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Ricci Curvature of
Internet Topology
Chien-Chun Ni
Stony Brook University
Join work with: Yu-Yao Lin1, Jie Gao1, David Gu1...
What is Internet Topology?
• Topology Graph: A node & edge relationship
INFOCOM 2015 2
P2P Network Router Network AS Netwo...
Why Internet Topology?
• Structural property:
• Robustness
• Vulnerability
• Connectivity
• Information flow:
• Congestion...
Analysis of Internet Topology?
• Graph Structure:
• Degree distribution
• Graph diameter
• Mean shortest path length
• Gra...
Curvature?
• A geometric property: Flatness of an object
INFOCOM 2015 5
=π
2D Plane
Zero Curvature
3D Sphere
Positive Curv...
Internet Curvature, Prior Discovery
• Internet has negative curvature[1-2]
• Def. by Gromov’s δ-hyperbolicity (thin triang...
Internet is Negatively Curved
A variety of data sets, both AS-level and router level
topologies, show δ-hyperbolicity for ...
Global v.s. Local Curvature
• Gromov δ-hyperbolicity: global measure
• Define one parameter on whole network graph
• No in...
Curvature in Geometry
• How do we know that the earth is round?
INFOCOM 2015 9
Sectional Curvature
Consider a tangent vector v = xy. Take another tangent vector
wx and transport it along v to be a tang...
Discrete Ricci Curvature
[Take the analog]: For an edge xy, consider the
distances from x’s neighbors to y’s neighbors and...
Discrete Ricci Curvature
• Issue: how to match x’s neighbors to y’s neighbors?
• Assign uniform distribution μ1 , μ2 on x’...
Discrete Ricci Curvature[1,2]
INFOCOM 2015 13
[1]: Ollivier, Y. (2007, January 31). Ricci curvature of Markov chains on me...
Example: Zero Curvature
• 2D grid
INFOCOM 2015 14
Example: Negative Curvature
• Tree: κ(x , y ) = 1/dx + 1/dy − 1, dx is degree of x.
INFOCOM 2015 15
Example: Positive Curvature
• Complete graph
INFOCOM 2015 16
Evaluation Datasets: Real Network
Curvature Distribution
• Negatively curved edges  “backbones”
INFOCOM 2015 18
Curvature Distribution (Conti.)
Negatively curved Positively curved
INFOCOM 2015 19
Curvature Distribution
Network Connectivity
• Adding edges with increasing/decreasing curvature
INFOCOM 2015 21
Robustness vs. Vulnerability
• Removing edges with increasing curvature
INFOCOM 2015 22
Curvature v.s. Centrality
• Negatively curved nodes are centrally located in the
network, but the correlation are weak.
Data Sets: Model Network
• Erdos-Renyi random graph (G(n,p) graph)
• Watt-Strogatz graph
• Random regular
• Preferential a...
Curvature Distribution (Model)
Network Connectivity (Model)
INFOCOM 2015 26
Conclusion and Discussion
• Another measure to understand the Internet topology
• Limitation of data sets
• Network embedd...
Thanks!
Questions and comments?
Contact: Chien-Chun Ni
Email: chni@cs.stonybrook.edu
INFOCOM 2015 28
Backup
INFOCOM 2015 29
Robustness vs. Vulnerability
INFOCOM 2015 30
Robustness vs. Vulnerability
(Model)
INFOCOM 2015 31
Network Congestion
• Curvature is correlated with betweenness centrality
(# shortest path through an edge).
Network Connectivity (Conti.)
INFOCOM 2015 33
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Ricci Curvature of Internet Topology

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Presented in INFOCOM 2015
http://www3.cs.stonybrook.edu/~chni/publication/ricci-curvature/
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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.

Published in: Science

Ricci Curvature of Internet Topology

  1. 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. 2. What is Internet Topology? • Topology Graph: A node & edge relationship INFOCOM 2015 2 P2P Network Router Network AS Network
  3. 3. Why Internet Topology? • Structural property: • Robustness • Vulnerability • Connectivity • Information flow: • Congestion control • Virus diffusion • Network Evolution INFOCOM 2015 3
  4. 4. Analysis of Internet Topology? • Graph Structure: • Degree distribution • Graph diameter • Mean shortest path length • Graph Geometry: • Curvature INFOCOM 2015 4
  5. 5. Curvature? • A geometric property: Flatness of an object INFOCOM 2015 5 =π 2D Plane Zero Curvature 3D Sphere Positive Curvature 3D Saddle Negative Curvature
  6. 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 INFOCOM 2015 6 [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. 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 INFOCOM 2015 7
  8. 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? INFOCOM 2015 8
  9. 9. Curvature in Geometry • How do we know that the earth is round? INFOCOM 2015 9
  10. 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. INFOCOM 2015 10 • Ricci Curvature: averaging over all directions w
  11. 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. INFOCOM 2015 11
  12. 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. INFOCOM 2015 12
  13. 13. Discrete Ricci Curvature[1,2] INFOCOM 2015 13 [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. 14. Example: Zero Curvature • 2D grid INFOCOM 2015 14
  15. 15. Example: Negative Curvature • Tree: κ(x , y ) = 1/dx + 1/dy − 1, dx is degree of x. INFOCOM 2015 15
  16. 16. Example: Positive Curvature • Complete graph INFOCOM 2015 16
  17. 17. Evaluation Datasets: Real Network
  18. 18. Curvature Distribution • Negatively curved edges  “backbones” INFOCOM 2015 18
  19. 19. Curvature Distribution (Conti.) Negatively curved Positively curved INFOCOM 2015 19
  20. 20. Curvature Distribution
  21. 21. Network Connectivity • Adding edges with increasing/decreasing curvature INFOCOM 2015 21
  22. 22. Robustness vs. Vulnerability • Removing edges with increasing curvature INFOCOM 2015 22
  23. 23. Curvature v.s. Centrality • Negatively curved nodes are centrally located in the network, but the correlation are weak.
  24. 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. 25. Curvature Distribution (Model)
  26. 26. Network Connectivity (Model) INFOCOM 2015 26
  27. 27. Conclusion and Discussion • Another measure to understand the Internet topology • Limitation of data sets • Network embedding: Euclidean, hyperbolic, hybrid? • Network evolution: Why? INFOCOM 2015 27
  28. 28. Thanks! Questions and comments? Contact: Chien-Chun Ni Email: chni@cs.stonybrook.edu INFOCOM 2015 28
  29. 29. Backup INFOCOM 2015 29
  30. 30. Robustness vs. Vulnerability INFOCOM 2015 30
  31. 31. Robustness vs. Vulnerability (Model) INFOCOM 2015 31
  32. 32. Network Congestion • Curvature is correlated with betweenness centrality (# shortest path through an edge).
  33. 33. Network Connectivity (Conti.) INFOCOM 2015 33

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