Presented in INFOCOM 2015 http://www3.cs.stonybrook.edu/~chni/publication/ricci-curvature/ -- Analysis of Internet topologies has shown that the Internet topology has negative curvature, measured by Gromov's ``thin triangle condition'', which is tightly related to core congestion and route reliability. In this work we analyze the discrete Ricci curvature of the Internet, defined by Ollivier et al., Lin et al., etc. Ricci curvature measures whether local distances diverge or converge. It is a more local measure which allows us to understand the distribution of curvatures in the network. We show by various Internet data sets that the distribution of Ricci curvature is spread out, suggesting the network topology to be non-homogenous. We also show that the Ricci curvature has interesting connections to both local measures such as node degree and clustering coefficient, global measures such as betweenness centrality and network connectivity, as well as auxiliary attributes such as geographical distances. These observations add to the richness of geometric structures in complex network theory.

1. Ricci Curvature of
Internet Topology
Chien-Chun Ni
Stony Brook University
Join work with: Yu-Yao Lin1, Jie Gao1, David Gu1, Emil Saucan2
1: Stony Brook University, 2: Technion, Israel Institute of Technology.

2. What is Internet Topology?
• Topology Graph: A node & edge relationship
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P2P Network Router Network AS Network

3. Why Internet Topology?
• Structural property:
• Robustness
• Vulnerability
• Connectivity
• Information flow:
• Congestion control
• Virus diffusion
• Network Evolution
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4. Analysis of Internet Topology?
• Graph Structure:
• Degree distribution
• Graph diameter
• Mean shortest path length
• Graph Geometry:
• Curvature
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5. Curvature?
• A geometric property: Flatness of an object
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=π
2D Plane
Zero Curvature
3D Sphere
Positive Curvature
3D Saddle
Negative Curvature

6. Internet Curvature, Prior Discovery
• Internet has negative curvature[1-2]
• Def. by Gromov’s δ-hyperbolicity (thin triangle property)
[Def.] For any triple a, b, c, the min distance from one
shortest path to the other two is no greater than δ
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[1]: Narayan, O., & Saniee, I. (2011). Large-scale curvature of networks. Physical Review E.
[2]: Shavitt, Y., & Tankel, T. (2004). On the Curvature of the Internet and its usage for Overlay Construction and Distance
Estimation. Infocom 2004.

7. Internet is Negatively Curved
A variety of data sets, both AS-level and router level
topologies, show δ-hyperbolicity for small constant δ.
• Properties:
• Tree-Like
• There is a ‘core’ in which all shortest path visit
• Congestion inside the core is high
• Preprocessing can speed up shortest path queries
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8. Global v.s. Local Curvature
• Gromov δ-hyperbolicity: global measure
• Define one parameter on whole network graph
• No information provided on each vertex and edge
• Local Curvature?
• Which edges are negatively curved?
• Local curvature v.s. network congestion?
• Local curvature v.s. node centrality?
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9. Curvature in Geometry
• How do we know that the earth is round?
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10. Sectional Curvature
Consider a tangent vector v = xy. Take another tangent vector
wx and transport it along v to be a tangent vector wy at y.
If |x’y’| < |xy| the sectional curvature is positive.
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• Ricci Curvature: averaging over all directions w

11. Discrete Ricci Curvature
[Take the analog]: For an edge xy, consider the
distances from x’s neighbors to y’s neighbors and
compare it with the length of xy.
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12. Discrete Ricci Curvature
• Issue: how to match x’s neighbors to y’s neighbors?
• Assign uniform distribution μ1 , μ2 on x’ and y’s neighbors.
• Use optimal transportation distance (earth-mover
distance) from μ1 to μ2: the matching that minimize the
total transport distance.
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13. Discrete Ricci Curvature[1,2]
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[1]: Ollivier, Y. (2007, January 31). Ricci curvature of Markov chains on metric spaces. arXiv.org.
[2]: Lin, Y., Lu, L., & Yau, S.-T. (2011). Ricci curvature of graphs. Tohoku Mathematical Journal, 63(4), 605–627.

14. Example: Zero Curvature
• 2D grid
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15. Example: Negative Curvature
• Tree: κ(x , y ) = 1/dx + 1/dy − 1, dx is degree of x.
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16. Example: Positive Curvature
• Complete graph
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17. Evaluation Datasets: Real Network

18. Curvature Distribution
• Negatively curved edges  “backbones”
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19. Curvature Distribution (Conti.)
Negatively curved Positively curved
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20. Curvature Distribution

21. Network Connectivity
• Adding edges with increasing/decreasing curvature
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22. Robustness vs. Vulnerability
• Removing edges with increasing curvature
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23. Curvature v.s. Centrality
• Negatively curved nodes are centrally located in the
network, but the correlation are weak.

24. Data Sets: Model Network
• Erdos-Renyi random graph (G(n,p) graph)
• Watt-Strogatz graph
• Random regular
• Preferential attachment
• Configuration model
• H(3,7) grid (Hyperbolic grid)

25. Curvature Distribution (Model)

26. Network Connectivity (Model)
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27. Conclusion and Discussion
• Another measure to understand the Internet topology
• Limitation of data sets
• Network embedding: Euclidean, hyperbolic, hybrid?
• Network evolution: Why?
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28. Thanks!
Questions and comments?
Contact: Chien-Chun Ni
Email: chni@cs.stonybrook.edu
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29. Backup
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30. Robustness vs. Vulnerability
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31. Robustness vs. Vulnerability
(Model)
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32. Network Congestion
• Curvature is correlated with betweenness centrality
(# shortest path through an edge).

33. Network Connectivity (Conti.)
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