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## GAME THEORY IN TOPOLOGY CONTROL

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### GAME THEORY. IN TOPOLOGY CONTROL. Robert P. Gilles. 11/18/05 ... algorithm design must account for selfish node behavior ! Non Cooperative Game Framework ... – PowerPoint PPT presentation

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Title: GAME THEORY IN TOPOLOGY CONTROL

1
GAME THEORY IN TOPOLOGY CONTROL
• Robert P. Gilles
• 11/18/05

2
Presentation Overview
• Motivation
• Problem Statement
• Game Theoretic Framework
• Topology Control Game
• Potential Game
• Convergence
• Results

3
Heterogeneous nodes- different functionality and
capability Wireless medium
4
Transmission pattern
• nodes can only transmit to other nodes within
• communication graph

5
Variable transmission range
nodes can adjust their transmission power to
conserve energy
self organizing network nodes route amongst
themselves
6
Abstraction
construction of symmetric communication graph
7
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)

8
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
9
Before Topology Control
10
After Topology Control
Topology Control- Choosing a subset of links and
nodes
11
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)
12
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

13
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 !

14
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
15
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

16
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
17
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

18
Corollary
• For the BR algorithm converges
to NE that is locally energy efficient and
preserves connectivity.
• Proof (Connectivity) Suppose not.
• Thus topology always connected at every stage.
So . Power minimization problem,
implies, locally energy efficient.

19
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.

20
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

21
Simulation results (I)
22
Simulation results
23
Simulation results
24
Simulation results