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Graphs over time: densification laws, shrinking diameters and possible explanations
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, 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.
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“Finally, the study of networks in a variety of applications assumes static networks, with some notable exceptions.21”
From A Big Data Guide to Understanding Climate Change: The Case for Theory-Guided Data Science (2014) · cites this paper as [21]
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Wehmuth, K., & Ziviani, A. (2012) · doi:10.1145/2184356.2184368 · Cited by 26
Metadata
- Authors
- Leskovec, J., Kleinberg, J., & Faloutsos, C.
- Affiliations
- Carnegie Mellon University (US)
- Cornell University (US)
- Venue
- Proceedings of the eleventh ACM SIGKDD international conference on Knowledge discovery in data mining
- Publisher
- ACM
- Published
- 2005-08-21
- Pages
- 177-187
- OpenAlex
- W2111708605
- Cited by
- 2,281
- References
- 31
- Citations by year
- 2026: 42 · 2025: 88 · 2024: 78 · 2023: 79 · 2022: 85 · 2021: 135 · 2020: 118 · 2019: 128 · 2018: 146 · 2017: 127 · 2016: 157 · 2015: 164 · 2014: 166 · 2013: 168 · 2012: 155
- FWCI
- 77.15
- Type
- conference paper
- Language
- en
- Open access
- closed
- Locations
- ACM SIGKDD Conference on Knowledge Discovery and Data Mining (KDD) (publishedVersion)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Physical Sciences › Computer Science › Computational Theory and Mathematics › Graph Theory and Algorithms
- Physical Sciences › Mathematics › Discrete Mathematics and Combinatorics › Graph theory and applications
- Keywords
- graph evolution, network densification, superlinear growth, forest fire model, social networks, technological networks, information networks, small-world effect, degree distribution, community structure
About these checks
Registry, retraction, Cites and statistics results are computed, not judged, with one exception: a printed reference that no index lists is read once by a model before it counts as a citation error. Claims are extracted by a model and kept only when their quote appears verbatim in the text; each cited source is then checked, with a model’s help, for whether it says what the claim relies on. Those verdicts can be wrong — read the claim’s quote and the cited source before relying on one.
Statistics and arithmetic are recomputed, not judged.
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