Graphs over Time: Densification Laws, Shrinking Diameters and Possible Explanations ∗ Jure Leskovec Jon Kleinberg Christos Faloutsos Carnegie Mellon University Cornell University Carnegie Mellon University
[email protected] [email protected] [email protected] duces graphs exhibiting the full range of properties observed both in prior work and in the present study. ABSTRACT How do real graphs evolve over time? What are “normal” Categories and Subject Descriptors growth patterns in social, technological, and information H.2.8 [Database Management]: Database Applications – networks? Many studies have discovered patterns in static Data Mining graphs, identifying properties in a single snapshot of a large network, or in a very small number of snapshots; these in- General Terms clude heavy tails for in- and out-degree distributions, com- munities, small-world phenomena, and others. However, Measurement, Theory given the lack of information about network evolution over long periods, it has been hard to convert these findings into Keywords statements about trends over time. Here we study a wide range of real graphs, and we observe densification power laws, graph generators, graph mining, some surprising phenomena. First, most of these graphs heavy-tailed distributions, small-world phenomena densify over time, with the number of edges growing super- linearly in the number of nodes. Second, the average dis- 1. INTRODUCTION tance between nodes often shrinks over time, in contrast In recent years, there has been considerable interest in to the conventional wisdom that such distance parameters graph structures arising in technological, sociological, and should increase slowly as a function of the number of nodes scientific settings: computer networks (routers or autonomous (like O(log n)orO(log(log n)).