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%)