GAME THEORY IN TOPOLOGY CONTROL

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GAME THEORY IN TOPOLOGY CONTROL

• Robert P. Gilles
• 11/18/05

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Presentation Overview

• Ad-Hoc Networks
• Motivation
• Problem Statement
• Game Theoretic Framework
• Topology Control Game
• Potential Game
• Convergence
• Results

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Elements of an ad-hoc network
Heterogeneous nodes- different functionality and
capability Wireless medium
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Transmission pattern

• nodes can only transmit to other nodes within
  link coverage

• communication graph

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Variable transmission range
nodes can adjust their transmission power to
conserve energy
self organizing network nodes route amongst
themselves
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Abstraction
construction of symmetric communication graph
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Motivation

• Issues
• Energy and Capacity
• Limiting resources in Ad-hoc networks.
• Improper topology
• Too sparse (high end-to-end delay, less robust to
  node failure)
• Too dense (limited spatial reuse, reduced
  capacity)

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Motivation

• Problem
• Connectivity
• a basic requirement
• How should the nodes select an appropriate
  transmit power?
• Underlying graph is connected
• Total energy consumption is minimized

Solution ? TOPOLOGY CONTROL
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Before Topology Control
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After Topology Control
Topology Control- Choosing a subset of links and
nodes
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Notations and Assumptions

• Network abstracted as undirected graph H(V,E)
• Let power required to support edge (i,j)
• Individual transmission power p(i)
• Connected graph with pi,max(with redundancies)
• Symmetric channel
• Multi-hop path between source and destination

p(i)
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Problem Statement

• The (design) problem of choosing per node
  (optimal) transmission ranges (variables) that
  preserves network connectivity (constraints).
• Formally
• Power assignment such that
  determines . Sub-graph G(V,E)
  must be efficient and preserve connectivity of
  H induced by

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Existing TC Algorithms

• nodes forced to cooperate to achieve global
  objective
• altruism may not hold when nodes competing for
  resources
• algorithm design must account for selfish node
  behavior !

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Non Cooperative Game Framework

• Each node independent, selfish
• Considers power level of other nodes when
  assigning transmission power somewhat fair
  (power level assignment)
• Best response to current state topology
  (overhead), but works even with local information
  (localized)
• Once a node determines its transmission power
  (utility maximizer), it will cooperate and
  forward.

Topology update Cycle Payoff better
connectivity,..
Actions (Power level)
Individual payoff
Neighborhood
Topology (Set of links)
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Framework

• Let
• Utility of node i
• Benefit of node i, , from being a
  part of topology g
• Cost to node i its transmission power pi

• Transmit power level vector
  induces a topology given by
• Let be the maximum-power- connected-graph
• Goal Generate that is energy
  efficient and preserves the connectivity of gmax

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Energy Efficiency

• A network gp is locally energy efficient if no
  node can reduce its transmission power without
  disconnecting the network

A network gp is globally energy efficient if sum
of every nodes transmission power is minimum
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Topology Control Game I

• Consider where
• and . f is the number of nodes
  that can be reached (multi-hop). Assume,
• This game is an OPG with

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Corollary

• For the BR algorithm converges
  to NE that is locally energy efficient and
  preserves connectivity.
• Proof (Connectivity) Suppose not.
• Contradiction.
• Thus topology always connected at every stage.
  So . Power minimization problem,
  implies, locally energy efficient.

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Corollary

• For the steady state topology is
  also globally energy efficient.
• Proof We have . Potential maximizer
  . But by previous corollary
  . Therefore,
• .Thus g(p) is globally efficient.

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Topology Control Game II

• Consider where
• This game is an EPG with EPF given by
• Utilities exhibit 0-1 around a power level
  threshold. So BR algorithm converges to a power
  profile just to the right of this threshold

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Simulation results (I)
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Simulation results
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Simulation results
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Simulation results