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

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Scale-free

Follows a power law

Number of collaborations

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Broad-scale

Follows a power law with sharp cutoff

Number of transmission lines

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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

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10-6

10-5

10-4

10-3

10-2

10-1

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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