the price of anarchy on complex networks
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
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
∝ 1/width
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)
∝ # of travelers
∝ length
CSSPL
Latency function (like time or cost per person)
lengthwidth
Latency travelersof #1
Network flow with congestion
CSSPL
Based on the model of Roughgarden & Tardos, 2000
S
Cost function on path i
Latency function
T
width of path i
length of path i
# of agent on path i
Given network with many agents going from S (source) to T (target), what will be the optimized distribution of agents for best performance??
Optimizations in physics
• Euler-Lagrange differential equation• minimal free energy in thermodynamic physics• Fitting experimental DATA with formula• Low temperature behavior of disordered magnets• …
Centralized controlMinimizing Global Cost
Centralized controlMinimizing Global Cost
Decentralized controlEach agent minimizes its own personal cost
Decentralized controlEach agent minimizes its own personal cost
User Optimization(Nash equilibrium)User Optimization(Nash equilibrium)
Global OptimizationGlobal Optimization
CSSPL
• There are two types of optimizations!!!
The “Price of Anarchy”
CSSPL
Koutsoupias & Papadimitriou, 1999
Price of Anarchy (Roughgarden & Tardos, 2000)
1 ≤
total cost of
Centralized controlMinimizing Global
Cost
Decentralized control
Each agent minimizes
its own personal cost
total cost ofGlobal Optimum
User OptimumPrice of AnarchyPrice of Anarchy
“Price we have to pay not being coordinated by central agency”“Price of being selfish”
Price of Anarchy: Contrived Example
CSSPL
Pigous’s example: Congestion sensitive network
S T
What will be the min total cost, i.e. Global Optimum = ?
10 agents want toGo from S to T.
If xa=x, then xb=10-x, ∴ total cost=10ᆞ x +
(10-x) ᆞ (10-x) = x2-10x+100=(x-5)2+75∴ xa=xb=5 with total cost 75
Price of Anarchy: Contrived Example
Global Optimum = 5x10 + 5x5 = 75
75/10 = 7.5min driving in average CSSPL
xa = xb =5
S T
Envy
BUTBUT
The upper agents get envious of people with lower costs!
What will be the User Optimum?
(Nash Equilibrium: everyone happy)CSSPL
Price of Anarchy: Contrived Example
xa = 5
xb = 5S T
CSSPL
user cost = 5 + 1 < 10
Price of Anarchy: Contrived Example
Move toLower path
+1 S T
xa = 5-1
xb = 5+1
CSSPL
Price of Anarchy: Contrived Example
again+1
S T
xa = 4-1
xb = 6+1
user cost = 6 + 1 < 10
CSSPL
Price of Anarchy: Contrived Example
again+1
S T
xa = 3-1
xb = 7+1
user cost = 7 + 1 < 10
CSSPL
Price of Anarchy: Contrived Example
again+1
S T
xa = 2-1
xb = 8+1
user cost = 8 + 1 < 10
CSSPL
Price of Anarchy: Contrived Example
User Optimum = 10 x10 = 100
avg 10min travel time > avg 7.5-min travel time
again+1
S T
xa = 1-1
xb = 9+1
User Optimum = 10 x10 = 100Global Optimum = 5x10 + 5x5 = 75
CSSPL
Price of Anarchy: Contrived Example
4/3 Price of Price of Anarchy!Anarchy!
S T
xa = 5 vs 0
xb = 5 vs 10
There is a gap between global optimum & user optimum!
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 w
idth, length, one-way etc)– Traffic from central square (S) to copley
square (T)
CSSPL
Boston Road Network
Start
End
CSSPL
(nodes 59, edges 108, regular-like)
Latency function = ax + b
lengthWidth
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
User Optimum Global Optimum
Number of Agents: 20
CSSPL
Congestion distribution on the edges
Reminder: POA = UE/GO
CSSPL
Variation of POA with Agent #
number of agents
Pri
ce o
f A
narc
hy
Why Price of Anarchy decreases?
CSSPL
• Fitness landscape for a simple case:
S T
l(xa)= 55
l(xb)= xb
Fitness for User Optimum
5
Strategy a
Strategy b
l(xb)= xb
Fitness for Global Optimum
cb (xb)= xb2
cb (xb)= 5xb
2.5
l(xa)= 5
l(xa)= 5
l(xb)= 2xb
2.5
POA too small??
• More general edge latency function– n > 1
CSSPL
- Roughgarden-Tardos
• Linear latency function:
S T
C=1
C(X) = X^3
When n=3UO = 1GO = 0.37*1 + (0.63)^4 = 0.528
POA = UA/GO = 1.894 Bigger than 4/3 (n=1)
1
0.63
0.37
Nash Equilibrium 4/3 x (Global Optimum)Nash Equilibrium 4/3 x (Global Optimum)
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): constr
uct more roads with tax money! Braess paradox
(counter-intuitive consequences)
Braess’s Paradox
T
x
x10
10
0: cost-free express road
User Optimum without middle arc = 150 = Global Optimum
CSSPL
Price of Anarchy = 200/150 = 4/34/3
increase
User Optimum with middle arc = 200
SS
Again 10 travelers want to move from S to T.
0 20 40 60 80 1000
500
1000
1500
2000
2500
0 10 20 30 40 50 60-4
-2
0
2
4
6
8
10
12congestionnegative NEpositive NE
NE
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
Affect of Arc Removal on User Optimum
edge index
Cost
in
crem
en
t
PoA=UO/GO
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
PoAC network representation
SW(N=100, r=3, p=0,0.1)
Regular Lattice (N=100)
Number of Agents = 60
CSSPL
PoAC network representation
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
Number of Agents = 60
CSSPL
Summary & Conclusion• Price of Anarchy on a network : price that a decentralized system s
hould 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
- network modification (Braess’s paradox)- Structural guidance of selfish users to the global optimiStructural guidance of selfish users to the global optimi
zed zed • Efficiency in traffic dynamics: RL>BA>ER>SW??• Correlation with topological properties? Degree?• More works are ongoing…
Flow from to Central Square to Copley Square could be improved by removing some streets (NOT adding new streets!)
CSSPL
Job opening at KAIST
• Funding: 2nd phase Brain Korea 21 Project• Several PostDoc & Research Professor positi
ons are available in many fields.• For more information, please contact
H. Jeong ([email protected])