Updates to Network Formation Simulation

1
Updates to Network Formation Simulation

• Ramakant Komali
• 03/15/05

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Added Components

• Topology Control
• - Connectivity model
• Belief Systems and Self Confirming Equilibrium
• - Network formation model

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TC- Problem Description

• Action Aipi pi,minltpi ltpi,max set of power
  levels
• These actions induce a network gg(p) -collection
  of links between nodes i and j such that
  k.dn(i,j)ltpi and/or k.dn(j,i)ltpj k is
  proportionality constant
• Network payoff
• ui(pi,g(p)) Mfi (p)-pi
• M is a constant
• fi is the number of nodes i is connected to
  directly or indirectly

Actions
Topology
Payoffs
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TC Simulation features

• Actions set- discrete power levels maximum-
  transmitCost (input parameter)
• Best Response- choose (minimum) power level that
  gives maximum connectivity
• Update function- Round robin BR
• Nodes connect to all nodes with a transmission
  radius
• Benefit only from bi-directional links
• Intermediate uni-directional links still possible
• Thus, if two nodes within each others transmit
  radius are not connected, a link is still
  possible.

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Network Payoff

• Benefit - the number of nodes to which a node is
  connected directly or indirectly (times a
  multiplier- input parameter)
• Effect of multiplier M on network topology (e.g.,
  if Mpi,max the game is OPG)
• Cost- power level
• Utility function benefit- cost

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Belief System- Problem Description

• Actions of node i is set of links, ij, to have
  with each node j Ailij lij 0 or 1
• Let the induced undirected network g(l)ij lij
  lji 1, for all i, j in N
• Utility function (connections model)
• bij benefit to node i from node j (input)
• delta decay factor 0,1 direct links more
  beneficial than indirect links (input)
• t(ij) geodesic shortest hop distance between i
  and j
• Cijlink cost set to TransmitCosti if euclidean
  distance (nth power) dn(i,j)ltTransmitCosti, and
  dn(i,j)ltTransmitCostj otherwise no link (n2
  to 4)

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Continued

• First order beliefs of node i, li (Gilles et
  al.)
• Forming link requires consent, deletion is one
  way

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Definitions (Gilles et al.)

• Network g is weakly monadically stable (WMS) if
• Network g is monadically stable (MS) if g is WMS
• and for every player i, and for every player j,
  
• (i.e., beliefs are confirmed)

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Claim

• Monadically stable (MS) networks are subset of
  Nash Equilibrium networks
• Sketch of proof g(l) MS gt g(l) WMS gt for every
  node i, li BR to li(beliefs of i about other
  nodes actions). Also, g(l) MS gtlil-igt li BR
  to l-i for all i gt g(l) is nash equilibrium
  network

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Belief Systems Simulation features

• Consent model gt only bi-directional links give
  benefit
• In the context of topology control- nodes can
  propose links only with nodes within its transmit
  radius (different from previous model)
• All link costs same transmitCost
• Every node has the same belief model
• Best response to beliefs (for e.g, node i will
  form link with node j, only if it believes node j
  will also form link and forming a link is in is
  benefit)

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Continued

• Implemented Connections model (Jackson- Wolinsky)
• Implemented consent model
• Round robin, simultaneous update
• Did not implement self confirming criteria
• I think Equilibrium is self confirming if links
  are bidirectional in equilibrium.
• If steady state network has uni-directional links
  then weakly monadically stable.

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Benefit of belief systems

• Fast convergence (since nodes know the utility
  functions of other nodes with which it wishes to
  form links)
• Generally, no empty networks in steady state
  (unlike Nash Equilibrium Networks)
• Networks similar to connections model networks
  (star, complete, ..)

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Matlab GUI

• Changed to include topology control/network
  formation model
• TransmitCost- Node property
• Changed to include Self Confirming/ Nash

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Matlab Interface
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I. Initial empty network
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I. Steady state network (WMS network)
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II. Initial empty network
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IIa. Steady state network (WMS network)
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IIb. Steady state network (NE network)
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III. Initial complete network
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IIIa. Steady state network (WMS network)
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IIIb. Steady state network (NE network)
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IV. Topology Control (Initial network)
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IV. Topology Control (Steady state)