networks: how information theory met the space and time · networks: how information theory met the...
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Networks: how Informationtheory met the space and time
Philippe JacquetINRIA
Ecole PolytechniqueFrance
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Plan of the talk
• History of networking and telecommunication• Physics, mathematics, computer science• Internet Routing complexity• Mobile ad hoc networking complexity• Wireless networking: Shannon’s law.• Information and space-time• Wireless capacity and space-time
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History of networking andtelecommunication
• 1900: Marconi on Eiffel tower: first wirelesstelecommunication
• 1948: Shannon and Information Theory,transistor: first telecommunication massiveoptimization
• 1986: Birth of Internet, World Wide Web: Firstmassive data, multimedia, multi-user network.
• 2000: the internet got wireless
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Rise of Telecommunications(1800-1900)
C. Chappe 1793 S. Morse 1837 T. Edison 1878 G. Marconi 1899
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From pigeon to light speed:the triumph over the matter
• Electromagnetic
• Journal paper
• Pigeon post
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From lamp to transistor: code andsignal processing revolution.
The triumph over the numbers• Telecommunications cross the border from
physics to mathematics
transistor 1947
enigma 1941
C. Shannon 1948
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The wonders of the digitalrevolution
C. Berrou 2000
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Internet: The triumph over thecomplexity
• Telecommunications cross the border ofcomputer science– Cold war 1970: the strategic network
identified as the weak point• Answer: ARPANET then INTERNET
– First, connect missiles sites– Then, universities– Every user
• The protocol (IP) binds heterogeneousnetworks (fiber, telephone, etc)
• Detects and avoids damaged parts in realtime.
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Facts and figure about internet
• internet:– 6.108 connected
machines– Connectivity degree 10-
100– 3.1010 web pages– 2.1015 online bits– speed sub c
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Internet is the most complicatedartefact
•Far from human brain:
•1011 neurons
•Connectivity degree :10,000 -100,000
•Speed: 100-400 m/s(scale with worldinternet)
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Histogram• Physics, math and computer sciences in
telecommunication
105
1010
1015
1900 1950 1980 2008
bit/s/100,000km2
1
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wireless performance fromMarconi to Wifi
• Traffic density– 1900:
• 10 bit/s/100.000 km2 ,• 1000 watt
– 2008:• 10,000,000 bit/s/ha,• 0.01 watt
A factor 1014
100,000,000,000,000
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Comparison: road traffic
Road traffic increase 1900-2008: < 105 =100,000?
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Physics,mathematics,computerscience
• The telecommunications without– The physics
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Physics,mathematics,computerscience
• The telecommunications without– The mathematics
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Physics,mathematics,computerscience
• The telecommunications without– The computer science
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Physics-math-computerscience
• Networks are together physics, mathand computer science
• Computer science is the science whichmakes a complex system a simpleobject– Gee, how hard sometimes to reach this
simplicity…
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Example: routing protocol
• Routing tables is like orientation mapsin every router
North-EastRoad
NorthRoad
South-EastRoadSouth
Road
South-WestRoad
North-WestRoad
RouterA
12880 kmNWBeijing
133 kmNParis
62 kmNErouterB
distanceexitdestination
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Example: routing protocol
• Two protocols;– RIP: run to neighbor protocol
• Deliver the routing table to local neighbor
– BGP: run theTour de France protocol• Deliver the local routing table to whole network
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Example: routing protocol
• RIP (distance vector):
– Complexity: NL (per refresh period)– Convergence time: network diameter
3 km
5 km
130 kmNParis
134 kmNParis
133 kmNEParis
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Example: routing protocol
• BGP (link state):
– Routing table computed on local link database– Complexity L2 (per refresh period)– Convergence: network diameter
3 km
5 km
A
B
C
D5 kmSEC
3 kmNEB
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Example: routing protocol
• Which is best:– Deliver whole table to local?
– Deliver local table to all?
•Divergence time makes the difference!– Network Diameter with BGP (symmetric)– At least Diameter × L with RIP (asymmetric)
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RIP failure
• Count to infinity (1983 ARPANETincident)
3 km
5 km1 km
3 km
3 km
2 km
4 km
5 km
!
"# L <15 # L
Diameter max limited to 15
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Wireless networksMobile ad hoc networks
– Mobility makes link failure a necessity• Refresh period 1 second• Automatic self-healing
– Local neighborhood is local space• Unlimited neighborhood size
– Stadium network: N=10,000, withaverage degree 1,000
• BGP needs 1014 links exchange per refreshtime
• BGP fails on Wifi networks with 20 users atwalking speed.
• Heavy density kills link state management.
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Wireless topology compression
• Optimized Link State Routing protocol– Advertize local table subset to whole
network– Local neighborhood subset is the
MultiPoint Relay (MPR) set of the node.– Every node receives a compressed
topology information.
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Wireless topology compression
• the MPR set covers the two-hopneighbor set
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BA
Wireless topology compression
• MPR sets forms a remote spanner– Nodes compute their routing table on
remote spanner plus their local topology.
• Topology compression is lossless– Optimal routes on remote spanner also
optimal without compression.
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Wireless topology compression
• In Erdoss-Renyi random graphs– Random selection gives compression rate
• In unit disk graph model– Greedy selection gives compression rate
• Stadium network– Compression rate 10-2
!
"r
=N3
L2logN
!
"r
= 3#N
L
$
% &
'
( )
2
3
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Dissemination compression
• Only MPR retransmit local routing table– Another compression of rate
• Stadium network– Overall compression 10-7!
" #r
N
L
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Wireless protocol compression
• 107 ratio is like– your car traveling at light speed
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Information theory in networks
• Originally for point to point codingA B
X: Emitted code Y: Received code
Channel capacity: I(X,Y)=entropy of Y - channel entropy
!
I(X,Y ) = h(Y ) " h(Y | X)
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Example: wireless transmission
• Emitted Symbol:• Received Symbol:
– Noise
– Capacity per symbol!
X an integer in [1,S]
!
Y = X + "
!
" an integer in [1,B]
!
h(Y ) = log(S + B)
h(Y | X) = logB
!
I(X,Y ) = log(1+S
B)
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Limitation of information theorywithin space and time
• Fake superluminal informationpropagation: the twin traveler paradox
100 km, same time X
X=Y
Y
time
!
I(X,Y ) > 0
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Causality in information theory• Twin traveler paradox resolution?.
– For a common fixed past X et Y should beindependent.
0))Past()Past(|),(( =! YX YXI
100 km, same time X
X=Y
Y
Past(X)Past(Y)
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Bell theorem• Information is non causal:
0))Past()Past(|),(( >! YX YXI
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Network: information andspace-time
• A Network is– Set of objects in physical
space– relay information
• from arbitrary source• to arbitrary destinations.
– Concept of router– Concept of information
propagation path
space
time
source
destination
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Space time relay versusinformation causality
• Can network of superluminal relays exist– Space time relay (A,B) definition
• Can relay anything– From any source in Past(A)– To any destination in Future(B)
A B
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Space time relay versusinformation causality
BA
time
B
A C
D
temporal loop: quantum unitarity violation
!
capacity I =|" |
log2
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A wireless model
• Emitters are distributed as Poissonprocess in the plan
– Signals sum
!
S = | z " zi|"#
i
$!
density "
!
z
!
zi
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A wireless model
• Signal distribution
!
E(e"#S) = exp("$%&(1" ')#' )
!
" =2
#
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Wireless information theory
• Wireless capacity,– emiters send independent information
– Computable and invariant even withrandom fading
!
I0 = E log2(1+| z " zi |
"#
| z " z j |"#
j$ i
%)
i
%
&
'
( ( (
)
*
+ + +
!
I0 ="
log2
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Wireless Shannon
• Invariant with dimensions
– Works also with fractal spaces• D=4/3
!
I0 ="
D(log2)#1
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Wireless Space capacity
• With signal over noise ratio Krequirement
• Average area of correct reception!
I(K) =sin("# )
"#K
$"
!
"(K) =I(K)
#
!
"(10) # 0.037066
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Wireless Space capacity
• Reception probability vsdistance
• Optimal routing radius
!
rm = argmaxr>0
{rp(r,",K)} =r1
"K
1
4
!
p(r,",K) = p(r "K#1
4 ,1,1)
!
p(r,1,1) = ("1)nsin(#n$)
#n
%&(n$)
n!r2n
!
p(r,1,1) =1" erf(r
2
2) when # = 4
r
!
z
!
z
!
" z
!
" z
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Wireless Space capacity
• Average number ofretransmissions
• Net traffic density
• True if neighborhood isdense enough
!
rm
!
z
!
" z
!
z " # z
rm p(rm )
!
"rm
2 N
A> logN
!
" = #rm p(rm )
E(| z $ % z |)
A
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Space capacity result (Gupta-Kumar 2000)
• The capacity increaseswith the density
• Massively densewireless networks
!
A" = # r12p1
A
E(| z $ % z |)
N
logN= O
N
logN
&
' (
)
* +
N
capacity
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Time capacity paradox
• Mobility can create capacity in sparse networks
• Delay Tolerant Networks
S
DX Xpathdisruption!
S D
End-to-end pathS
D
X
Xpathdisruption!
nodelink
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Information propagation speed
• Unit disk graph model• Random walk mobility
model
!
z !
" z
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Time capacity paradox
• Mobility creates capacitycapacity
time
capacity
timePermanently disconnected Permanently connected
Information propagation time
!
T( " z )
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Information propagation speed
• Upper bound of information propagationspeed– Any quantity c such that
– Is the smallest ratio in the kernel of
!
"
#
!
D(",#) = ($ + #)2 % "2v 2 % $ %2&'sI
0(")
1%&'2
"I1(")
!
Ik() are modified Bessel functions
!
lim " z #$ P(T( " z ) <| z % " z |
c) = 0
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Information propagation speed
space
time
!
speed s =1
turn rate " = 0.1
node density # = 0.25
theory