Graphs Over Time: Densification Laws, Shrinking Diameters, Explanations And Realistic Generators

Jure Leskovec

Abstract

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; 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. We studied a wide range of real graphs, and we observed 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)). What underlying process causes a graph to systematically densify, and to experience a decrease in effective diameter even as its size increases? The existing graph generation models do not exhibit these types of behavior, even at a qualitative level. In most cases they are also very complicated to analyze mathematically. So we propose a graph generator that is mathematically tractable and matches this collection of properties. The main idea is to use a non-standard matrix operation, the Kronecker product, to generate graphs that we refer to as "Kronecker graphs". We show that Kronecker graphs naturally obey all the properties; in fact, we can rigorously prove that they do so. We also provide empirical evidence showing that they can mimic very well several real graphs.

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