Networks: how Information theory met the space and time Philippe - - PowerPoint PPT Presentation

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Networks: how Information theory met the space and time Philippe - - PowerPoint PPT Presentation

Oct 21, 2023 404 likes •926 views

Networks: how Information theory met the space and time Philippe Jacquet INRIA Ecole Polytechnique France Plan of the talk History of networking and telecommunication Physics, mathematics, computer science Internet Routing

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SLIDE 1

Networks: how Information theory met the space and time

Philippe Jacquet INRIA Ecole Polytechnique France

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SLIDE 2

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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SLIDE 3

History of networking and telecommunication

  • 1900: Marconi on Eiffel tower: first wireless

telecommunication

  • 1948: Shannon and Information Theory,

transistor: first telecommunication massive

  • ptimization
  • 1986: Birth of Internet, World Wide Web: First

massive data, multimedia, multi-user network.

  • 2000: the internet got wireless
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SLIDE 4

Rise of Telecommunications (1800-1900)

  • C. Chappe 1793
  • S. Morse 1837
  • T. Edison 1878
  • G. Marconi 1899
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SLIDE 5

From pigeon to light speed: the triumph over the matter

  • Electromagnetic
  • Journal paper
  • Pigeon post
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SLIDE 6

From lamp to transistor: code and signal 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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SLIDE 7

The wonders of the digital revolution

  • C. Berrou 2000
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SLIDE 8

Internet: The triumph over the complexity

  • Telecommunications cross the border of

computer 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 heterogeneous

networks (fiber, telephone, etc)

  • Detects and avoids damaged parts in real

time.

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SLIDE 9

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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SLIDE 10

Internet is the most complicated artefact

  • Far from human brain:
  • 1011 neurons
  • Connectivity degree :

10,000 -100,000

  • Speed: 100-400 m/s

(scale with world internet)

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SLIDE 11

Histogram

  • Physics, math and computer sciences in

telecommunication

105 1010 1015 1900 1950 1980 2008 bit/s/100,000km2 1

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SLIDE 12

wireless performance from Marconi 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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SLIDE 13

Comparison: road traffic

Road traffic increase 1900-2008: < 105 =100,000?

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SLIDE 14

Physics,mathematics,computer science

  • The telecommunications without

– The physics

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SLIDE 15

Physics,mathematics,computer science

  • The telecommunications without

– The mathematics

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SLIDE 16

Physics,mathematics,computer science

  • The telecommunications without

– The computer science

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SLIDE 17

Physics-math-computer science

  • Networks are together physics, math

and computer science

  • Computer science is the science which

makes a complex system a simple

  • bject

– Gee, how hard sometimes to reach this simplicity…

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SLIDE 18

Example: routing protocol

  • Routing tables is like orientation maps

in every router

North-East Road North Road South-East Road South Road South-West Road North-West Road

RouterA

12880 km NW Beijing 133 km N Paris 62 km NE routerB distance exit destinati

  • n
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SLIDE 19

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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SLIDE 20

Example: routing protocol

  • RIP (distance vector):

– Complexity: NL (per refresh period) – Convergence time: network diameter

3 km 5 km 130 km N Paris 134 km N Paris 133 km NE Paris

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SLIDE 21

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 D

5 km SE C 3 km NE B

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SLIDE 22

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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SLIDE 23

RIP failure

  • Count to infinity (1983 ARPANET

incident)

3 km 5 km 1 km 3 km 3 km 2 km 4 km 5 km

L <15 L

Diameter max limited to 15

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SLIDE 24

Wireless networks

Mobile 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, with average degree 1,000

  • BGP needs 1014 links exchange per refresh

time

  • BGP fails on Wifi networks with 20 users at

walking speed.

  • Heavy density kills link state management.
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SLIDE 25

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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SLIDE 26

Wireless topology compression

  • the MPR set covers the two-hop

neighbor set

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B A

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

  • ptimal 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 = N 3 L2 logN r = 3 N L

  • 2

3

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SLIDE 29

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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SLIDE 30

Wireless protocol compression

  • 107 ratio is like

– your car traveling at light speed

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SLIDE 31

Information theory in networks

  • Originally for point to point coding

A 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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SLIDE 32

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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SLIDE 33

Limitation of information theory within space and time

  • Fake superluminal information

propagation: the twin traveler paradox

100 km, same time X X=Y Y time I(X,Y) > 0

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SLIDE 34

Causality in information theory

  • Twin traveler paradox resolution?.

– For a common fixed past X et Y should be independent.

)) Past( ) Past( | ) , (( =

  • Y

X Y X I

100 km, same time X X=Y Y Past(X) Past(Y)

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SLIDE 35

Bell theorem

  • Information is non causal:

)) Past( ) Past( | ) , (( >

  • Y

X Y X I

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SLIDE 36

Network: information and space-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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SLIDE 37

Space time relay versus information 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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SLIDE 38

Space time relay versus information causality

B A

time

B A C D temporal loop: quantum unitarity violation capacity I = | | log2

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SLIDE 39

A wireless model

  • Emitters are distributed as Poisson

process in the plan

– Signals sum

S = | z zi |

i

  • density

z zi

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SLIDE 40

A wireless model

  • Signal distribution

E(eS) = exp((1 ) )

= 2

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SLIDE 41

Wireless information theory

  • Wireless capacity,

– emiters send independent information – Computable and invariant even with random fading

I0 = E log2(1+ | z zi | | z z j |

ji

  • )

i

  • I0 =
  • log2
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SLIDE 42

Wireless Shannon

  • Invariant with dimensions

– Works also with fractal spaces

  • D=4/3

I0 = D (log2)1

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SLIDE 43

Wireless Space capacity

  • With signal over noise ratio K

requirement

  • Average area of correct reception

I(K) = sin()

  • K

(K) = I(K)

  • (10) 0.037066
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SLIDE 44

Wireless Space capacity

  • Reception probability vs

distance

  • Optimal routing radius

r

m = argmax r>0

{rp(r,,K)} = r

1

  • K

1 4

p(r,,K) = p(r K

1 4,1,1)

p(r,1,1) = (1)n sin(n)

  • n
  • (n)

n! r2n p(r,1,1) =1 erf(r2 2 ) when = 4

r

z z

  • z
  • z
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SLIDE 45

Wireless Space capacity

  • Average number of

retransmissions

  • Net traffic density
  • True if neighborhood is

dense enough

r

m

z

  • z

z z r

m p(r m)

r

m 2 N

A > logN

= r

m p(r m)

E(| z z |)

A

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SLIDE 46

Space capacity result (Gupta- Kumar 2000)

  • The capacity increases

with the density

  • Massively dense

wireless networks

A = r

1 2p1

A E(| z z |) N logN = O N logN

  • N

capacity

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SLIDE 47

Time capacity paradox

  • Mobility can create capacity in sparse networks
  • Delay Tolerant Networks

S D

X X

path disruption!

S D

End-to-end path

S D

X X

path disruption!

node link

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SLIDE 48

Information propagation speed

  • Unit disk graph model
  • Random walk mobility

model

z

  • z
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SLIDE 49

Time capacity paradox

  • Mobility creates capacity

capacity time capacity time

Permanently disconnected Permanently connected

Information propagation time

T( z )

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SLIDE 50

Information propagation speed

  • Upper bound of information propagation

speed

– Any quantity c such that – Is the smallest ratio in the kernel of

  • D(,) =

( + )2 2v 2 2sI0() 1 2 I1()

Ik() are modified Bessel functions

lim

z P(T(

z ) < | z z | c ) = 0

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SLIDE 51

Information propagation speed

space time

speed s =1 turn rate = 0.1 node density = 0.25

theory