Optimal Gateway Selection in Multi-domain Wireless Networks: A Potential Game Perspective

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1. Motivation 2. Gateway Selection Game 3. Equilibrium Selection Learning 4. Evaluation 5. Conclusions

Optimal Gateway Selection in Multi-domain Wireless

Networks: A Potential Game Perspective

Yang Song, Starsky H.Y. Wong, and Kang-Won Lee Wireless Networking Research Group

IBM T. J. Watson Research Center

Mobicom 2011

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1

Motivation

2

Gateway Selection Game

3

Equilibrium Selective Learning

4

Performance Evaluation

5

Conclusions

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Coalition Networks with Multiple Domains

Scenario:

- Coalition networks withheterogenousgroups.

- Inter-connected via wireless links, e.g., IEEE 802.11, WiMAX, UAV, satellite, 3G/4G etc.

Example:

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Interoperability Issue

Problems:

Inter-domain communication is non-trivial for heterogenous domains

Different network protocol, security schemes, policies Security and policy

enforcement, traffic analysis

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Problems:

Solution:

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Interoperability Issue

Problems:

Solution:

Designate

gateway

nodes

Gateways

Domain B Domain A

D1 S1

S2

D2

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Gateways

Domain B

Domain C Domain A

Destination

Source

Each pair of nodes has a cost,

e.g., routing metric cost, such as

hop count, RIP, AODV etc.

Euclidean distance ETX, ETT, RTT Energy consumption etc.

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Cost Efficient Gateway Selection

Gateways

Domain B

Destination

Source

Each pair of nodes has a cost,

e.g., routing metric cost, such as

For a single domain

Intra-domain cost

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Gateways

Domain B

Destination

Source

Each pair of nodes has a cost,

e.g., routing metric cost, such as

For a single domain For the network

Intra-domain cost Inter-domain backbone cost

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Cost Efficient Gateway Selection

Gateways

Domain B

Destination

Source

Each pair of nodes has a cost,

e.g., routing metric cost, such as

For a single domain For the network

Intra-domain cost + Inter-domain backbone cost

Question:

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Gateways

Domain B

Destination

Source

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Challenges

Gateways

Domain B

Destination

Source

Combinatorial nature of solution space

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Gateways

Domain B

Destination

Source

Distributed solution

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Challenges

Gateways

Domain B

Destination

Source

Each domain may designate gateway for its own benefit (self-interested / lack of coordination)

Distributed solution

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Gateways

Domain B

Destination

Source

Distributed solution Equilibrium efficiency

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Challenges

Gateways

Domain B

Destination

Source

Reluctance in revealing its own intra-domain topology

Distributed solution Equilibrium efficiency

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Gateways

Domain B

Destination

Source

Distributed solution Equilibrium efficiency Local information only

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Challenges

Gateways

Domain B

Destination

Source

Distributed solution Equilibrium efficiency Local information only

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M : the set of domains in the coalition network

Nm: the set of nodes in the domain

g_mⁱ = 1: node i is selected as the gateway node and gmⁱ = 0 o.w. and bim= argmax_i∈N_mg_mⁱ be the selected gateway node gm= {g_m¹, g_m²,· · · , gm^|N^m^|}: the gateway selection strategyof

s = {g1,g2,· · · , g_|M|}: the jointgateway selection profileof the network Satellite/UAV/3G/4G link:

cost η (expensive), to enforce always-on connectivity A pair of node i and j:

c(i, j) ≥ 0 is the associated symmetric link cost, c(i, j) = η if out of range

′ , min (c (i, j) , η)

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Gateway Selection Game

For each single domain

Minimize (Localinformation and observation only)

Um(gm,g−m) = X

i6= bim,i ∈Nm

c i,ibm

+ X

n6=m,n∈M

c^′ ibm, bin

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Um(gm,g−m) = X

i6= bim,i ∈Nm

c i,ibm

+ X

n6=m,n∈M

c^′ ibm, bin

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Gateways

Domain B

Destination

Source

Player: each domain m ∈ M Strategy space: Nm

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Gateway Selection Game

For each single domain

Um(gm,g−m) = X

i6= bim,i ∈Nm

c i,ibm

+ X

n6=m,n∈M

c^′ ibm, bin

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Gateways

Domain B

Destination

Source

Questions

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Um(gm,g−m) = X

i6= bim,i ∈Nm

c i,ibm

+ X

n6=m,n∈M

c^′ ibm, bin

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Gateways

Domain B

Destination

Source

Questions

– Agreement? ⇐⇒Existenceof NE

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Gateway Selection Game

For each single domain

Um(gm,g−m) = X

i6= bim,i ∈Nm

c i,ibm

+ X

n6=m,n∈M

c^′ ibm, bin

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Gateways

Domain B

Destination

Source

Questions

– Agreement? ⇐⇒Existenceof NE – Performance? ⇐⇒Efficiencyof NE

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Um(gm,g−m) = X

i6= bim,i ∈Nm

c i,ibm

+ X

n6=m,n∈M

c^′ ibm, bin

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Gateways

Domain B

Destination

Source

Questions

– Agreement? ⇐⇒Existenceof NE – Performance? ⇐⇒Efficiencyof NE

For overall network

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Existence of Nash Equilibrium

Theorem

The gateway selection game has a Nash equilibrium, which

minimizes, either locally or globally, the following function

F(s) = X

m

X

i6= bim,i∈Nm

c

i , b i

_m

+ X

(

ⁱ^b^m^{, b}ⁱⁿ

)

^∈CCG(s)

c

^′

b

i

_m

, b i

_n

. (3)

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Theorem

The gateway selection game has a Nash equilibrium, which

minimizes, either locally or globally, the following function

F(s) = X

m

X

i6= bim,i∈Nm

c

i , b i

_m

+ X

(

ⁱ^b^m^{, b}ⁱⁿ

)

^∈CCG(s)

c

^′

b

i

_m

, b i

_n

. (3)

Nash equilibrium may not be unique

Multiple Nash equilibria have different performance

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Existence of Nash Equilibrium

Theorem

The gateway selection game has a Nash equilibrium, which

minimizes, either locally or globally, the following function

F(s) = X

m

X

i6= bim,i∈Nm

c

i , b i

_m

+ X

(

ⁱ^b^m^{, b}ⁱⁿ

)

^∈CCG(s)

c

^′

b

i

_m

, b i

_n

. (3)

Nash equilibrium may not be unique

Multiple Nash equilibria have different performance

To capture the (in)efficiency of Nash equilibrium,Price of Anarchy andPrice of Stabilityare introduced

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For two player gateway selection games, the best Nash Equilibrium is the global network optimum solution, i.e., the price of stability is 1.

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Efficiency of Nash Equilibria

For |M| = 2

For |M| ≥ 3

For |M| ≥ 3, if the link cost metric c(a, b) satisfies the triangle inequality, the price of stability is always 1.

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For |M| ≥ 3

For |M| ≥ 3, if the link cost metric c(a, b) satisfies the triangle inequality, the price of stability is always 1.

All else

If the triangle inequality does not hold, the price of stability of an

|M|-player gateway selection game is at most (1 + δ), where

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B-logit: Binary Logit Algorithm

B-logit: For every time slot t:

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Randomly select one of the players, say m, to update its gateway selection while other domains remain unchanged.

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B-logit: Binary Logit Algorithm

B-logit: For every time slot t:

Denote the current gateway selection of domain m as gm(t).

Domain m randomly selects a node in its domain as the gateway candidate. Denote the candidate gateway selection strategy bygfm. Domain m updates as

Pr (gm(t + 1) =gfm) (5)

= exp^−U^m(gfm,g−m(t))^/τ

exp^−U^m(gfm,g−m(t))^/τ+ exp^−U^m(^gm(t),g−m(t))^/τ and

Pr (gm(t + 1) = gm(t)) = 1− Pr (gm(t + 1) =gfm) (6) where τ is a small positive constant, a.k.a., thesmoothing factor

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Denote the current gateway selection of domain m as gm(t).

Domain m randomly selects a node in its domain as the gateway candidate. Denote the candidate gateway selection strategy bygfm. Domain m updates as

Pr (gm(t + 1) =gfm) (5)

= exp^−U^m(gfm,g−m(t))^/τ

exp^−U^m(gfm,g−m(t))^/τ+ exp^−U^m(^gm(t),g−m(t))^/τ and

Pr (gm(t + 1) = gm(t)) = 1− Pr (gm(t + 1) =gfm) (6) where τ is a small positive constant, a.k.a., thesmoothing factor

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Proof (sketch)

1,1

x y x y1,2 x y1,3 x y1,_{c l}×

⋯ ⋯

c l,c l

x_×y_×

,2

xc l_×y xc l_×,y3

⋯ ⋯

2,1

x y

3,1

x y

,1

xc l_×y

2,2

x y x y2,3 ⋯ ⋯ x y2,c l×

Note Pr (s^′→ s^′′)

1

|M| 1

|N_m|

exp^{−U(s′′ )/τ}

exp^−Um(gm ,g−m(t)^f )^/τ_{+ exp}−Um(gm (t),g−m(t))^/τ Verify

π(s^′) = exp^{−F (s}^′^)/τ P

s∈Sexp^{−F (s)/τ} satisfies thedetailed balance equation, i.e., π(s^′) Pr (s^′→ s^′′) = π(s^′′) Pr (s^′′→ s^′) B-logitalgorithm induces a reversible,

irreducible, and aperiodic Markov chain and it is theuniquesteady state distribution.

By taking τ→ 0, we have

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Generalization of B-logit

γ-logit algorithm family (Γ):

γ-logit shares the same structure as B-logit except in (5), where the probability is calculated as

Pr (gm(t + 1) =gfm) = exp^−U^m(gfm,g−m(t))^/τ

γ(s^′,s^′′) (7) where s^′={gm(t), g−m(t)} and s^′′={fgm,g−m(t)} are two gateway selection profiles inS, and γ satisfies

1 Symmetry

γ(s^′,s^′′) = γ(s^′′,s^′),∀s^′∈ S, s^′′∈ S,

2 Feasibility γ(s^′,s^′′)≥ max

exp^−U^m^(s^′^)/τ,exp^−U^m^(s^′′^)/τ . B-logitis aspecial caseof γ-logit algorithm with

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converging to the global minimizer of the potential function

asymptotically.

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Theorem

Every γ-logit algorithm in Γ is equilibrium selective, i.e.,

converging to the global minimizer of the potential function

asymptotically.

Which isbetter?

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converging to the global minimizer of the potential function

asymptotically.

Which isbetter?

Each γ-logit algorithm induces a Markov chain with different transition probability matrix, where

Pi,j(γ), Pr sⁱ → s^j

= 1

|M|

1

|Nm|

exp^{−U (s}^j^)/τ γ(sⁱ,s^j)

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Theorem

Every γ-logit algorithm in Γ is equilibrium selective, i.e.,

converging to the global minimizer of the potential function

asymptotically.

Which isbetter?

Each γ-logit algorithm induces a Markov chain with different transition probability matrix, where

Pi,j(γ), Pr sⁱ → s^j

= 1

|M|

1

|Nm|

exp^{−U (s}^j^)/τ γ(sⁱ,s^j)

The mixing rate of a Markov chain is determined by the second largest eigenvalue modulus (SLEM), i.e.,

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For every time slot t:

Denote the current gateway selection of domain m as gm(t). Domain m randomly selects a node in its domain as the gateway candidate. Denote the candidate gateway selection strategy bygfm. Domain m updates as

Pr (gm(t + 1) =gfm) = exp^−U^m(gfm,g−m(t))/τ

max (exp^−U^m^(s^′^)/τ,exp^−U^m^(s^′′^)/τ).

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Solution: MAX-logit Algorithm

MAX-logit:

For every time slot t:

Denote the current gateway selection of domain m as gm(t). Domain m randomly selects a node in its domain as the gateway candidate. Denote the candidate gateway selection strategy bygfm. Domain m updates as

Pr (gm(t + 1) =gfm) = exp^−U^m(gfm,g−m(t))/τ

max (exp^−U^m^(s^′^)/τ,exp^−U^m^(s^′′^)/τ). Denote µ^MAX as the second largest eigenvalue modulus associated with MAX-logit algorithm.

Theorem

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|M| domains where each domain has |N | nodes

For each domain, nodes are randomly deployed in a round area with radius 125m, centered at a random point within the square field of 1000× 1000m²

Link cost:

1 Euclidean distance: Network optimum solution is the best Nash (γ-logit algorithms converge to the network optimum solution)

2 Random cost: γ-logit algorithm converges to the approximate 1 + δ solution (Nash equilibrium)

3 Randomly select p% of the links in the network and add random cost offset which is uniformly distributed between 0 and 5% of the original cost

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Euclidean Distance Scenarios

p% = 0%

2, 3, 4 domains where each domain has 20 nodes

0 20 40 60 80 100

3000 3100 3200 3300 3400

Iteration steps

Global network cost

MAX−logit B−logit

OPT

0 50 100 150 200

5000 5500 6000 6500

Iteration steps

OPT

0 50 100 150 200

8000 8500 9000 9500 10000 10500

Iteration steps

OPT

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0 20 40 60 80 100

3000 3100 3200 3300 3400

Iteration steps

OPT

0 50 100 150 200

5000 5500 6000 6500

Iteration steps

OPT

0 50 100 150 200

8000 8500 9000 9500 10000 10500

Iteration steps

OPT

Nodes per domain 2 domains 3 domains 4 domains

5 nodes 16.06% 24.52% 33.85%

10 nodes 25.00% 29.81% 28.55%

20 nodes 11.96% 20.19% 20.36%

30 nodes 5.87% 16.46% 17.60%

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Random Cost Scenarios

p= 50, i.e., 50% of the links in the network are associated with random link cost

0 50 100 150 200

2500 3000 3500 4000 4500 5000

Iteration steps

OPT B−logit MAX−logit

0 50 100 150 200

4000 4500 5000 5500 6000

Iteration steps

BOUND

OPT

0 50 100 150 200

6000 7000 8000 9000 10000 11000 12000

Iteration steps

BOUND

OPT

MAX−logit

B−logit

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cost

0 50 100 150 200

2500 3000 3500 4000 4500 5000

Iteration steps

OPT B−logit MAX−logit

0 50 100 150 200

4000 4500 5000 5500 6000

Iteration steps

BOUND

OPT

0 50 100 150 200

6000 7000 8000 9000 10000 11000 12000

Iteration steps

BOUND

OPT

MAX−logit

B−logit

Nodes per domain 2 domains 3 domains 4 domains

5 nodes 21.84% 24.46% 27.38%

10 nodes 21.00% 21.44% 21.56%

20 nodes 9.54% 9.13% 5.47%

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Conclusions

Interactive gateway selection by multiple domains in coalition networks

In a potential game framework, the existence and inefficiency of Nash equilibria are characterized (two domains, multi-domains) Equilibrium selective learning: generalized B-logit into γ-logit, or Γ

Propose MAX-logit which converges to the best Nash equilibrium at the fastest speed in Γ

Other applications of potential games in power control, channel allocation, spectrum sharing content distribution etc.

Optimal Gateway Selection in Multi-domain Wireless Networks: A Potential Game Perspective