LCA
Connectivity in sensor
networks
Patrick Thiran (joint work with Olivier Dousse,
François Baccelli and Martin Hasler)
LCA-ISC-I&C EPFL
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Patrick.Thiran@epfl.ch
http://lcawww.epfl.ch
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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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 intensity l.
 Position of base stations at nodes of a honey-comb grid, all
connected to each other by a wired network.
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
1
The Boolean model with circular grains
 Fixed radio range r
 Nodes i and j, at positions xi and xj are directly connected
iff
xi

xj
 r
 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.
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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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 l, 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))

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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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Full or partial connectivity ?
 Long range connectivity appears much before full connectivity
because of a phase transition mechanism (percolation)
(lr2)
λr2
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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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 
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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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).

Pconnectivity
Pcoverage =
1-exp(-lr2)
λr2
(λr2)c
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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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) 
x / r 

i 0
( le
 lr
)
(
 x / r 1  le  lr ( x  (i + 1) r )
( x  ir )
 lr
e

i!
i!
i 0
i
)
i
 Pc(x)  exp(-l(x-r)e-lr) decreases exponentially fast with x
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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”)

sub-critical (r slightly < rc)
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Pc(x)
(λr2)c
λr2
super-critical (r slightly > rc)
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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Bottlenecks are unavoidable
N A ( p )  min # {edges intersecte d by C}
CC A ( p )
CA(p)
p 
p 0
A
C
p 0
Let P be the number of alternate paths
between any pair of nodes A and B.
Thm: min(NA, NB)  P  max(NA0, NB0)
N A ( p) / N A0
subcritical
X
p 0
N A0  lim N A ( p )  sup N A ( p )
p
N A ( p) / N A0
N A  lim N A ( p )  inf N A ( p )
supercritical
p
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p
Patrick Thiran
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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
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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Beyond the Boolean model: the physical
(STIRG) model
 Signal to Noise Ratio at Node j receiving from Node i is
SNRi  j 
–
–
–
–
PL( xi
N0 + 

xk
 PL( x
k i , j
i
)
 xk
)
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
min{ SNRi  j , SNR j i }  
 Reference:
O. Dousse, F. Baccelli, P. Thiran, « Impact of Interferences on
Connectivity in ad hoc networks », Infocom 2003.
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Interferences can destroy connectivity…
=0
>0
(l)
=0
>0
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l
Patrick Thiran
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… but not always.
=0
>0
 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/. A necessary condition for percolation is thus   1/.

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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
Patrick Thiran
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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)
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
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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, …
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The First COST-IST(EU)-NSF(USA) Workshop on EXCHANGES & TRENDS IN NETWORKING
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