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Optimal Gateway Selection in Multi-domain Wireless Networks: A - - PowerPoint PPT Presentation

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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

slide-1
SLIDE 1
  • 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

Research was sponsored by US Army Research and UK Ministry of Defense under W911NF-06-3-0001. 1 / 19 IBM Research

slide-2
SLIDE 2
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Overview

1

Motivation

2

Gateway Selection Game

3

Equilibrium Selective Learning

4

Performance Evaluation

5

Conclusions

2 / 19 IBM Research

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

Coalition Networks with Multiple Domains

Scenario:

  • Coalition networks with heterogenous groups.
  • Inter-connected via wireless links, e.g., IEEE 802.11, WiMAX, UAV, satellite,

3G/4G etc.

Example:

Joint military missions, US-UK Disaster rescue teams, fire-fighters and police officers Wireless sensor networks of different organizations, e.g., Internet of Things (IoT), Smart Planet Solutions

3 / 19 IBM Research

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

Interoperability Issue

4 / 19 IBM Research

slide-5
SLIDE 5
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Interoperability Issue

Problems:

Inter-domain communication is non-trivial for heterogenous domains Different network protocol, security schemes, policies Security and policy enforcement, traffic analysis

4 / 19 IBM Research

slide-6
SLIDE 6
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Interoperability Issue

Problems:

Inter-domain communication is non-trivial for heterogenous domains Different network protocol, security schemes, policies Security and policy enforcement, traffic analysis

Solution:

4 / 19 IBM Research

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SLIDE 7
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Interoperability Issue

Problems:

Inter-domain communication is non-trivial for heterogenous domains Different network protocol, security schemes, policies Security and policy enforcement, traffic analysis

Solution: Designate gateway nodes

Gateways Domain B Domain A D1 S1 S2 D2

4 / 19 IBM Research

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SLIDE 8
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Cost Efficient Gateway Selection

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.

5 / 19 IBM Research

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SLIDE 9
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Cost Efficient Gateway Selection

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.

For a single domain Intra-domain cost

5 / 19 IBM Research

slide-10
SLIDE 10
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Cost Efficient Gateway Selection

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.

For a single domain For the network Intra-domain cost Inter-domain backbone cost

5 / 19 IBM Research

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SLIDE 11
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Cost Efficient Gateway Selection

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.

For a single domain For the network Intra-domain cost + Inter-domain backbone cost Question: How to select the set of gateways s.t. the overall cost is minimized?

5 / 19 IBM Research

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SLIDE 12
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

6 / 19 IBM Research

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SLIDE 13
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

Combinatorial nature of solution space

6 / 19 IBM Research

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SLIDE 14
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

Combinatorial nature of solution space

Distributed solution

6 / 19 IBM Research

slide-15
SLIDE 15
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

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

Distributed solution

6 / 19 IBM Research

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SLIDE 16
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

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

Distributed solution Equilibrium efficiency

6 / 19 IBM Research

slide-17
SLIDE 17
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

Combinatorial nature of solution space Each domain may designate gateway for its own benefit (self-interested / lack of coordination) Reluctance in revealing its own intra-domain topology

Distributed solution Equilibrium efficiency

6 / 19 IBM Research

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SLIDE 18
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

Combinatorial nature of solution space Each domain may designate gateway for its own benefit (self-interested / lack of coordination) Reluctance in revealing its own intra-domain topology

Distributed solution Equilibrium efficiency Local information only

6 / 19 IBM Research

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SLIDE 19
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Challenges

Gateways Domain B Domain C Domain A Destination Source

Combinatorial nature of solution space Each domain may designate gateway for its own benefit (self-interested / lack of coordination) Reluctance in revealing its own intra-domain topology

Distributed solution Equilibrium efficiency Local information only potential game theory & equilibrium selective learning

6 / 19 IBM Research

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SLIDE 20
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Network Model

M : the set of domains in the coalition network Nm: the set of nodes in the domain g i

m = 1: node i is selected as

the gateway node and g i

m = 0

  • .w. and

im = argmaxi∈Nmg i

m

be the selected gateway node gm = {g 1

m, g 2 m, · · · , g |Nm| m

}: the gateway selection strategy of domain m s = {g1, g2, · · · , g|M|}: the joint gateway selection profile of 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 c′ (i, j) min (c (i, j) , η)

7 / 19 IBM Research

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SLIDE 21
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Gateway Selection Game

For each single domain

Minimize (Local information and observation only) Um (gm, g−m) =

  • i=

im,i∈Nm

c

  • i,

im

  • +
  • n=m,n∈M

c′

  • im,

in

  • (1)

8 / 19 IBM Research

slide-22
SLIDE 22
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Gateway Selection Game

For each single domain

Minimize (Local information and observation only) Um (gm, g−m) =

  • i=

im,i∈Nm

c

  • i,

im

  • +
  • n=m,n∈M

c′

  • im,

in

  • (1)

Gateways Domain B Domain C Domain A Destination Source

Player: each domain m ∈ M Strategy space: Nm

8 / 19 IBM Research

slide-23
SLIDE 23
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Gateway Selection Game

For each single domain

Minimize (Local information and observation only) Um (gm, g−m) =

  • i=

im,i∈Nm

c

  • i,

im

  • +
  • n=m,n∈M

c′

  • im,

in

  • (1)

Gateways Domain B Domain C Domain A Destination Source

Player: each domain m ∈ M Strategy space: Nm Questions

8 / 19 IBM Research

slide-24
SLIDE 24
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Gateway Selection Game

For each single domain

Minimize (Local information and observation only) Um (gm, g−m) =

  • i=

im,i∈Nm

c

  • i,

im

  • +
  • n=m,n∈M

c′

  • im,

in

  • (1)

Gateways Domain B Domain C Domain A Destination Source

Player: each domain m ∈ M Strategy space: Nm Questions

– Agreement? ⇐ ⇒ Existence of NE

8 / 19 IBM Research

slide-25
SLIDE 25
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Gateway Selection Game

For each single domain

Minimize (Local information and observation only) Um (gm, g−m) =

  • i=

im,i∈Nm

c

  • i,

im

  • +
  • n=m,n∈M

c′

  • im,

in

  • (1)

Gateways Domain B Domain C Domain A Destination Source

Player: each domain m ∈ M Strategy space: Nm Questions

– Agreement? ⇐ ⇒ Existence of NE – Performance? ⇐ ⇒ Efficiency of NE

8 / 19 IBM Research

slide-26
SLIDE 26
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Gateway Selection Game

For each single domain

Minimize (Local information and observation only) Um (gm, g−m) =

  • i=

im,i∈Nm

c

  • i,

im

  • +
  • n=m,n∈M

c′

  • im,

in

  • (1)

Gateways Domain B Domain C Domain A Destination Source

Player: each domain m ∈ M Strategy space: Nm Questions

– Agreement? ⇐ ⇒ Existence of NE – Performance? ⇐ ⇒ Efficiency of NE

For overall network

Minimize (intra-domain cost + cost of backbone communication links) R(s) =

  • m
  • i=

im,i∈Nm

c

  • i,

im

  • +
  • (

im, in)∈MCG(s)

c′

  • im,

in

  • .

(2)

8 / 19 IBM Research

slide-27
SLIDE 27
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Existence of Nash Equilibrium

Theorem The gateway selection game has a Nash equilibrium, which minimizes, either locally or globally, the following function F(s) =

  • m
  • i=

im,i∈Nm

c

  • i,

im

  • +
  • (

im, in)∈CCG(s)

c′

  • im,

in

  • .

(3)

9 / 19 IBM Research

slide-28
SLIDE 28
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Existence of Nash Equilibrium

Theorem The gateway selection game has a Nash equilibrium, which minimizes, either locally or globally, the following function F(s) =

  • m
  • i=

im,i∈Nm

c

  • i,

im

  • +
  • (

im, in)∈CCG(s)

c′

  • im,

in

  • .

(3)

Nash equilibrium may not be unique Multiple Nash equilibria have different performance

9 / 19 IBM Research

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SLIDE 29
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Existence of Nash Equilibrium

Theorem The gateway selection game has a Nash equilibrium, which minimizes, either locally or globally, the following function F(s) =

  • m
  • i=

im,i∈Nm

c

  • i,

im

  • +
  • (

im, in)∈CCG(s)

c′

  • im,

in

  • .

(3)

Nash equilibrium may not be unique Multiple Nash equilibria have different performance To capture the (in)efficiency of Nash equilibrium, Price of Anarchy and Price of Stability are introduced Price of Stability = value of best equilibrium value of optimal solution

9 / 19 IBM Research

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SLIDE 30
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Efficiency of Nash Equilibria

For |M| = 2

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

10 / 19 IBM Research

slide-31
SLIDE 31
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Efficiency of Nash Equilibria

For |M| = 2

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

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.

10 / 19 IBM Research

slide-32
SLIDE 32
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Efficiency of Nash Equilibria

For |M| = 2

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

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 δ = η

  • |M|

2

+

1 |M| − 3 2

  • minm∈M mingm
  • i=

im(gm),i∈Nm c

  • i,

im(gm) . (4)

10 / 19 IBM Research

slide-33
SLIDE 33
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

B-logit: Binary Logit Algorithm

B-logit: For every time slot t:

11 / 19 IBM Research

slide-34
SLIDE 34
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

B-logit: Binary Logit Algorithm

B-logit: For every time slot t: Randomly select one of the players, say m, to update its gateway selection while other domains remain unchanged.

11 / 19 IBM Research

slide-35
SLIDE 35
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

B-logit: Binary Logit Algorithm

B-logit: For every time slot t: Randomly select one of the players, say m, to update its gateway selection while other domains remain unchanged. 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 by

gm. Domain m updates as Pr (gm(t + 1) = gm) (5) = exp−Um(

gm,g−m(t))/τ

exp−Um(

gm,g−m(t))/τ + exp−Um(gm(t),g−m(t))/τ

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

  • f the algorithm.

11 / 19 IBM Research

slide-36
SLIDE 36
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

B-logit: Binary Logit Algorithm

B-logit: For every time slot t: Randomly select one of the players, say m, to update its gateway selection while other domains remain unchanged. 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 by

gm. Domain m updates as Pr (gm(t + 1) = gm) (5) = exp−Um(

gm,g−m(t))/τ

exp−Um(

gm,g−m(t))/τ + exp−Um(gm(t),g−m(t))/τ

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

  • f the algorithm.

It is known that as τ → 0, B-logit converges to the best Nash equilibrium with arbitrarily high probability.

11 / 19 IBM Research

slide-37
SLIDE 37
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Proof (sketch)

1 1

, x y

1 2

, x y

1 3

, x y

1, c l

x y ×

⋯⋯

, c l c l

x y

× × , 2 c l

x y

× , 3 c l

x y

×

⋯ ⋯

2, 1

x y

3, 1

x y

, 1 c l

x y

× 2, 2

x y

2, 3

x y

2, c l

x y ×

⋯⋯

Note Pr (s′ → s′′)

1 |M| 1 |Nm| exp−U(s′′)/τ exp−Um(

gm,g−m(t))/τ + exp−Um(gm(t),g−m(t))/τ

Verify π(s′) = exp−F(s′)/τ

  • s∈S exp−F(s)/τ

satisfies the detailed balance equation, i.e., π(s′) Pr (s′ → s′′) = π(s′′) Pr (s′′ → s′) B-logit algorithm induces a reversible, irreducible, and aperiodic Markov chain and it is the unique steady state distribution. By taking τ → 0, we have π(s∗) → 1, where s∗ = argmins∈S F(s)

12 / 19 IBM Research

slide-38
SLIDE 38
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Generalization of B-logit

13 / 19 IBM Research

slide-39
SLIDE 39
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

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) = gm) = exp−Um(

gm,g−m(t))/τ

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

1

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

2

Feasibility γ(s′, s′′) ≥ max

  • exp−Um(s′)/τ, exp−Um(s′′)/τ

. B-logit is a special case of γ-logit algorithm with γ

  • s′, s′′

= γ

  • s′′, s′

= exp−Um(s′)/τ + exp−Um(s′′)/τ .

13 / 19 IBM Research

slide-40
SLIDE 40
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Theorem Every γ-logit algorithm in Γ is equilibrium selective, i.e., converging to the global minimizer of the potential function asymptotically.

14 / 19 IBM Research

slide-41
SLIDE 41
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Theorem Every γ-logit algorithm in Γ is equilibrium selective, i.e., converging to the global minimizer of the potential function asymptotically.

Which is better?

14 / 19 IBM Research

slide-42
SLIDE 42
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Theorem Every γ-logit algorithm in Γ is equilibrium selective, i.e., converging to the global minimizer of the potential function asymptotically.

Which is better? Each γ-logit algorithm induces a Markov chain with different transition probability matrix, where Pi,j(γ) Pr

  • si → sj

= 1 |M| 1 |Nm| exp−U(sj)/τ γ(si, sj)

14 / 19 IBM Research

slide-43
SLIDE 43
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Theorem Every γ-logit algorithm in Γ is equilibrium selective, i.e., converging to the global minimizer of the potential function asymptotically.

Which is better? Each γ-logit algorithm induces a Markov chain with different transition probability matrix, where Pi,j(γ) Pr

  • si → sj

= 1 |M| 1 |Nm| exp−U(sj)/τ γ(si, sj) The mixing rate of a Markov chain is determined by the second largest eigenvalue modulus (SLEM), i.e., µ (P(γ)) = max

  • |λ2 (P(γ)) |, |λ|S| (P(γ)) |
  • .

The smaller µ (P(γ)) is, the faster.

14 / 19 IBM Research

slide-44
SLIDE 44
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Solution: MAX-logit Algorithm

MAX-logit:

For every time slot t: Randomly select one of the players, say m, to update its gateway selection while other domains remain unchanged. 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 by

  • gm. Domain m updates as

Pr (gm(t + 1) = gm) = exp−Um(

gm,g−m(t))/τ

max (exp−Um(s′)/τ, exp−Um(s′′)/τ).

15 / 19 IBM Research

slide-45
SLIDE 45
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Solution: MAX-logit Algorithm

MAX-logit:

For every time slot t: Randomly select one of the players, say m, to update its gateway selection while other domains remain unchanged. 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 by

  • gm. Domain m updates as

Pr (gm(t + 1) = gm) = exp−Um(

gm,g−m(t))/τ

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

Theorem

Denote µ (P(γ)) as the second largest eigenvalue modulus induced by an arbitrary γ-logit algorithm in Γ. We have µMAX ≤ µ (P(γ)) .

15 / 19 IBM Research

slide-46
SLIDE 46
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Evaluation setup

|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

  • riginal cost

Global link cost η = 500, |M| = 2, 3, 4 τ = 0.0001

16 / 19 IBM Research

slide-47
SLIDE 47
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Euclidean Distance Scenarios

p% = 0% 2, 3, 4 domains where each domain has 20 nodes

20 40 60 80 100 3000 3100 3200 3300 3400

Iteration steps Global network cost MAX−logit B−logit OPT

50 100 150 200 5000 5500 6000 6500

Iteration steps Global network cost MAX−logit B−logit OPT

50 100 150 200 8000 8500 9000 9500 10000 10500

Iteration steps Global network cost MAX−logit B−logit OPT 17 / 19 IBM Research

slide-48
SLIDE 48
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Euclidean Distance Scenarios

p% = 0% 2, 3, 4 domains where each domain has 20 nodes

20 40 60 80 100 3000 3100 3200 3300 3400

Iteration steps Global network cost MAX−logit B−logit OPT

50 100 150 200 5000 5500 6000 6500

Iteration steps Global network cost MAX−logit B−logit OPT

50 100 150 200 8000 8500 9000 9500 10000 10500

Iteration steps Global network cost MAX−logit B−logit 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% Average over 5000 sample runs Performance improvement declines when no. of nodes increases

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slide-49
SLIDE 49
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Random Cost Scenarios

p = 50, i.e., 50% of the links in the network are associated with random link cost 2, 3, 4 domains where each domain has 20 nodes

50 100 150 200 2500 3000 3500 4000 4500 5000

Iteration steps Global network cost OPT B−logit MAX−logit

50 100 150 200 4000 4500 5000 5500 6000

Iteration steps Global network cost BOUND OPT MAX−logit B−logit

50 100 150 200 6000 7000 8000 9000 10000 11000 12000

Iteration steps Global network cost BOUND OPT MAX−logit B−logit 18 / 19 IBM Research

slide-50
SLIDE 50
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

Random Cost Scenarios

p = 50, i.e., 50% of the links in the network are associated with random link cost 2, 3, 4 domains where each domain has 20 nodes

50 100 150 200 2500 3000 3500 4000 4500 5000

Iteration steps Global network cost OPT B−logit MAX−logit

50 100 150 200 4000 4500 5000 5500 6000

Iteration steps Global network cost BOUND OPT MAX−logit B−logit

50 100 150 200 6000 7000 8000 9000 10000 11000 12000

Iteration steps Global network cost 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% 30 nodes 1.90% 1.93% 2.24%

Table: Convergence rate improvement by MAX-logit when p = 50.

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slide-51
SLIDE 51
  • 1. Motivation
  • 2. Gateway Selection Game
  • 3. Equilibrium Selection Learning
  • 4. Evaluation
  • 5. Conclusions

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,

  • r 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.

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