PowerLaw Internet Topology and Topology Generators

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Power-Law Internet Topology and Topology
Generators

• Presented by Songlin Cai

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Why Topology is important?

• Design Efficient Protocols
• Create Accurate Model for Simulation
• Derive Estimates for Topological Parameters
• Study Fault Tolerance and Anti-Attack Properties

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Two Levels of Internet Topology

• Router Level and AS Level

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Random Graph
Erdös-Rényi model (1960)
Pál Erdös (1913-1996)
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About Erdos

• Hungarian
• One of the greatest mathematicians of our era
• Had no home, no wife or child, and no property or
  money to speak of.
• Some French socialist said that private property
  was theft. I say that private property is a
  nuisance.
• Coauthored more than 1,500 papers.

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Power-Law, Zipf, Pareto

• Power-Law
• PX x  x-(k1) x-a
• Pareto
• PX gt x  x-k
• Zipf
• y  r-b

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Internet Instances

• Three Snapshots of Internet

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Power-Laws 1
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Rank Plots
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Rank Plots
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Power-Law 2
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Outdegree Plots
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Outdegree Plots
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Two Power Laws to be One?

• At first, it appears that we have discovered two
  separate power laws, one produced by ranking the
  variables, the other by looking at the frequency
  distribution. Some papers even make the mistake
  of saying so Refer to Faloutsos et al.

http//ginger.hpl.hp.com/shl/papers/ranking/rankin
g.html
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Approximation 1 Hop Plot
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Hop plots
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Hop plots
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Power Law3 Eigenvalue
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Eigenvalue plots
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Eigenvalue plots
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What cause power law?

• The richer get rich,
• The poor get prison(hungry).

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Barabasi-Albert Model

• Network Evolution
• Add new nodes, Add new links, Rewire links
• Linear Preference
• Add a new nodes
• An existing node with to be selected to
  connect to is based on

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PLRG (W.Aiello, F. Chung, L. Lu)

• Suppose there are y vertices of degree x where x
  and y satisfy
• What can be calculate
• The maximum degree of the graph
• The number of vertices
• The number of edges

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Large Scale Properties

• What property is large scale?
• Power Law degree distribution
• Hierarchical Structure
• What else?

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A list of properties

• Neighborhood size (or expansion)
• Resilience, the size of a cut-set for a balanced
  bi-partition
• Distortion, or the minimum communication cost
  spanning tree
• Node diameter distribution
• Eigenvalue distribution
• Size of a vertex cover
• Biconnectivity
• The average pairwise shortest path between nodes
  in the largest component under random failure or
  under attack.

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A list of Topology Generators
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Three Distinguishing Properties

• Neighborhood size (or expansion)
• Resilience, the size of a cut-set for a balanced
  bi-partition
• Distortion, or the minimum communication cost
  spanning tree

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Ball-growing

• Measure the quantiy in a ball of radius h and
  then consider how that quantity grows as a
  function of h.

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Rate of spreading Expansion

• Calculate the size of the reachable set for each
  node in the graph, average the result, and then
  normalize by the total number of nodes in the
  graph
• The number of sites you can reach by travering h
  hops.
• For tree, the number of sites grows exponentially
  in h.

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Expansion
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Existence of alterate paths Resillience

• The minimum cut-set size for a balanced
  bi-partition of a graph.

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Resilience
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Tree-like behavior Distortion

• Consider any spanning tree T on a graph G, and
  compute the average distance on T between any two
  vertices that share an edge in G.
• How many extra hops are required to go from one
  side of an edge in G to the other, if we are
  restricted to using T.

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Distortion
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Discussion1
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Weighted Vertex Cover

• Do the degree-based generator produce networks
  with hierarchy and , if so , how?

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Link Value Rank Distribution(x-axis on log scale)
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Link value rank distribution(x-axis on linear
scale)
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Conclusion 2

• Degree-based generators capture the large-scale
  structure of the measured network surpringly
  well, according to their metrics.
• The hierarchy present in the measured networks is
  looser and less strict than in the structural
  generators, and this is well captured by the
  hierarchical structure in degree-based
  generators.
• The hierarchy in degree-based generators arises
  from the long-tailed distribution of degrees, and
  the backbone links are merely the links
  connecting two high-degree nodes.

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Reference

• On Power-Law Relationships of the Internet
  Topology
• Zipf, Power-laws, and Pareto a ranking tutorial
• http//ginger.hpl.hp.com/shl/papers/ranking/rankin
  g.html
• Network Topology Generators Degree-Based vs.
  Structural
• A Random Graph Model for Massive Graphs
• Emergence of Scaling in Random Networks