Convergence in Sensor Network Formation

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Convergence in Sensor Network Formation

• James Neel
• July 22, 2004

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Presentation Overview

• Connections Model Overview
• Problem setup
• Proof of best response convergence for ? 1

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Connections Model

• Slightly modified from Jackson 1996
• Each node i assigns a benefit bij to having
  another node j in network g
• Benefit is degraded by a factor ? raised to the
  length of the shortest path from i to j.
• Additional costs for supporting link can be
  accrued
• Utility Function
• tij shortest path length between i and j - 1

Jackson, M.O. and A. Wolinsky A Strategic Model
of Social and Economic Networks, Journal of
Economic Theory 71, 44-74.
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Sensor Network Scenario
Source

• Network of Sources and Sink
• Only benefit from connection to sink
• Costs function of transmitted distance (transmit
  power limited)
• Net benefit function of raw benefit and hops
• Models delay, link reliability

Sink
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Best Response Characterization

• Best Response is a single link or no link
• Proof Additional links that beneficially connect
  to sink add no benefit, links that are not
  beneficial is not preferable to no link
• In a NN, no paths to the sink branch (requires
  two outbound link).
• In NN, no paths lead out from source
• Implication All Nash Networks are trees for all ?

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Some Terminology

• Trimmed network A network, n, such that all
  nodes maintain one link or no link. (not
  necessarily a stable network)
• Poison path A profitable path for node j that is
  not profitable for a node after j.
• Pruned network a trimmed network with no poison
  paths
• Healthy network a pruned network in which every
  node has positive utility

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Best response characterization

• After a complete round-robin best response, any
  starting network must be in a trimmed network.
• Proof Best response is a single link or no link.
  After each node has had a chance to play, each
  node must have a single link or no link.
• Trimmed networks playing a best response do not
  create poison paths (for ?1)
• Proof Creating a new poison path requires a node
  to node to create a link that is not in its
  self-interest, thus not a best response.

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Best response characterization

• In a pruned network, all nodes not in a healthy
  network must be totally disconnected.
• Proof Suppose a node has a link and is not in
  the healthy network. Then this link must be
  unprofitable else would be in the healthy
  network. But there can be no unprofitable links
  in a pruned network (no poison paths).
  Contradiction.

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Convergence for ?1

• After two complete best-response round-robins,
  the network is pruned for ?1
• Proof After the first round, the network is
  trimmed, and no node maintains a direct link (not
  path) to the source that is not profitable. Thus
  the only poison paths are disconnected poison
  paths. All prior poison paths are no longer
  profitable in the second round, so all prior
  poison paths are abandoned. Further no new
  poisoned paths can be made. With no poisoned
  paths, the trimmed network is a pruned network.

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Convergence for ?1

• For ?1, all nodes in a healthy network, remain
  in a healthy network when playing a round-robin
  best response (or better response).
• Proof A node falls out of a healthy network if
  it chooses to disconnect or a node ahead in its
  path disconnects. As disconnecting drops the
  utility to 0 (or worse), this is never preferable.

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Convergence for ?1

• For ?1, the number of nodes in a healthy network
  is nondecreasing for a round robin best (or
  better) response.
• In a pruned network, for ?1 the number of nodes
  not in a healthy subnet is nonincreasing.

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Convergence for ?1

• For ?1, all healthy networks converge when
  following a round-robin best response.
• Proof Consider the function
  
• As this is a nondecreasing function on a finite
  action space (and f is finite), play must
  converge. Note that any profitable deviation
  increases the value of f.

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Convergence for ?1

• For ?1, all pruned sensor networks converge.
• Proof Pruned sensor networks consist of nodes in
  H and H (not in H). Nodes in H converge. When
  profitable, nodes from H transition to H (and
  thus converge). Nodes in H that cannot
  profitably join H remain disconnected (and thus
  are at their steady-state once pruned).

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Convergence for ?1

• For ?1 the sensor network scenario converges
  under a round-robin best response from every
  starting network.
• Proof Initial network necessarily converges to a
  pruned network. Pruned network necessarily
  converges.
• For ?1 the sensor network scenario has weak FIP.
• Proof Best response is everywhere convergent.

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Spanning Trees for ? 1

• If there exists a numbering scheme such that
  ck,k-1 lt b, then a spanning tree (of sorts) NN
  exists and is converged to from the empty
  network.
• Proof Choose sink as node 0. Node 1 joins N
  (healthy network) with its best response. If node
  k is in N then node k1 will join in its next
  iteration (if not before). By induction, all
  nodes will eventually join N, forming a spanning
  tree.

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Other convergence results

• Cant establish convergence by asynchronous
  convergence theorem (fails synchronous
  convergence theorem even if a box condition could
  be found)
• Due to weak FIP
• Updating errors can be made Friedman.
• Will converge random round-robin.

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Summary

• Showed that sensor network is always convergent
  for ? 1 when implementing a best response round
  robin dynamic.
• Showed network has weak FIP (a rather novel
  concept in network formation)
• Gave condition for formation of spanning tree.