The Price of Anarchy on Complex Networks

Transcription

1 The Price of Anarchy on Complex Networks NetSci. Conference May 22, 2006 HyeJin Youn, Hawoong Jeong Complex Systems & Statistical Physics Lab. (Dept. of Physics, KAIST, Korea)

2 Importance of networks & dynamics Protein Marriage map between 100 richest people in Korea Interaction Internet routing Traffic Networkon roads WWW H.Jeong et al (2001)

3 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 1 Latency = travelers + width [# of ] length 1/width # of travelers

4 Network flow with congestion Cost function on path i Latency function 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?? Based on the model of Roughgarden & Tardos, 2000

5 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 Global Optimization Decentralized control Each agent minimizes its its own personal cost User Optimization (Nash equilibrium)

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

7 Price of Anarchy: Contrived Example Pigous s example: Congestion sensitive network 10 agents want to Go from S to T. S T What will be the min total cost, i.e. Global Optimum =? If x a =x, then x b =10-x, total cost=10ᆞx + (10-x) ᆞ(10-x) = x 2-10x+100=(x-5) x a =x b =5 with total cost 75

8 Price of Anarchy: Contrived Example Envy The upper agents get envious of people with lower costs! x a = x b =5 S T BUT Global Optimum = 5x10 + 5x5 = 75 75/10 = 7.5min driving in average

9 Price of Anarchy: Contrived Example x a = 5 S x b = 5 T What will be the User Optimum? (Nash Equilibrium: everyone happy)

10 Price of Anarchy: Contrived Example Move to Lower path x a = 5-1 x b = S T user cost = < 10

11 Price of Anarchy: Contrived Example again x a = x b = 6+1 S T user cost = < 10

12 Price of Anarchy: Contrived Example again x a = x b = 7+1 S T user cost = < 10

13 Price of Anarchy: Contrived Example again x a = x b = 8+1 S T user cost = < 10

14 Price of Anarchy: Contrived Example again x a = x b = 9+1 S T User Optimum = 10 x10 = 100 avg 10min travel time > avg 7.5-min travel time

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

16 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)

17 Boston Road Map

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

19 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

20 User Optimum Global Optimum Number of traveler =1

21 User Optimum Global Optimum Number of traveler =2

22 User Optimum Global Optimum Number of traveler =3

23 User Optimum Global Optimum Number of traveler =4

24 User Optimum Global Optimum Number of traveler =5

25 User Optimum Global Optimum Number of traveler =6

26 User Optimum Global Optimum Number of traveler =7

27 User Optimum Global Optimum Number of traveler =8

28 User Optimum Global Optimum Number of traveler =9

29 User Optimum Global Optimum Number of traveler =10

30 User Optimum Global Optimum Number of traveler =11

31 User Optimum Global Optimum Number of traveler =12

32 User Optimum Global Optimum Number of traveler =13

33 User Optimum Global Optimum Number of traveler =14

34 User Optimum Global Optimum Number of traveler =15

35 User Optimum Global Optimum Number of traveler =16

36 User Optimum Global Optimum Number of traveler =17

37 User Optimum Global Optimum Number of traveler =18

38 User Optimum Global Optimum Number of traveler =19

39 Congestion distribution on the edges User Optimum Global Optimum Number of Agents: 20

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

41 Why Price of Anarchy decreases? Fitness landscape for a simple case: Strategy a c b (x b )= xb 2 S l(x a )= 5 l(x b )= xb T 5 5 l(x b )= xb l(x a )= 5 c b (x b )= 5xb 2.5 Strategy b Fitness for User Optimum Fitness for Global Optimum l(x b )= 2xb 2.5 l(x a )= 5

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

43 Upper bound of POA More general edge latency function POA GO n n Possibility of getting higher order of POA if using latency function with higher exponents

44 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)

45 Braess s Paradox Again 10 travelers want to move from S to T. x 10 SS 0: cost-free express road T 10 x User Optimum without middle arc = 150 = Global Optimum increase User Optimum with middle arc = 200 Price of Anarchy = 200/150 = 4/3

46 Boston Road Network Start End

47 Affect of an Arc Removal on User Optimum Start End negative

48 Affect of Arc Removal on User Optimum Cost increment congestion negative NE positive NE 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)

49 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?

50 PoAC network representation Number of Agents = 60 Regular Lattice (N=100) SW(N=100, r=3, p=0,0.1)

51 PoAC network representation Number of Agents = 60 ER(N=100, k=6) BA(N=100, m3) Thick and black edge: x+10 (wide and long) Thin and grey: 10x+1 size of node: PoAC method of spreading: using Kamada-Kawai(free) in Pajek except SW

52 PoA (s-t pair) Number of agents = 60 Number density <PoA> s-t pairs

53 PoAC distribution Number of agents = 60 Number density PoA centrality

54 <PoA> all S-T pairs, network <PoA> SW worst in POA! ER bad! SF good! RL best! Number of Agents

55 PoA (s-t pair) BC correlation Number of agents = 60 <PoA>bc(s)bc(t) BC(s)*BC(t)

56 PoA (S-T pair) degree correlation Number of agents = 60 <PoA>k(S)k(T) k(s)*k(t)

57 Summary & Conclusion Price of Anarchy on a network : price that a decentralized system should pay for not being coordinated, can be understood as a measure of inefficiency of the system. Price of Anarchy on a real world (Boston Road Network) - It is small, but it does exist! Reducing the Price of Anarchy Flow from to Central Square to Copley Square could be improved by removing some streets (NOT adding new streets!) - network modification (Braess s paradox) - Structural guidance of selfish users to the global optimized Efficiency in traffic dynamics: RL>BA>ER>SW?? Correlation with topological properties? Degree? More works are ongoing

58 Job opening at KAIST Funding: 2 nd phase Brain Korea 21 Project Several PostDoc & Research Professor positions are available in many fields. For more information, please contact H. Jeong (hjeong@kaist.ac.kr)