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The Price of Anarchy on Complex Networks

The Price of Anarchy on Complex Networks. KIAS Conference July, 2006. HyeJin Youn, Hawoong Jeong Complex Systems & Statistical Physics Lab. (Dept. of Physics, KAIST, Korea). CSSPL. Marriage map between 100 richest people in Korea. Importance of networks & dynamics. CSSPL.

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The Price of Anarchy on Complex Networks

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  1. The Price of Anarchy on Complex Networks KIAS Conference July, 2006 HyeJin Youn, Hawoong Jeong Complex Systems & Statistical Physics Lab. (Dept. of Physics, KAIST, Korea) CSSPL

  2. Marriage map between 100 richest people in Korea Importance of networks & dynamics CSSPL

  3. Protein Interaction Network Internet routing H.Jeong et al (2001) Importance of networks & dynamics

  4. ∝ 1/width ∝ # of travelers Network Dynamics • “States” of both the nodes and the edges can change • Dynamics of the networks : The topology of the network itself often evolves in time • Dynamics on the networks : Agents are moving on the networks (E.g. Zero-range process, Contact process, Cascading failure, Shortest paths & OPTIMAL PATH) Latency function (like time or cost per person) ∝ length CSSPL

  5. Network flow with congestion Latency function Cost function on path i length of path i # of agent on path i width of path i S T Given network with many agents going from S (source) to T (target), what will be the optimized distribution of agents for best performance?? CSSPL Based on the model of Roughgarden & Tardos, 2000

  6. Optimizations in physics • Euler-Lagrange differential equation • minimal free energy in thermodynamic physics • Fitting experimental DATA with formula • Low temperature behavior of disordered magnets • … • There are two types of optimizations!!! Centralized control Minimizing Global Cost GlobalOptimization Decentralized control Each agent minimizes its own personal cost UserOptimization(Nash equilibrium) CSSPL

  7. The “Price of Anarchy” Decentralized control Each agent minimizes its own personal cost total cost of User Optimum Price of Anarchy total cost of Global Optimum Centralized control Minimizing Global Cost Koutsoupias & Papadimitriou, 1999 Price of Anarchy 1 ≤ (Roughgarden & Tardos, 2000) “Price we have to pay not being coordinated by central agency” “Price of being selfish” CSSPL

  8. S T Price of Anarchy: Contrived Example Pigous’s example: Congestion sensitive network 10 agents want to Go from S to T. If xa=x, then xb=10-x, ∴ total cost=10ᆞx + (10-x) ᆞ(10-x) = x2-10x+100=(x-5)2+75 ∴ xa=xb=5 with total cost 75 What will be the min total cost, i.e. Global Optimum = ? CSSPL

  9. Envy S T Price of Anarchy: Contrived Example The upper agents get envious of people with lower costs! xa = xb =5 BUT Global Optimum = 5x10 + 5x5 = 75  75/10 = 7.5min driving in average CSSPL

  10. S T Price of Anarchy: Contrived Example xa = 5 xb = 5 What will be the User Optimum? (Nash Equilibrium: everyone happy) CSSPL

  11. S T Price of Anarchy: Contrived Example xa = 5-1 Move to Lower path xb = 5+1 +1 user cost = 5 + 1 < 10  CSSPL

  12. S T Price of Anarchy: Contrived Example xa = 4-1 again +1 xb = 6+1 user cost = 6 + 1 < 10  CSSPL

  13. S T Price of Anarchy: Contrived Example xa = 3-1 again +1 xb = 7+1 user cost = 7 + 1 < 10  CSSPL

  14. S T Price of Anarchy: Contrived Example xa = 2-1 again +1 xb = 8+1 user cost = 8 + 1 < 10  CSSPL

  15. S T Price of Anarchy: Contrived Example xa = 1-1 again +1 xb = 9+1 User Optimum = 10 x10 = 100  avg 10min travel time > avg 7.5-min travel time CSSPL

  16. S T Price of Anarchy: Contrived Example There is a gap between global optimum & user optimum! xa = 5 vs 0 xb = 5 vs 10 User Optimum = 10 x10 = 100 Price of Anarchy! 4/3 Global Optimum = 5x10 + 5x5 = 75 CSSPL

  17. More realistic/complex example • Assumption: traffic reaches at equilibrium • Price of Anarchy on a real world • the Boston Road Network • (with real geometrical information like width, length, one-way etc) • Traffic from central square (S) to copley square (T) CSSPL

  18. Boston Road Map CSSPL

  19. Start Boston Road Network End (nodes 59, edges 108, regular-like) Latency function = ax + b Width length CSSPL

  20. More realistic/complex example • Assumption: traffic reaches at equilibrium • Price of Anarchy on a real world • the Boston Road Network • (with real geometrical information) • Global optimum : mapping to Min-cost Max-flow problem • User optimum ~ approximate optimum: Metropolis Algorithm and Annealing method to find out the optimum configurations CSSPL

  21. User OptimumGlobal Optimum Number of traveler =1 CSSPL

  22. User OptimumGlobal Optimum Number of traveler =2 CSSPL

  23. User OptimumGlobal Optimum Number of traveler =3 CSSPL

  24. User OptimumGlobal Optimum Number of traveler =4 CSSPL

  25. User OptimumGlobal Optimum Number of traveler =5 CSSPL

  26. User OptimumGlobal Optimum Number of traveler =6 CSSPL

  27. User OptimumGlobal Optimum Number of traveler =7 CSSPL

  28. User OptimumGlobal Optimum Number of traveler =8 CSSPL

  29. User OptimumGlobal Optimum Number of traveler =9 CSSPL

  30. User OptimumGlobal Optimum Number of traveler =10 CSSPL

  31. User OptimumGlobal Optimum Number of traveler =11 CSSPL

  32. User OptimumGlobal Optimum Number of traveler =12 CSSPL

  33. User OptimumGlobal Optimum Number of traveler =13 CSSPL

  34. User OptimumGlobal Optimum Number of traveler =14 CSSPL

  35. User OptimumGlobal Optimum Number of traveler =15 CSSPL

  36. User OptimumGlobal Optimum Number of traveler =16 CSSPL

  37. User OptimumGlobal Optimum Number of traveler =17 CSSPL

  38. User OptimumGlobal Optimum Number of traveler =18 CSSPL

  39. User OptimumGlobal Optimum Number of traveler =19 CSSPL

  40. Congestion distribution on the edges User OptimumGlobal Optimum Number of Agents: 20 CSSPL

  41. Variation of POA with Agent # Reminder: POA = UE/GO Price of Anarchy number of agents CSSPL

  42. S T Why Price of Anarchy decreases? •Fitness landscape for a simple case: cb (xb)= xb2 Strategy a l(xb)= xb cb (xb)= 5xb l(xa)= 5 5 5 l(xa)= 5 l(xb)= xb 2.5 Strategy b Fitness for User Optimum Fitness for Global Optimum l(xb)= 2xb l(xa)= 5 2.5 CSSPL

  43. 1 0.63 C=1 Nash Equilibrium 4/3 x (Global Optimum) T S 0.37 C(X) = X^3 POA too small?? •Linear latency function: • More general edge latency function • n > 1 - Roughgarden-Tardos When n=3 UO = 1 GO = 0.37*1 + (0.63)^4 = 0.528 POA = UA/GO = 1.894 Bigger than 4/3 (n=1) CSSPL

  44. Where to use?? • To write a paper … • Network design for better traffic?

  45. Making network more efficient without central government?? • Lower PoA ~ better(?) system (∵even w/o central control, user optimum is closer to global optimum, better!) • Let’s make better network with lower PoA • Simple thought (by stupid government): construct more roads with tax money!  Braess paradox (counter-intuitive consequences)

  46. increase Braess’s Paradox Again 10 travelers want to move from S to T. x 10 S 0: cost-free express road T 10 x User Optimum without middle arc = 150 = Global Optimum Price of Anarchy = 200/150 = 4/3 User Optimum with middle arc = 200 CSSPL

  47. Start Boston Road Network End CSSPL

  48. Start Affect of an Arc Removal on User Optimum End negative CSSPL

  49. Affect of Arc Removal on User Optimum Cost increment PoA=UO/GO edge index 53 out of 108 edges are identified as deteriorating inefficiency! (ΔPoA<0) 19 out of 53 edges are found having made the decentralised system cost more! (ΔNE<0) CSSPL

  50. More systematic approaches • Model network analysis • Regular Lattice • Erdos-Renyi Network • Small-world Network • Scale-free Network • Multiple Sources & Targets • Any correlation between PoA and other topological quantities? CSSPL

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