Multiscale Network Processes:

Fractal and p-Adic analysis

Vladimir Zaborovsky, Technical University, Robotics Institute,

Saint-Petersburg, Russiae-mail vlad@neva.ru

February 2003Tahiti

10th International Conference on TelecommunicationsICT’2003

Content

• Introduction

• Basic questions and

experimental background

• Fractional analysis

• Wavelet decomposition

• p-adic and constructive

analysis   

• Conclusion

Keywords:packet traffic, long-range dependence, self-similarity, wavelet, p-adic analysis.

computer network and

network processes

Appl 1

Appl 2

Appl n

Appl i

characteristics:• number of nodes and links• performance (bps and pps )• applications, • control protocols, etc.

feature:• fractal or 1/f spectrum• heavy-tailed correlation structure• self similarity• etc.

Introduction

Packet traffic - discrete positive process with a singular internal structure.

trend

multiplicativecascades

Spatial-Temporal features:

spectralcomponents

1. Common questions:

a) metrics and dimension of state

space;

b) statistical or dynamical

approaches;

c) predictable or chaotic behaviors of

congested periods.

2. Relationship between:

d) line bit speed and virtual line

throughput

e) microscopic packet dynamics and

heavy-tailed statistical distributions

f) fractal properties and QoS issues

Basic aspects

Experimental data flows in spectral and statistical domain

“tailbehavior”

“tailbehavior”

realdata

classicalnormaldistribution

=1

Spectral domain – 1/f process

Second-order statistics domainlog{varRTT(m)}

logm

frequency

<1

Correlation Structure in power law scale time intervals

ICMP packets. Autocorrelation function of number of packets

aggregation period T=pmL0 ;

T = 64 ms = 25 ms

T = 8 ms = 23 ms

T = 2 ms = 21 ms

- which modelis “right”?

T= 4ms = 22msT= 2ms = 21msT= 1ms = 20ms

- what feature is important

p – 2,3,5,…

m = 0,1,2,3

L0 = time scale

virtualgrid

channelstructure

physicalnetwork

(IP address, port)

node nnode 1

Virtual channel:

node nnode 1

Channel signal:

01001101

(MAC frame)

Physical signal:(signal and noise value levels)1

0

Network environment and logical structure

protocol application

macroscopic processes

microscopic processes

R(k)~Ak–b and

• Fractal process and power low correlation decays:

, )t;x(F )x(nd )(f )t; 1x( n)t;x( nt

00

1.1 bkA~)mk(R 1.2

Basic equation (continuous time approximation):

1.3

Models and features

peer-to-peervirtual connection

node n(1,t) node n(2,t) node n(x,t) … node n(m,t)

number of noden(x,t) – number of packets, at node x, at time t

signal propagation

t1 t2 titn

new comer packetsnumber of packets that already exist in the node x

P(n(x;t)<n0) F(x,t)

n(x;t) – number of packets n(x; t) at node number x at the time moment t

where

Packet delay/drop processes in virtual channel.

a)End-to-End model

(discrete time scale)

b)Node-to-Node model

(real time scale)

c)Jump model

(fractal time scale)

Common and Fine Structure of the packet traffic.

Spatial-Temporal Microscopic Process

nodes:

• Common packets loss condition: each packet can be lost, so

. dt)t(ftt}t{M0

1.4

sourceintermediate

node x destination

node 1 node n“t”

Basic model of the packet “dissipation”

);t(Fdt

d)t(f F(t) – distribution function

virtual channel

Functional equation for scale invariant or “stable” distribution function

2.0 ; 1cc

)c/t(F)c/t(F)t(F

21

21

this packet never come to the destination node

10 ,)t1(

)t(f1

Take into account

expression for can be written as

.1dt)t(f ;0)t(f0

1.5

Resume:

1. For the t>>1 density function f(t) has a scale-

invariant property and power low decay like (1.1)

2. Virtual connection can be characterized by

dynamics equation (1.3) and statistical (1.4)

condition.

Simple F(t) approximation

)t(Fdt

d)t(f

Features:• Space measure [1/sec 1/sec sec] = [1/sec]• Fractal time scale

microscopic dynamics

[Sec] fractal time scale or network signal time propagation measure

1/[ms] nominal channel bit rate measure (real number)

1/[ms] effective bandwidthmeasure

X

virt

ual c

hann

el 1

virt

ual c

hann

el 3

virt

ual c

hann

el 4

possiblepacket loss

virt

ual c

hann

el 2

Y

X0 Z

State Space of the Network Process

one-to-one reflection

macroscopic dynamics

RTT signal

raw signal: Curve of Embedding Dimension:

n >> 1 (white nose)

network signal

wavelet approximation

wavelet image: Curve of Embedding Dimension:

n=58(fractal structure)

Micro Dynamics of packets (network signal)

Resume:• Dynamics of network process has limited value (n=58) of embedded dimension parameters (or signal has internal structure).

• Temporal fractality associated to p-adic time scale, where T=pmL0, L0 – time scale.

Generalized Fractal Dimension Dq Multifractal Spectrum f()

Network signal (RTT signal) and its:

Fractal measure

The fractional equation of packet flow: (spatial-temporal virtual channel)

where – fractional derivative of function n(x;t),

– Gamma function,

n(x; t) – number of packets in node number x at time t;

– parameter of density function (1.5)

tD

)1(

t

)x(n

x

)t;x(n)]t;x(n[D)1( 0

t4.1

Fractal Model of Network Signal (packet flow)

Why fractional derivative?

Operator - take into account a possible loss

of the packets;

tD

The dependence of packets number n(k,100)/n0 for

different values of parameter at the time moment t=100

Equation (4.1) has solution

.t

1

)(

)1(

t

1

)21(

)1(k

t

1n)t;k(n

12

2

0

4.2

number of node

Initial conditions

n(0;t)=n0(t):

.t

1)32(

)1(m

)22(1

t)1(n)t;m(c 2120

The time evolution of c(m,t)/n02

4.3

Spatial-temporal co-variation function

2-Adic Wavelet Decomposition

0j

2

0njnjnn0

1j

)t(WV)t(x

а) network traffic

b) Wavelet coefficients and their maxima/minima lines

P-adic analyze: Basic ideas

p-adic numbers

(p is prime: 2,3,5,…)can be regarded as a completion of the rational numbers using norm

|x|p = 0 if x = 0

|xy|p = |x|p |y|p

|xy|p max {|xp| , |yp|} |x|p + |y|p

The distance function d(x,y)=|xy|p possesses a general

property called ultrametricity

d(x,z) max {d(x,y),d(y,z)}

p-Adic decomposition:x and y belong to same class if the distance between x and y satisfies the condition

d(x,y) < D

Classes form a hierarchical tree.

p-Adic Fractality

Basic feature:

• p-adic norm for a sum of p-adic numbers cannot be larger than the maximum of the p-adic norm for the items

• the canonical identification

mapping p-adics to real

• i:th structural detail appears in finite region of the fractal structure is:infinite as a real numberand has finite norm as a p-adic number

This norm – p-adic invariant of the fractal.

Nm

mm

Nmp

mm RpxRpxx

The wavelet basis in L2(R+)

is 2-adic multiscale basis

P-adic field structure

)x()x()x(1,

21

21

,0

pn

np ZpZ

)...)pZa(pa(Z pkik,j,i

p

cluster ,

where

{0} …p2Zp pZp Zp p-1Zp …Qp ,

Zn Z, ),nx2(2 2n

)R(LVU

...VVVV...

2jZj

2101

p-Adic Self-Similar Feature of Power Low Function

Power low functions as f(x)=xn are self-similar in p-adic sence:

the value of the function at interval (pk,pk+1) determines the function completely

function y=x2

p = 2 p = 3

p = 11p = 7

Inputprocess

Outputprocess

PPS

virtual channel

RTT

Experimental data:RTT spatial-temporal integral characteristic

Location:

packets per second

t, sec

Constructive analysis: hidden periods and spectrum

PPS differential characteristic

Basic Idea:

• Natural Basis of Signal is defined by Signal itself

• Constructive Spectrum of the Signal consist of blocks with different numbers of minimax values

MiniMax Process Decomposition

PPS

time scale

blocks sequence

analyzing process: packet-per-second curve

time

Constructive Components of the Analyzing Process

Source RTT process

and its constructive components:

sec

number of “max” in each block

Network Process: Constructive Spectrum

Dynamic Reflection diagram RTT(t)/RTT(t+1)

Transitive curve: block length=4 to block length=8

RTT(t)

RTT(t+1)

2-Adic Analysis of hidden period:

Source signal:

Filtered signal: block length=5

number of time interval

number of time interval

detailed structure

Quasi Turbulence Network Structure

Multiscale Forecasting Algorithm: application aspect

• The features of processes in computer networks correspond to the multiscale chaotic dynamic systems process .

• Fractional equations and wavelet decomposition can be used to describe network processes on physical and logical levels.

• Concept of p-adic ultrametricity in computer network emerges as a possible renormalized distance measure between nodes of virtual channel .

• Constructive analysis p-adic of network process allows correctly describe the multiscale traffic dynamic with limited numbers of parameters.

Conclusion