Summary from Previous Lecture

1
Summary from Previous Lecture

• Real networks
• AS-level
• N 12709, M27384 (Jan 02 data)
• route-views.oregon-ix.net, hhtp//abroude.ripe.net
  /ris/rawdata
• router-level
• Traceroute servers, www.traceroute.org
• www graph NO(billions) (1999 data)
• movie stars N300K 150K movies
• power networks N4941
• collaboration network N70K, E200K (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) ) ? gt0 ?lt
  1
• Exponential P(u,v) ? exp (-d/( L-d) )
• Locality P(u,v) ? if d lt 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.
Stub domain u,v are in the same domain
Gateway router
Backbone router
Transit domain u,v are in different domains
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 (p0 regular,
  p1 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
BRITEhttp//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.)

?
- ?-1
?
F(n) (? k n ) / 1-(k/P)
PLSxLSxD, kgt1

• 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
INEThttp//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

• Waxman88B. 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
  SIGCOMM99, August 1999.
• BA99A. 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.
• AB00R. Albert, A. Barabasi, Topology of
  evolving Network local events and Universality,
  Phys. Rew. Letters 855234-5237, 2000.