Classes of small-world networks
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We study the statistical properties of a variety of diverse real-world networks. We present evidence of the occurrence of three classes of small-world networks: (a) scale-free networks, characterized by a vertex connectivity distribution that decays as a power law; (b) broad-scale networks, characterized by a connectivity distribution that has a power law regime followed by a sharp cutoff; and (c) single-scale networks, characterized by a connectivity distribution with a fast decaying tail. Moreover, we note for the classes of broad-scale and single-scale networks that there are constraints limiting the addition of new links. Our results suggest that the nature of such constraints may be the controlling factor for the emergence of different classes of networks.
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The degree distribution is an important network property that can be used to distinguish between networks.
“The degree distribution is the histogram of node degrees for the overall network and is an important network property 5 that can be used to distinguish between networks (described below).”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [5]
The absence of scale-free degree distribution in brain networks is not entirely unexpected given the space constraints involved; it would be difficult to accommodate superconnected regions in the extreme tail of the degree distribution within the closed space of the skull.
“This is not entirely unexpected given the space constraints involved.In other words, it would be difficult to imagine having sufficient room for a brain growing inside a closed space like the skull to accommodate superconnected regions that would reside in the extreme tail of the degree distribution.5”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [5]
Other models include the aging of vertices, whereby a certain number of nodes disappear over time, a process that replicates scale-free degree distributions.
“Other models include the aging of vertices, whereby a certain number of nodes disappear over time, a process that replicates scale-free degree distributions.5”
From Graph theory analysis of complex brain networks: new concepts in brain mapping applied to neurosurgery (2016) · cites this paper as [5]
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- Authors
- Amaral, L. A. N., Scala, A., Barthélémy, M., & Stanley, H. E.
- Affiliations
- Center for Polymer Studies and Department of Physics, Boston University, Boston, MA 02215
- Venue
- Proceedings of the National Academy of Sciences
- Publisher
- National Academy of Sciences
- Published
- 2000-09-26
- Volume
- 97
- Issue
- 21
- Pages
- 11149-11152
- ISSN
- 0027-8424 (ISSN-L)
- 1091-6490
- PMC
- PMC17168
- OpenAlex
- W2104085672
- Cited by
- 3,051
- References
- 3
- Citations by year
- 2026: 26 · 2025: 54 · 2024: 56 · 2023: 55 · 2022: 71 · 2021: 95 · 2020: 102 · 2019: 96 · 2018: 140 · 2017: 110 · 2016: 137 · 2015: 139 · 2014: 172 · 2013: 164 · 2012: 197
- FWCI
- 42.50
- Type
- journal article
- Language
- en
- Open access
- green
- www.ncbi.nlm.nih.gov
- Locations
- Proceedings of the National Academy of Sciences (publishedVersion)
- Europe PMC (PubMed Central) (publishedVersion)
- PubMed Central (publishedVersion)
- PubMed (publishedVersion)
- arXiv (Cornell University) (submittedVersion)
- OpenBU (Boston University) (submittedVersion)
- arXiv (Cornell University)
- Funders
- U.S. Public Health Service (P41 RP13622)
- National Center for Research Resources (P41RR013622)
- Topics
- Physical Sciences › Physics and Astronomy › Statistical and Nonlinear Physics › Complex Network Analysis Techniques
- Social Sciences › Economics, Econometrics and Finance › Economics and Econometrics › Complex Systems and Time Series Analysis
- Physical Sciences › Mathematics › Geometry and Topology › Graph theory and applications
- Keywords
- small-world networks, scale-free networks, degree distribution, power-law decay, connectivity distribution, real-world networks, network topology
- MeSH terms
- Interpersonal Relations, Nerve Net, Nonlinear Dynamics
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