1. Rivers of the World are Small-World Networks Carlos J. Anderson David G. Jenkins John F. Weishampel

2. What is a small-world network? Example: Six degrees of separation –But there are over six billion people in the world!

3. More small-world networks (a) Proteins (b) Neurons (c) Internet (a)(c) (b)

4. Classes of small-world networks Number of website links Cumulative connectivity distribution Scale-free Follows a power law Number of collaborations Cumulative connectivity distribution Broad-scale Follows a power law with sharp cutoff Number of transmission lines Cumulative connectivity distribution Single-scale Fast decaying tail

5. Random vs. scale-free Random Scale-free

6. Attack tolerance of networks Random network Scale-free network

7. 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”

8. 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?

9. 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

11. Network construction River systemNetwork

12. 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 way

13. 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

14. 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

15. 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

16. Acknowledgments Department of Biology University of Central Florida