Connectivity Analysis of Wireless Ad Hoc Networks

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Connectivity Analysis of Wireless Ad Hoc Networks

• Xiaobo Long
• CSCI 4260 MATH 4150 course presentation

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Wireless ad hoc network

• No fixed infrastructure.
• Each node acts as a router, forwarding data
  packets for other nodes.
• Each node has a transmission range.
• Node v can receive signal from node u if node v
  is within the transmission range of the sender u.
• Otherwise, two nodes communicate through
  multi-hop wireless link by using intermediate
  nodes to relay the messages.

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A wireless ad hoc network
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Network model

• GG(V,E)
• V represents network-enabled ad hoc devices.
• E represents the wireless communication links.
• Assumptions
• each node has same transmission range r0
• two-dimensional simulation area A.
• n nodes have i.i.d distributions on A
• uniform random distribution is used.
• node density
• Undirected graph G.
• Only bidirectional links considered.

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Geometric random graph

• nodes are mobile in wireless ad hoc networks.
• some links will be broken and new links will be
  established.
• graph G (V,E) is a dynamic system.
• Assume At all times t, V (t) V (G) remains the
  same.
• but E(t) is arbitrary subject to the connectivity
  of G.

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A simple model for the analytical study of
connectivity on ad hoc networks

• Use of random geometric graphs g(n r)
• n vertices uniformly and independently
  distributed on a square field A.
• there exists an edge between two vertices iff the
  distance between them is less or equal to the
  transmission range r.
• for a given pair of parameters (n r),
  connectivity arises with a probability P(n,r).

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Goal of connectivity probability analysis

• To achieve a connected ad hoc network
• for any node there must be a multi-hop path to
  any other node.
• Given n and
• What is the minimum r0 required to achieve
• An connected G with certain probability?
• An ad hoc network in which no node is isolated?
• A K-connected G with certain probability?
• An ad hoc network that will still be connected if
  any k-1 nodes fail
• Or given r0
• P(n, r0)probability of G being connected ?
• probability of G being K-connected ?

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Connectivity and minimum degree

• Relationship of K-connectivity and minimum
  degree
• The event dmingt0 is necessary (but not
  sufficient) condition for G being connected.
• P(G is k-connected) P(dmin K)
• G is k-connected)? dmin K
• Converse not necessarily true
• P(G is disconnected) 1-P(dmingt0)

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Penroses theorem 1

• P(G is k-connected)P(dmin K)
• For n sufficiently large
• for P(dmin K) almost 1
• Penrose proved that
• With high probability,
• Starts with a trivial graph,
• Adds the corresponding links as r0 increases,
• Resulting graph becomes k-connected at the moment
  it achieves minimum degree dmin of k.

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Minimum node degree

• Theorem 1 (A probabilistic bound for the minimum
  node degree of a homogeneous ad hoc network) 2
• --can be proved by stochastic process methods

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Express connectivity probability as function of r0

• As a special case
• if n01, for n sufficiently large.
• Connectivity probability P(n,r0) P(dmin 1)
  with high probability.
• Thus given n nodes, we have expressed
  connectivity probability of ad hoc networks as
  function of transmission range.

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Conclusion

• Connectivity of wireless ad hoc network is
  investigated.
• Model the network as a geometric random graph.
• Use analytical expression to find transmission r0
  that creates, for a given node density, an almost
  surely k-connected network.

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References

• 1 M.D. Penrose, On k-connectivity for a random
  graphs, Wiley Random Structures and Algorithms,
  vol. 15, no. 2, pp. 145-164, 1999
• 2 Christian Bettstetter, On the Minimum Node
  Degree and Connectivity of a Wireless Multihop
  Network, MOBIHOCs, June 2002
• 3 Xiangy Li, Algorithmic, geometric and graphs
  issues in wireless networks, Journal of Wireless
  Communications and Mobile Computing (WCMC), Wiley
  Publications. vol. 3, no. 2, pp. 119 140,
  September 2003
• 4 Calomme and Guy Leduc, The Critical
  Neighbourhood Range for Asymptotic Overlay
  Connectivity in Dense Ad Hoc Networks, June 2005