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# Phoenix: A Weight-based Network Coordinate System Using Matrix Factorization

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Yang Chen, Xiao Wang, Cong Shi, Eng Keong Lua, Xiaoming Fu, Beixing Deng, Xing Li. Phoenix: A Weight-based Network Coordinate System Using Matrix Factorization. IEEE Transactions on Network and Service Management, 2011, 8(4):334-347.

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• ### Phoenix: A Weight-based Network Coordinate System Using Matrix Factorization

1. 1. Phoenix: A Weight-BasedNetwork Coordinate System Using Matrix Factorization Yang Chen Department of Computer Science Duke University ychen@cs.duke.edu
2. 2. Outline• Background• System Design• Evaluation• Perspective Future Work 2
3. 3. BACKGROUND 3
4. 4. Internet Distance What? • Round-trip propagation / transmission delay between two Internet nodes Why? • Strong indicator of network proximity • Relatively stable How? • Measurement tool “Ping” is with major operating systems 50msAlice Bob 4
5. 5. Use Cases• Knowledge of Internet distance is useful for… – P2P content delivery (file sharing/streaming) – Online/mobile games – Overlay routing – Server selection in P2P/Cloud – Network monitoring 5
6. 6. Scalability• Huge number of end-to-end paths in large scale systems N nodes N ´ N measurements SLOW and COSTLY when the system becomes large! 6
7. 7. Network Coordinate (NC) Systems (5, 10, 2) (-3, 4, -2)Alice Bob Distance Function 22ms • Scalable measurement: N2  NK (K << N) • Every node is assigned with coordinates • Distance function: compute the distance between two nodes without explicit measurement 7 [Ng et al, INFOCOM’02]
8. 8. Deployments They are all usingNetwork Coordinate Systems! 8
9. 9. Basic models• Euclidean Distance-based NC (ENC) – Modeling the Internet as a Euclidean space – Systems: Vivaldi [Dabek et al., SIGCOMM’04], GNP [Ng et al, INFOCOM’02], NPS [Ng et al., USENIX ATC’04], PIC [Costa et al., ICDCS’04]…• Matrix Factorization-based NC (MFNC) – Factorizing an Internet distance matrix as the product of two smaller matrices – Systems: IDES [Mao et al., JSAC’06], Phoenix, … 9
10. 10. Modeling the Internet as a Euclidean space d=3• In a d-dimensional Euclidean space, each node will be mapped to a position• Compute distances based on coordinates using Euclidean distance 10
11. 11. Triangle Inequality Violation 29.9 > 5.6+3.6 Czech Republic 5.6 ms 29.9 ms Slovakia 3.6 ms Hungary A Triangle Inequality Violation (TIV)Predicted distances in example in GEANT networkEuclidean space must satisfy triangle inequality Lots of TIVs in the Internet due sub-optimal routing!! 11 [Zheng et al, PAM’05]
12. 12. Correlation in Internet Distance Matrices Distance measurement using PlanetLab nodes Duke UNC Yale Aachen Oxford Toronto THU NUSDuke - 3 24 107 122 37 219 252UNC 3 - 24 106 109 38 219 253 Internet paths with nearby end nodes are often overlap!! Rows in different Internet distance matrices are large correlated (low effective rank) [Tang et al, IMC’03], [Lim et al, ToN’05], [Liao et al, CoNEXT’11] 12
13. 13. Factorization of an Internet Distance Matrix  N columns { d columns  » N rows ´ M X Y T        X7 = [ 1 0 3 ],Y2 = [ 2 0 5 ] M ij » Xi ×Yj    M 72 » X7 ×Y2 =1´ 2 + 0 + 3´ 5 =17 [Mao et al., JSAC’06] 13
14. 14. Matrix Factorization-Based NC  N columns  X2 { d columns    Y2 » N rows ´ M X Y T• Each node i has an outgoing vector Xi and an incoming vector Yi• Distance function is the dot product. 14 No triangle inequality constrain in this model!
15. 15. SYSTEM DESIGN 15
16. 16. Goals• Substantial improvement in prediction accuracy• Decentralized and scalable• Robust to dynamic Internet 16
17. 17. Workflow of Phoenix System Peer Scalable CoordinatesInitialization Discovery Measurement Calculation 17
18. 18. System Initialization Measured Distance Predicted Distance (X1,Y1) (X2,Y2) H1 H1 H2 H2 H4 H4H3 H3 (X4,Y4) (X3,Y3) • Early nodes (N<K): Full-mesh measurement • Compute coordinates of early nodes by minimizing the overall discrepancy between predicted distances and measured distancesNonnegative matrix factorization: [D. D. Lee and H. S. Seung, Nature, 401(6755):788–791, 181999.]
19. 19. Dynamic Peer Discovery Tracker H2 H3 H5 H3 H4 H6 H1 H2H2 H3 H4 H5 H6 H1 H3 H4 H5 H6 Gossip among nodes • N>K, all nodes become ordinary nodes 19
20. 20. Reference Node Selection• Every new node randomly selects K existing nodes as reference nodes 20
21. 21. Measurement and Bootstrap Coordinates Calculation Measured Distance Predicted Distance (X2,Y2) (XK,YK) (X1,Y1) R1 R2  RK R1 R2  RK H new H new (Xnew,Ynew)• Node Hnew computes its own coordinates by minimizing the overall discrepancy between predicted distances and measured distances (Non-negative least squares) 21
22. 22. Accuracy of Reference CoordinatesNode N (XA,YA) … Node ANode 3 Predicted Distance Measured distanceNode 2Node 1 0 50 100 150 Distance between Node A and every other node 22
23. 23. Accuracy of Reference Coordinates (cont.)Node N (XB,YB) … Node B Misleading the nodesNode 3 referring to Node B!! Predicted Distance Measured DistanceNode 2Node 1 0 20 40 60 80 100 120 Distance between Node B and every other node 23
24. 24. Referring to Inaccurate Coordinates (X2,Y2) (XK,YK)(X1,Y1) R1 R2  RK Error Propagation: Hnew may mislead nodes refer to it H new (Xnew,Ynew) Give preference to Minimize accurate reference the impact coordinates of RK 24
25. 25. Heuristic Weight AssignmentRK Predicted Distance Measured distance…R3 Enhanced Coordinates Bootstrap CoordinatesR2R1 H new Updating coordinates 0 50 100 150 200 regularly Distance between Hnew and every reference node 25
26. 26. EVALUATION 26
27. 27. Evaluation Setup• Data sets – PL: 169 PlanetLab nodes – King: 1740 Internet DNS servers• Metric – Relative Error (RE) MeasuredDist - PredictedDist RE = min(MeasuredDist, PredictedDist) 27
28. 28. Evaluation: Relative Error 90th Percentile Relative Error Phoenix Phoenix Vivaldi IDES (Simple) 0.63 0.91 0.83 0.89 28
29. 29. Evaluation (cont.)• Other findings through evaluation – Robust to node churn – Fast convergence – Robust to measurement anomalies – Robust to distance variation 29
30. 30. FUTURE WORK 30
31. 31. Perspective Topics• NC systems in mobile-centric environment – Access latency, host mobility, host churn• Scalable Prediction of other important network parameters – Available bandwidth, shortest-path distance in social graph 31
32. 32. Software• NCSim – Simulator of Decentralized Network Coordinate Algorithms – http://code.google.com/p/ncsim/• Phoenix – Original Phoenix simulator in IEEE TNSM paper – http://www.cs.duke.edu/~ychen/Phoenix_TNS M_2011.zip 32