On Power-Law Relationships of the Internet Topology
CSCI 780, Fall 2005
Outline How does network topology look
like? Random Graph?
Properties of Network Topology Degree distribution
Power law (this paper) Structure Structure
Hierarchical StructureHierarchical Structure
Network Topology On Router Level
Topology Graph = (V, E) Each node denotes a router Edge is the physical link between two
routers On AS level
Topology Graph = (V, E) Each node denotes an AS Edge is AS pair which have a BGP session
between them
Two Levels of Internet Topology Router-level: nodes are routers AS-level: nodes are domains
Why Topology Is Important?
Design Efficient Protocols Create Accurate Model for
Simulation Derive Estimates for Topological
Parameters Study Fault Tolerance and Anti-
Attack Properties
Key Findings We observe power-laws of the
Internet topology
Distributions are skewed, so average can be misleading
The log-log plots are linear
Power-Law, Zipf, Pareto Power-Law (probability distribution function)
P[X = x] ~ x-(k+1) = x-a
Pareto (cumulative distribution function)P[X > x] ~ x-k
Zipf ( size vs. rank )y ~ r-b
They are different ways of looking at the same thing
Internet Instances Three Snapshots at AS-level, one at Router-level
(95)
Power Law Properties (degree vs. rank)
Power Law 1: (rank exponent) The degree, dv , of a node v, is
proportional to the rank of the node, rv, to the power of a constant, R:
dv rvR
(Rank is the index of in order of decreasing out-degree)
Rank Plots Log-Log scale graph
X axis is rank, Y axis is out-degree
Power Law Properties (frequency vs. degree)
Power Law 2: (Out-degree exponent) The frequency, fd, of an out-degree, d,
is proportional to the out-degree to the power of a constant, O:
fd dO
Out-degree Plots Log-log scale graph
X axis is out-degree, Y axis is frequency
Out-degree Plots (cont’d)
Neighborhood Size of neighborhood within some
distance P(h): total number of pairs of nodes
within h hops
Hop-plot exponent
Ph hĦ
Average Neighborhood Size
Eigenvalue of Graph
Power Law Properties(eigenvalues)
Power Law 3: The eigenvalues, i, of a graph are
proportional to the order,i,to the power of a constant,
i i
Eigenvalues of a graph are the eigenvalues for the adjacency matrix of this graph
Eigenvalue plots Log-log scale graph
X axis is the order of eigenvalue Y axis is the eigenvalue
Discussion Describing the Internet topology
Power-low exponents are more descriptive than average
Protocol performance analysis Estimate useful graph metrics
(neighborhood) Predication
Answer what-if questions Realistic-graph generation
Connectivity does not Mean Reachability
Now we know properties of connectivity
But connectivity DOES NOT=reachability! Commercial agreement Routing policy
An annotated topology….
Route Propagation Policy Constrained by contractual commercial
agreements between administrative domains
Regional ISP A
Regional ISPB
University C
e.g., An AS does not provide transit services between its providers
AS Commercial Relationships Provider-customer:
customer pays its provider for transit services Peer-peer:
exchange traffic between customers no exchange of money
Sibling-sibling: have mutual transit agreement merging ISPs, Internet connection backup
However, AS relationships are not public!
AS Relationship Graph
AS1
AS3AS2
AS5AS4
AS7
AS6
provider-to-customer edge
peer-peer edge
sibling-sibling edge
Route Propagation Rule
An AS or a set of ASes with sibling relationship does not provide transit services between any two of its providers and peers
BGP routing table entries have certain patterns
Internet Architecture Hierarchical structure
Backbone Edge network
AS2
AS1
AS3
Hierarchical Topology Based on AS
relationship Tiers Provider/
Customer
Hierarchical Topology The number of ASes in different tiers on
2001/05, there are 11038 ASes Tier 1: 22 (0.20%) Tier 2: 5228 (47.37%) Tier 3: 4193 (37.99%) Tier 4: 1396 (12.64%) Tier 5: 174 (1.67%) Tier 6: 19 (0.17%) Tier 7: 6 (0.05%)
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