insocial, technological, and information networks? Many studies have discoveredpatterns in static graphs, identifying properties in a single snapshot of alarge network, or in a very small number of snapshots; these include heavytails for in- and out-degree distributions, communities, small-world phenomena,and others. However, given the lack of information about network evolution overlong periods, it has been hard to convert these findings into statements abouttrends over time. Here we study a wide range of real graphs, and we observe some surprisingphenomena. First, most of these graphs densify over time, with the number ofedges growing super-linearly in the number of nodes. Second, the averagedistance between nodes often shrinks over time, in contrast to the conventionalwisdom that such distance parameters should increase slowly as a function ofthe number of nodes (like O(log n) or O(log(log n)). Existing graph generation models do not exhibit these types of behavior, evenat a qualitative level. We provide a new graph generator, based on a ``forestfire'' spreading process, that has a simple, intuitive justification, requiresvery few parameters (like the ``flammability'' of nodes), and produces graphsexhibiting the full range of properties observed both in prior work and in thepresent study. We also notice that the ``forest fire'' model exhibits a sharp transitionbetween sparse graphs and graphs that are densifying. Graphs with decreasingdistance between the nodes are generated around this transition point.
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