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Ricci Curvature of Internet Topology

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Chien-Chun Ni
Chien-Chun Ni

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.

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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, Emil Saucan2 1: Stony Brook University, 2: Technion, Israel Institute of Technology.
What is Internet Topology? • Topology Graph: A node & edge relationship INFOCOM 2015 2 P2P Network Router Network AS Network
Why Internet Topology? • Structural property: • Robustness • Vulnerability • Connectivity • Information flow: • Congestion control • Virus diffusion • Network Evolution INFOCOM 2015 3
Analysis of Internet Topology? • Graph Structure: • Degree distribution • Graph diameter • Mean shortest path length • Graph Geometry: • Curvature INFOCOM 2015 4
Curvature? • A geometric property: Flatness of an object INFOCOM 2015 5 =π 2D Plane Zero Curvature 3D Sphere Positive Curvature 3D Saddle Negative Curvature
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.
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
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
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 tangent vector wy at y. If |x’y’| < |xy| the sectional curvature is positive. INFOCOM 2015 10 • Ricci Curvature: averaging over all directions w
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
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
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.
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 attachment • Configuration model • H(3,7) grid (Hyperbolic grid)
Curvature Distribution (Model)
Network Connectivity (Model) INFOCOM 2015 26
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
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

More Related Content

Ricci Curvature of Internet Topology

Editor's Notes

  1. Social: node-> ppl, edge-> relation In different resolution: the meaning of edge and node changes
  2. How easy to cut a tree into half, min max flow Retweet or like, predict multicast: windows update, Infrastructure upgrade priority How internet evolve
  3. Result like power law, short diameter
  4. Draw a tree Imaging high way Euclidean =>inf Hyperbolic plain => constant
  5. Delta => hopcoun reft value grow slower
  6. How do we know it’s negative? Two graph, are they identical
  7. Local Curvauture Hit equater
  8. Negatively curved edges are like “backbones”, maintaining the connectivity of clusters, in which edges are mostly positively curved.
  9. Core! Robustness Backbone effect Negatively curved edges are well connected. Adding edges with increasing/decreasing curvature: few/many connected components.
  10. Removing edges with increasing curvature: size of largest connected component drops quickly.
  11. Positive edge => cluster happen
  12. Core of network H37 => hyperbolic core backbone effect Small world property is not relevant; graph hyperbolicity and power law degree distribution appear to be more relevant.
  13. No edge weight Different network => different curvature distribution Negatively curved edges: Tend to act as backbones of graph Related to robustness of graph Positively curved edges: Act as cluster or boundary leaves
  14. X