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Connectivity in sensor networks

This paper discusses the issue of connectivity in wireless ad hoc networks, sensor networks, and hybrid networks. It explores the position of nodes and base stations, as well as connectivity in packet radio networks. The concept of percolation theory is also introduced to analyze connectivity in different network domains.

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Connectivity in sensor networks

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  1. LCA Connectivity in sensor networks Patrick Thiran (joint work with Olivier Dousse, François Baccelli and Martin Hasler) LCA-ISC-I&C EPFL Patrick.Thiran@epfl.ch http://lcawww.epfl.ch -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  2. Ad-hoc <-> sensor <-> hybrid <-> cellular • Connectivity is an essential issue in wireless • ad hoc networks (many to many, global) • sensor networks (many to one, global) • hybrid (multi-hop cellular) networks (many to many, local) • Position of nodes is a (often homogeneous Poisson) spatial point process, with intensityl. • Position of base stations at nodes of a honey-comb grid, all connected to each other by a wired network. -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  3. Fixed radio range r • Nodes i and j, at positions xi and xjare directly connected iff The Boolean model with circular grains • References: • E. N. Gilbert, « Random plane networks », SIAM Journal, 1961. • R. Meester and R. Roy, « Continuum percolation », Cambridge University Press, 1996. • O. Dousse, P. Thiran, M. Hasler, « Connectivity in ad hoc and hybrid networks », Infocom 2002. -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  4. Connectivity in packet radio networks • Finite domain: is the network fully connected ? • Kleinrock & Silvester (1978) « Optimum transmission radii in packet radio networks or why six is a magic number » : r2l = 5.89 is a good value for throughput • Philips, Panwar, Tantawi (1989) « Connectivity properties of a packet radio network » : r2l must grow logarithmically with the area of the domain • Gupta & Kumar (1998) « Critical power for asymptotic connectivity in wireless networks » : for ,r2l = log l + K(l) where K(l)  . • Shakkottai, Srikant, Shroff (2003) « Unreliable sensor grids: Coverage, Connectivity and Diameter»: r2l log l/p(l) where p(l) is the node failure prob. • Infinite domain: is there an infinite connected component (percolation theory): • Continuum percolation: there exist a finite (r2l)c, below which all connected components are a.s. bounded and above which there is a unique infinite connected component (Gilbert (1961), Hall (1985), Zuev & Siderenko (1985), Menshikov (1986), Meester & Roy (1990, 1994)) • 1.64 < (r2l)c < 17.9 (Gilbert (1961)) • 2.195 < (r2l)c < 10.526 (Philips, Panwar, Tantawi (1989)) • (r2l)c  4.5 (numerical value, Quintanilla, Torquato, Ziff (2000)) -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  5. (lr2) λr2 Full or partial connectivity ? • Long range connectivity appears much before full connectivity because of a phase transition mechanism (percolation) -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  6. Ad hoc or sensor network ? • Ad hoc network : multiple transmissions, many to many. Connectivity metric = probability that an arbitrary pair of nodes is connected to the rest of the network Pc • Sensor network : many to one (the base station collecting data). Connectivity metric = probability that one arbitrary node is connected to the base station  -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  7. Pconnectivity Pcoverage = 1-exp(-lpr2) λr2 (λr2)c Ad hoc or sensor network ? • Ad hoc network : • connectivity • Sensor network : • connectivity (probability that an arbitrary node is connected to the base station) and • coverage (probability that an arbitrary point is covered by a node). -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  8. Network on a line x • Let Pc(x) be the probability that two nodes distant of x space units are connected, given r and l. • In 1-dim: Full connectivity  Full coverage • In 1=dim: • Pc(x)exp(-l(x-r)e-lr) decreases exponentially fast with x  -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  9. Pc(x) λr2 (λr2)c sub-critical (r slightly < rc) super-critical (r slightly > rc) Network on a plane • Percolation theory: Let(r, l) be the probability that an arbitrary node belongs to an infinite cluster (percolation probability). Then there is(lr2)c such that • (r, l) = 0 if r2l< (r2l)c (“sub-critical”) • (r, l) > 0 if r2l> (r2l)c (“super-critical”) -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  10. CA(p) A p Bottlenecks are unavoidable Let P be the number of alternate paths between any pair of nodes A and B. Thm: min(NA, NB)  P  max(NA0, NB0) C supercritical subcritical p p -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  11. Beyond the Boolean model with circular grains: irregularity helps • Percolation occurs sooner for elongated shapes (Penrose (1993), Booth, Bruck, Cook, Franceschetti (2003)) • Possible advantage of directional antennas • Uni-directional links -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  12. Beyond the Boolean model: the physical (STIRG) model • Signal to Noise Ratio at Node j receiving from Node i is • P = Emitting power • L(d) = Attenuation function at distance d (e.g., L(d) = d-a ) • N0 = Background thermal noise • γ = degree of orthogonality of the code (γ = 1 for a narrowband system, 0 γ < 1 for a CDMA system) • Nodes i and j are directly connected iff • Reference: O. Dousse, F. Baccelli, P. Thiran, « Impact of Interferences on Connectivity in ad hoc networks », Infocom 2003. -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  13. (l) g = 0 g > 0 l Interferences can destroy connectivity… g = 0 g > 0 -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  14. … but not always. • Differences with the boolean model: • The existence of an edge depends on every other node’s position • The node degree is bounded (it was Poisson for the Boolean model) by 1 + 1/bg. A necessary condition for percolation is thus g < 1/b. g = 0 g > 0 -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  15. Beyond connectivity: MAC, Broadcast, routing ? • Simple TDMA ensures as much connecitivty as performant CDMA shemes (Dousse, Baccelli, Thiran 2003) • Routing is more challenging close to percolation threshold (graph is looks like a labyrinth) (Kuhn, Wattenhofer, Zollinger 2003) • Probabilistic broadcast (Sasson, Cavin and Schiper 2003) -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

  16. Conclusion • Percolation theory is very useful for sensor and ad hoc networks: connectivity and beyond. • Boolean model with circular grains • Importance of dimensionality of the area covered by the network • In 2 dim: connectivity  coverage • Phase transition is key to explain connectivity in 2 dim at low density • Beyond Boolean model with circular grains : • Elongated grains (directional antennas) helpful for connectivity • Physical model (STIRG): Percolation occurs despite interferences  • Proof for L(x) with a compact support. Probably not a necessary condition. • Beyond connectivity : MAC, routing, broadcast, … -NeXtworking’03- Chania, Crete, Greece, June 23-25,2003 The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES &TRENDS INNETWORKING

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