1. Summary from Previous Lecture Real networks: –AS-level N= 12709, M=27384 (Jan 02 data) route-views.oregon-ix.net, hhtp://abroude.ripe.net/ris/rawdata – router-level –Traceroute servers, www.traceroute.org – www graph N=O(billions) (1999 data) –movie stars N=300K 150K movies –power networks N=4941 –collaboration network N=70K, E=200K (1991-98 data) Similar characteristics: –Small world property (six degrees of separation) D= 0.35 + 2 log N (log is 10 base) www should have diameter 19 –Clustering property (circle of friends) –Preferential attachment –Power laws: rank, outdegree, hop-plot, eigen value. How to generate graphs that has internet-like properties?

2. Global Metrics (Graph) Min, max and Average node degree Diameter Hop-plot value, hop-plot exponent Effective diameter Frequency of node degrees Characteristic path length Clustering coefficient Size of the giant component Eigenvalue exponent Expansion Resilience Distortion

3. Local Metrics (Node) Degree Rank Clustering coefficient Eccentricity Significance Betweenness Closeness

4. Internet Topology Generators Motivation: performance of network protocols/algorithms may vary depending on the methods. Flat Random Methods: –Place the nodes on a plane randomly –[ER]: p, n –[Waxman88]: P(u,v)=  exp (-d/(  L) )  >0  <= 1 –Exponential: P(u,v)=  exp (-d/( L-d) ) –Locality: P(u,v) =  if d < r;  o.w. [Gatech97] Hierarchical Methods –N-level: Place nodes on Euclidean Plane randomly Divide the plane into equal size square sectors (scale parameter S1 for level 1) Assign each node to a square out of S1xS1 squares Subdivide each square with a node using S2 Edge lengths are determined by the level

5. Hierarchical Methods Cont. Transit domain: u,v are in different domains Stub domain: u,v are in the same domain Backbone router Gateway router Parameters: T, Nt, K, Ns

6. Incremental Models Watts and Strogatz [WS98] –Start with a ring lattice of n nodes and k edges per node. –Rewiring process with probability p (p=0 regular, p=1 random) Barabasi and Albert [BA99] –Start with a small network core –At each step choose randomly between Adding a new node with m link Adding m links without a new node Linear preferential attachment –F(d) power law exponents are much less than the measured ones -(2.18 vs –3) Albert and Barabasi [AB00] –Consider rewiring of m links with linear preferential attachment –Not always a connected graph.

7. BRITE http://www.cs.bu.edu/fac/matta/software.html Node assignment: –High level square (HS), Low level squares (LS); HSxHS, LSxLS –Choose a LS and drop a node in there –For each HS pick a number “n” of nodes randomly from the following distribution (I.e., bounded Pareto dist.) F(n)= (  k n ) / 1-(k/P) P=LSxLSxD, k>=1  -  -1  Link assignment : parameter ‘m” #of links per new node New node v is connected to i with P(v,i)= wi di/  wj dj j  Cv where Cv is the set of candidate nodes for v di is the degree of node i

8. INET http://topology.eece.umich.edu/inet Number of nodes N and the fraction k of N that has outdegree of 1 Assumes exponential growth rate and computes number of months (t) it would take the internet to grow from its size in Nov 1997 to N. Compute outdegree-frequency and rank-outdegree distributions Construct the network in three steps –Form a spanning tree of nodes with degree at least 2 –Attach nides with degree 1 to the tree –Connect all the remaining nodes to satisfy their degree properties With probability k/K K= sum of outdegrees of all nodes already in G

9. References [Waxman88]:B. M. Waxman, “Routing of multipoint connections,” IEEE JSAC, 6(9):1617-1622. [Gatech97]: E. Zagura et al, “A quantitative Comparision of Graph-Based Model for Internet Topology”, ACM/IEEE ToN 5(6):770-783, Dec. 1997. [FFF99] M. Faloutsos et al, “On power-Law relationships of the Internet Topology”. ACM SIGCOMM’99, August 1999. [BA99]:A. Barabasi, R. Albert, “Emerging of scaling in random networs,” Science, 509-512, October 1999. [WS98]: D.J.Watts and S.H. Strogatz, “Collective Dynamics of small-world” networks,” Nature 393,440-442, 1998. [AB00]:R. Albert, A. Barabasi, “Topology of evolving Network: local events and Universality”, Phys. Rew. Letters 85:5234-5237, 2000.