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Rivers of the World are Small-World Networks
Carlos J. AndersonDavid G. Jenkins
John F. Weishampel
What is a small-world network?
• Example: Six degrees of separation
– But there are over six billion people in the world!
More small-world networks
(a) Proteins (b) Neurons (c) Internet
(a) (c)
(b)
Classes of small-world networks
Number of website links
Cu
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necti
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y d
istr
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Scale-free
Follows a power law
Number of collaborations
Cu
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Broad-scale
Follows a power law with sharp cutoff
Number of transmission lines
Cu
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necti
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Single-scale
Fast decaying tail
Random vs. scale-free
Random Scale-free
Attack tolerance of networks
Random networkScale-free network
Emergence of a scale-free network
• Growth– Starting with two connected nodes, add a new
node to the network one at a time
• Preferential attachment– New nodes prefer to attach to the more
connected nodes
“Rich get richer”
We asked…
• Are river systems small-world networks?
• If so, which class?
– Single-scale, broad-scale, scale-free?
• How can our results be used in aquatic ecology and conservation?
Methods
• Analyzed USGS HYDRO1k data– http://lpdaac.usgs.gov/gtopo30/hydro– Flow accumulation data for Africa, Asia,
Europe, South America, and North America
• Custom Java code– Analyzed flow accumulation data– Counted all connecting tributaries per stream
Network construction
River system Network
Results• Nearly scale-free
• Larger scales show greatest variation
• Curvature suggests constraints, such as continental edge
• All river systems of the world are connected in the same wayLinks
100 101 102 103 104
Cum
ulat
ive
Pro
babi
lity
10-7
10-6
10-5
10-4
10-3
10-2
10-1
100
AfricaAsiaEuropeNorth AmericaSouth America
Predictions
• How should a river system respond to disturbance (e.g., invasive species)?– Random attacks should not disrupt the
connectivity of the system– System should be vulnerable to attacks on
major streams (hubs)
• All river systems are created equal– Studies on river topology should apply to all
rivers of the world
Conservation
• Focus conservation efforts on major streams (hubs)– Routine check on boats that enter major
streams (may carry invasive species)
• Restoration of migratory pathways– Removal of certain dams that block salmon
from their reproductive sites
Summary
• Scale-free networks are tolerant to random attacks, but not to directed attacks
• All rivers of the world are scale-free
• Management and conservation policies may be revised based on this topology
Acknowledgments
Department of Biology
University of Central Florida