HY-483 Presentation On power law relationships of

the internet topology

A First Principles Approach to Understanding the Internet’s Router-level Topology

On natural mobility models

On power law relationships of the internet topology

Michalis Faloutsos U.C. Riverside Dept. of Comp. Science

Michalis@cs.ucr.edu Petros Faloutsos

U. of Toronto Dept. of Comp. Science pfal@cs.toronto.edu

Christos Faloutsos Carnegie Mellon Univ. Dept. of Comp.

Science christos@cs.cmu.edu

Previous work

Heavy tailed distributions used to describe LAN and WAN traffic

Power laws describe WWW traffic

There hasn't been any work on power laws with respect to topology.

Dataset & Methodology Three inter-domain level instances of

the internet (97-98), in which the topology grew by 45%.

Router-level instance of the internet in 1995

Min,Max and Means fail to describe skewed distributions

Linear Regression & correlation coeficients, to fit a plot to a line

First power law: the rank exponent R

Lemma1:

Lemma2:

The rank exponent in AS and router level

Second power law: the outdegree exponent

O

Test of the realism of a graph metric follows a power law exponent is close to realistic numbers

The outdegree exponent O

Approximation: the hop-plot exponent H

Lemma 3:

Definition deff:

Lemma 4: O (d·h^H)

Previous definition O(d^h)

The hop-plot exponent H

Average Neighborhood size

Third power law: the eigen exponent ε

The eigen value λ of a graph is related with the graph's adjacency matrix A (Ax = λx) diameter the number of edges the number of spanning trees the number of CCs the number of walks of a certain length

between vertices

The eigenvalues exponent ε

Contributions-Speculations

Exponents describe different families of graphs

Deff improved calculation complexity from previous O(d^h) to O(d·h^H)

What about 9-20% error in the computation of E?

A First Principles Approach to Understanding the Internet’s

Router-level Topology

Lun Li California Institute of Technology lun@cds.caltech.edu

David Alderson California Institute of Technology

alderd@cds.caltech.edu Walter Willinger

AT&T Labs Research walter@research.att.com

John Doyle California Institute of Technology

doyle@cds.caltech.edu

Previous work

Random graphs Hierarchical structural models Degree-based topology generators.

Preferential attachment General model of random graphs

(GRG) Power Law Random Graph (PLRG)

A First Principles Approach

Technology constraints Feasible region

Economic considerations End user demands

Heuristically optimal networks Abilene and CENIC

Evaluation of a topology Current metrics are inadequate and lack

a direct networking interpretation Node degree distribution Expansion Resilience Distortion Hierarchy

Proposals Performance related Likelihood-related metrics

Abilene-CENIC

Comparison of simulated topologies with power law degree distributions

and different features

Performane-Likelihood Comparison

Contributions-Speculations Different graphs generated by degree-based

models, with average likelihood, are Difficult to be distinguished with macroscopic

statistic metrics Yield low performance

Simple heuristically design topologies High performance Efficiency

Robustness not incorporated in the analysis Validation with real data

On natural mobility models

Vincent Borrel Marcelo Dias de Amorim Serge Fdida

LIP6/CNRS – Université Pierre et Marie Curie 8, rue du Capitaine Scott – 75015 – Paris – France {borrel,amorim,sf}@rp.lip6.fr

Previous work Individual mobility models

Random Walk Random Waypoint Random Direction model Boundless Simulation Gauss-Markov model, City Section model,

Group mobility models Reference Point model, Exponential Correlated Pursue model

Aspects of real-life networks Scale free property and high clustering

coefficient Biology Computer networks Sociology

Proposal: Gathering Mobility (1/2)Why?

Current group mobility models Rigid Unrealistic

Match reality using scale free distributions Human behavior Research on Ad-hoc inter-contacts

Proposal: Gathering Mobility (2/2)

The model

Individuals Cycle behavior

Attractors Appear-dissapear

Probability an individual to choose an attractor

Attractiveness of an attractor

ExperimentScale-free spatial distribution

Scale-free Population growth

Contributions-Speculations A succesive merge of individual and group

behavior: Individual movement No explicit grouping

Vs Strong collective behaviour Influence by other individuals Gathering around centers of interest of varying

popularity levels

Determination of maintenance of this distribution in case of population decrease and renewal