Shortest Paths & Link Weight Structure in Networks Piet Van - - PowerPoint PPT Presentation

• P. Van Mieghem 1

Faculty of Electrical Engineering, Mathematics and Computer Science

Shortest Paths & Link Weight Structure in Networks

Piet Van Mieghem CAIDA WIT (May 2006)

• P. Van Mieghem 2

Faculty of Electrical Engineering, Mathematics and Computer Science

Introduction The Art of Modeling Conclusions Outline

• P. Van Mieghem 3

Faculty of Electrical Engineering, Mathematics and Computer Science

Telecommunication: e2e

• Main purpose: Transfer information from A B
• Nearly always: Transport of packets along shortest (or optimal) paths
• Optimality criterion: in terms of Quality of Service (QoS) parameters

(delay, loss, jitter, distance, monetary cost,...)

• Broad focus:

What is the role of the graph/network on e2e QoS ? What is the collective behavior of flows, the network dynamics ?

A B

NETWORK G(N,L)

• P. Van Mieghem 4

Faculty of Electrical Engineering, Mathematics and Computer Science

Fractal Nature of the Internet

• P. Van Mieghem 5

Faculty of Electrical Engineering, Mathematics and Computer Science

Large Graphs

protein

Internet observed by RIPE boxes

• P. Van Mieghem 7

Faculty of Electrical Engineering, Mathematics and Computer Science

Ad-hoc network

Despite node movements, at any instant in time the network can be considered as a graph with a certain topology Physics of the evolution

• f Ad-hoc graphs

• P. Van Mieghem 8

Faculty of Electrical Engineering, Mathematics and Computer Science

Introduction The Art of Modeling:

The Basic Model

Conclusions Outline

• P. Van Mieghem 9

Faculty of Electrical Engineering, Mathematics and Computer Science

Network Modeling

• Any network can be representated as a graph G with N nodes

and L links:

– adjaceny matrix, degree (=number of direct neighbors), connectivity, etc... – link weight structure: importance of a link

• Link weights: also known as quality of service (QoS)

parameters

– delay, available capacity, loss, monetary cost, physical distance, etc...

• Assumption that transport follows shortest paths

– correct in more than 70% cases in the Internet

• P. Van Mieghem 10

Faculty of Electrical Engineering, Mathematics and Computer Science

Network Modeling Confinement to:

• Hop Count (or number of hops) of the shortest path:

– Apart from end systems, QoS degradation occurs in routers (= node). – QoS measures (packet delay, jitter, packet loss) depend on the number of traversed routers and not on the ‘length’ of shortest path. – Relatively easy to measure (trace-route utility) and to compute (initial assumption)

• Weight (End-to-end delay) of the shortest path:

– perhaps the most interesting QoS measure for applications – difficult to measure accurately – capacity of a link:

• measurement is related to a delay measurement (dispersion of two

IP packets back-to-back)

• P. Van Mieghem 11

Faculty of Electrical Engineering, Mathematics and Computer Science

Simple Topology Model

• Link weights w(i→j )>0:

– unknown, but very likely not constant, w(i→j )≠1 – assumption: i.i.d. uniformly/exponentially distributed – bi-directional links: w(i→j )= w(j→i )

• Complete graph KN
• One level of complexity higher: ER Random graphs of class

Gp(N)

– N: number of nodes – p: link density or probability of being edge i→j equals p – only connected graphs: p > pc ~ logN/N

5 2 4 3 1

2 2/3 3 1 5/2 1

Is this the right structure? Are exponential weights reasonable? Not a good model for the Internet graph, but reasonably good for Peer to Peer networks (Gnutella, KaZaa) and Ad Hoc networks

• P. Van Mieghem 12

Faculty of Electrical Engineering, Mathematics and Computer Science

Markov Discovery Process in the complete graph KN

3) transition rates for KN:

) (

1 ,

n N n

n n

− =

+

λ

1

time

t

node A

tt

t t2 t3 T node A B

1

time

( )

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ =

∑

= ≤ ≤ n j j j n j

a Exp a Exp

1 1

) ( min

1) property of i.i.d. exponential r.v.’s 2) memoryless property of exponential distribution

• P. Van Mieghem 13

Faculty of Electrical Engineering, Mathematics and Computer Science

Discovery times: where τj is exponentially distributed with mean

Markov Discovery Process and Uniform Recursive Tree

∑

=

=

k j j k

v

1

τ

time 1 2 3 4 5 6 7 8 1 2 6 3 4 7 8 5

corresponding URT Markov Discovery Process

v1 v2 v3 v4 v5 v6 v7 v8

[ ]

) ( 1 j N j E

j

− = τ H = 0 H = 1 H = 2 H = 3

Growth rule URT: Every not yet discovered node has equal probability to be attached to any node of the URT. Hence, the number of URTs with N nodes equals (N-1)!

τ6

• P. Van Mieghem 14

Faculty of Electrical Engineering, Mathematics and Computer Science

Uniform Recursive Tree

1 3 6 18 2 22 11 7 5

Root

12 4 9 10 15 20 17 24 14 19 21 8 13 16 23 25

1

) (

=

N

X 5

) 1 ( = N

X 9

) 2 (

=

N

X 7

) 3 (

=

N

X 4

) 4 (

=

N

X

26

[ ]

[ ]

N k if Pr

) ( ) (

≥ = = =

k N k N N

X N X E k y

[ ] [ ]

∑

− = −

=

1 ) 1 ( ) ( N k m k m k N

m X E X E

Recursion from Growth rule URT: Recursion for generating function: (

) ( )

) ( ) ( 1

1

z z N z N

N N

ϕ ϕ + = +

+

Solution:

[ ]

) 1 ( ) 1 ( ) ( ) ( + Γ + Γ + Γ = = z N z N z E z

N

y N

ϕ

• P. Van Mieghem 15

Faculty of Electrical Engineering, Mathematics and Computer Science

Hopcount of the shortest path in KN with exp. link weights

• Shortest path (SP) described via Markov discovery process.
• Shortest path tree (SPT) = Uniform recursive tree (URT)
• The generating function of the hopcount HN of shortest path

between different nodes is with from which (via Taylor expansion around z = 0 and Stirling’s approximation) follows that and

[ ]

⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = N z N N z E

N H N

1 ) ( 1 ϕ ) 1 ( ) 1 ( ) ( ) ( + Γ + Γ + Γ = z N z N z

N

ϕ

[ ] ∑

= −

− ≈ =

k m m k m N

m k N N c k H P )! ( log

[ ]

1 log − + ≈ γ N H E

N

[ ]

6 log var

2

π γ − + ≈ N H N

close to Poisson

• P. Van Mieghem 16

Faculty of Electrical Engineering, Mathematics and Computer Science

The Weight of SP

[ ] ∏

= −

− + − =

k n zv

n N n z n N n e E

k

1

) ( ) (

The k-th discovered node is attached at time vk = τ1+τ2+...+τk where τn is exponentially distributed with rate n(N-n) and the τ’s are independent (Markov property):

[ ] [ ] [

]

in URT node attached th

• k

is endnode Pr

1

∑

= − −

=

N k zv zW

k N

e E e E

The weight WN of the SP in the complete graph with exponential link weights is From this pgf, the mean weight (length) is derived as

[ ]

∑

− =

− = − =

1 1 '

1 1 1 ) (

N n W N

n N W E

N

ϕ

[ ]

∑∏

= = −

− + − − = =

N k k n zW W

n N n z n N n N e E z

N N

1 1

) ( ) ( 1 1 ) ( ϕ

• r

time 1 2 3 4 5 6 7 8

Markov Discovery Process

v1 v2 v3 v4 v5 v6 v7 v8

τ6

• P. Van Mieghem 17

Faculty of Electrical Engineering, Mathematics and Computer Science

Additional Results

• The asymptotic distribution of the weight of the SP is
• The flooding time (= minimum time to inform all

nodes in network) is

• The weight WU of the shortest path tree can be

computed explicitly.

• Other instances of trees (see Multicast).
• The degree distribution (= number of direct

neighbors) in the URT can be computed explicitly,

[ ]

x

e N N

e x N NW

−

− ∞ →

= ≤ − log Pr lim

∑

− = − = 1 1 1 N j j N

v τ

[ ]

⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + = =

− − 2 1

log 2 Pr N N O k D

k k

• P. Van Mieghem 18

Faculty of Electrical Engineering, Mathematics and Computer Science

Introduction The Art of Modeling:

Extension & Measurements

Conclusions Outline

• P. Van Mieghem 19

Faculty of Electrical Engineering, Mathematics and Computer Science

Extending the Basic Model

• From the complete graph KN to the Erdös-Rényi random

graph Gp(N): from p = 1 to p < 1.

• Result (which can be proved rigorously): for large N and fixed

p > pc ~ logN/N holds that the SPT in the class of Gp(N) with exponential link weights is a URT. Intuitively: Gp(N) with p > pc is sufficiently dense (there are enough links) and the link weights can be arbitrarily small such that the thinning effect of the link weights is precisely the same in all connected random graphs.

• Implications:

– basic model (SPT = URT) seems more widely applicable! – influence of the link weight structure is important

• P. Van Mieghem 20

Faculty of Electrical Engineering, Mathematics and Computer Science Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Border Router

Test Box D Test Box B Test Box C Test Box A

MySQL Database

Central Point

The results

ISP A ISP C ISP D ISP B

Probe-packets Border Router Border Router Border Router Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Border Router

Test Box D Test Box B Test Box C Test Box A

MySQL Database

Central Point

The results

ISP A ISP C ISP D ISP B

Probe-packets Border Router Border Router Border Router Internal networks Internal networks Internal networks Internal networks Internal networks Internal networks Border Router

Test Box D Test Box B Test Box C Test Box A

MySQL Database

Central Point

The results

ISP A ISP C ISP D ISP B

Probe-packets Border Router Border Router Border Router

RIPE measurement configuration

The traceroute data provided by RIPE NCC (the Network Coordination Centre of the Réseaux IP Européen) in the period 1998-2001.

• P. Van Mieghem 21

Faculty of Electrical Engineering, Mathematics and Computer Science

Model and Internet measurements

0.10 0.08 0.06 0.04 0.02 0.00 25 20 15 10 5 Measurement from RIPE fit(Log(N))=13.1 Pr[H = k] E[h]=13 Var[h]=21.8

α=E[h]/Var[h]=0.6

hop k

[ ] ∑

= −

− ≈ =

k m m k m N

m k N N c k H P )! ( log

2000

• P. Van Mieghem 22

Faculty of Electrical Engineering, Mathematics and Computer Science

Model and Internet measurements

0.10 0.08 0.06 0.04 0.02 0.00 Pr[H = k] 30 25 20 15 10 5 hop k

Asia Europe USA fit with log(NAsia) = 13.5 fit with log(NEurope) = 12.6 fit with log(NUSA) = 12.9

• ctober 2004

Hop k Pr[H = k]

• P. Van Mieghem 23

Faculty of Electrical Engineering, Mathematics and Computer Science

But…

Power law graph τ = 2.4 G 0.013(300)

• Internet: power law graph for AS ?
• No consistent model for hopcount, although seemingly success
• f rg with exp. or unif. link weight structure

• P. Van Mieghem 24

Faculty of Electrical Engineering, Mathematics and Computer Science

Degree graph

Scaling law for hopcount in Degree Graph defined by Pr[D > x] = c x1-τ with mean µ = E[D]. For τ > 3, α > 0, and ak = [Logνk]-Logνk Pr[HN – [LogνN] = k] = Pr[RaN = k] + O(logN)−α where the random variable Ra Pr[Ra > k] = E[ exp{- κ νa+kW1W2}|W1W2 > 0] with and ν = E[D(D-1)]/E[D] and κ = E[D]/(ν−1) and where W1 and W2 are independent normalized copies of a branching

• process. [Work in collaboration with R. van der Hofstad and G. Hooghiemstra]

Importance: (a) for each N and M where aN = aM (i.e. M=N/νk ) Pr[HN>k] follows by a shift over k hops from Pr[HM>k]. Hence, the hopcount for arbitrary large degree graphs (e.g. Internet?) can be simulated. (b) Currently most accurate hopcount formula

[ ] ( ) [ ]

ν ν ν µ γ ν log | log 2 log 1 log log 2 1 log log > − − − + − + ≈ W W E N H E

N

• P. Van Mieghem 25

Faculty of Electrical Engineering, Mathematics and Computer Science

Introduction The Art of Modeling:

The Link Weight Structure

Conclusions Outline

• P. Van Mieghem 26

Faculty of Electrical Engineering, Mathematics and Computer Science

Extreme Value Index

• SP is mainly determined by smallest link weights in the

distribution Fw(x) = Pr[w < x]

• Taylor:

Fw(x) = Fw(0) +F’w(0).x+O(x2) = fw(0).x+O(x2) SP is not changed by scaling of link weights, hence, fw(0), can be considered as scaling factor:

• 1. fw(0) > 0 is finite: regular distribution
• 2. fw(0) = 0: link weights cannot be arbitrarily small;

more influence of topology

• 3. fw(0) → ∞ : increasing prob. mass at x = 0
• Polynomial distribution:

Fw(x) = xα x in [0,1] = 1 x > 1 where α is the extreme value index

• 1. α = 1: regular distribution
• 2. α > 1: decreasing influence w
• 3. α < 1: increasing influence w

1 1 x

Fw(x)

α = 1 α > 1 α < 1

• P. Van Mieghem 27

Faculty of Electrical Engineering, Mathematics and Computer Science

Link Weight Distribution

Fw(x) x

α = 1 α > 1 α < 1

1 ε

larger scale

1 For the shortest path, only the small region around zero matters!

• P. Van Mieghem 28

Faculty of Electrical Engineering, Mathematics and Computer Science

Three regimes

฀ α → ∞ :

– all link weights are equal to w = 1 – no influence of link weights; only the topology matters

฀ α → 1 :

– link weights are regular (e.g. uniform, exponential) – SPT = URT if underlying graph is connected

฀ α → 0 :

– strong disorder: heavily fluctuating link weights in region close to x = 0 – union of all SPT is the minumum spanning tree (MST)

• P. Van Mieghem 29

Faculty of Electrical Engineering, Mathematics and Computer Science

Average Hopcount

30 25 20 15 10 5 Average Number of Hops

3 4 5 6 7 8 9

100

2 3 4 5 6 7 8 9

1000 Number of Nodes N Limit: α = 0 α = 0.05 α = 0.1 α = 0.2 α = 0.4 α = 0.6 α = 0.8 p = 2 ln N/N

( ) [ ] ( ) [ ]

( )

for 1 around for ln

3 / 1

→ = ≈ α α α α α N O H E N H E

N N

• P. Van Mieghem 30

Faculty of Electrical Engineering, Mathematics and Computer Science

MST and URT

7

89 93 78 98 27 57 90 77 76 3 45 82 26 32 51 15 36 91 42 33 56 81 73 84 53 67 61 52 40 63 54 80 17 28 100 22 65 10 9 85 6 72 75 19 12 18 37 24 44 97 2 8 34 46 47 13 66 69 58 95 68 23 60 49 99 1 11 71 92 38 83 20 29 86 55 41 79 59 5 96 31 25 70 64 43 39 88 16 21 94 50 74 48 14 35 87 4 30 62

7

64 54 16 91 4 34 47 79 19 97 33 85 95 39 99 80 41 45 42 8 9 31 48 57 70 56 40 81 29 43 86 100 46 62 11 71 52 72 63 28 53 89 49 50 69 25 30 20 90 14 82 77 12 37 38 13 60 66 67 36 35 44 2 84 58 3 83 22 55 15 23 6 21 74 1 98 26 17 65 75 27 5 94 59 51 92 10 73 98 18 24 76 88 68 96 78 38

(a) (b)

MST (α → 0 limit) URT (α → 1 limit) N = 100 nodes

• P. Van Mieghem 31

Faculty of Electrical Engineering, Mathematics and Computer Science

Phase Transition

1.0 0.8 0.6 0.4 0.2 0.0 Pr[GUspt(α) = MST] 0.01 0.1 1 10 α/αc N = 25 N = 50 N = 100 N = 200

URT-like MST-like ∆α αc = O(N-β) ∆α = O(N-β) β around 2/3

Van Mieghem, P. and S. M. Magdalena, "A Phase Transition in the Link Weight Structure of Networks", Physical Review E, Vol. 72, November, p. 056138, (2005).

• P. Van Mieghem 32

Faculty of Electrical Engineering, Mathematics and Computer Science

Phase Transition: implications

• Nature: superconductivity

– T < Tc : macroscopic quantum effect (R = 0) – T > Tc : normal conductivity (R > 0)

• Networks:

– Artifically created phase transition – if link weight structure can be changed independently of topology, control of transport: α < αc : almost all over critical backbone (MST) α > αc : spread over more paths; load balanced – critical backbone has minimum possible number

• f links N – 1: only these links need to be secured

• P. Van Mieghem 33

Faculty of Electrical Engineering, Mathematics and Computer Science

GUspt: Union of SPTs Observable Part of a Network

• Minimum spanning tree belongs to GUspt
• Degree distribution in the overlay GUspt on the

complete graph with exp. link weights is exactly known.

• Conjecture: For large N, the overlay GUspt on the ER

random graph Gp(N) with i.i.d. regular weights and any p > pc is a connected ER graph Gpc(N).

– we have good arguments, not a rigorous proof – important for Peer-to-peer networks!

• P. Van Mieghem 34

Faculty of Electrical Engineering, Mathematics and Computer Science

Introduction The Art of Modeling: Conclusions Outline

• P. Van Mieghem 35

Faculty of Electrical Engineering, Mathematics and Computer Science

Summarizing

• Basic SPT model: URT

– simple, analytic computations possible – reasonable first order model to approximate ad-hoc networks, less accurate for Internet (degree!)

• Link weight structure is important: much room for

research:

– what is link weight structure of a real network? – what are relevant weights (delay, loss, distance, etc...?) – how to update link weights ? – if link weights can be chosen independent of topology, a phase transition exists: steering of traffic in two modes

• P. Van Mieghem 36

Faculty of Electrical Engineering, Mathematics and Computer Science

Summary

• Articles:http://www.nas.e

wi.tudelft.nl/people/Piet

• Book: Performance

Analysis of Computer Systems and Networks, Cambridge University Press (2006)