1. 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

2. On power law relationships of the internet topology Michalis Faloutsos U.C. Riverside Dept. of Comp. Science Michalis@cs.ucr.edu Michalis@cs.ucr.edu Petros Faloutsos U. of Toronto Dept. of Comp. Science pfal@cs.toronto.edu pfal@cs.toronto.edu Christos Faloutsos Carnegie Mellon Univ. Dept. of Comp. Science christos@cs.cmu.edu christos@cs.cmu.edu

3. 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.

4. 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

5. First power law: the rank exponent R Lemma1: Lemma2:

6. The rank exponent in AS and router level

7. 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

8. The outdegree exponent O

9. Approximation: the hop-plot exponent H Lemma 3: Definition deff: Lemma 4: O (d·h^H) Previous definition O(d^h)

10. The hop-plot exponent H

11. Average Neighborhood size

12. 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

13. The eigenvalues exponent ε

14. 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?

15. A First Principles Approach to Understanding the Internet’s Router-level Topology Lun Li California Institute of Technology lun@cds.caltech.edulun@cds.caltech.edu David Alderson California Institute of Technology alderd@cds.caltech.edu alderd@cds.caltech.edu Walter Willinger AT&T Labs Research walter@research.att.comwalter@research.att.com John Doyle California Institute of Technology doyle@cds.caltech.edu doyle@cds.caltech.edu

16. Previous work Random graphs Hierarchical structural models Degree-based topology generators. Preferential attachment General model of random graphs (GRG) Power Law Random Graph (PLRG)

17. A First Principles Approach Technology constraints Feasible region Economic considerations End user demands Heuristically optimal networks Abilene and CENIC

18. 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

19. Abilene-CENIC

20. Comparison of simulated topologies with power law degree distributions and different features

21. Performane-Likelihood Comparison

22. 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

23. 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

24. 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

25. Aspects of real-life networks Scale free property and high clustering coefficient Biology Computer networks Sociology

26. 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

27. Proposal: Gathering Mobility (2/2) The model Individuals Cycle behavior Attractors Appear-dissapear Probability an individual to choose an attractor Attractiveness of an attractor

28. Experiment Scale-free spatial distribution Scale-free Population growth

29. 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