On Power-Law Relation-ships of the Internet

TopologyMichalis FaloutsosPetros Faloutsos

Christos Faloutsos

Evolution of Network Models

• Erdos-Renyi random graph model (1959) Prob-lem: existence of clustering (Granoveter, 1972)

• Watts-Strogatz model (1998) Problem: existence of hub(=connector)(Barabasi, 1998)

• Power-law

Existence of Hub

• Skewed topology of web• Visibility of a web page - # of incoming links• Nd Web case – 325,000 pages

270,000 pages (82%): ≤3 incoming links 42 pages: ≥1,000 incoming links

• Extended observation – 203,000,000 pages90%: ≤10 incoming links3 pages: ≥1,000,000 incoming links

• e.g. Amazon, Yahoo, Google …• The large-scale organization of metabolic networks• Protein P53 network• The phone call graph

Power law distribution• Bell curve(random) / power law(unevenness)• Tail: bell – exponentially decay

power law – Not exponentially decay Existence of hub

Power law distribution

• Random network – average links, peak scale of the network

• Network w/ powel law distribution – no character-istic node, no intrinsin scale scale-free network

• y ∝ xα • Observation of log-log plot

On power-law relationships of the internet topology

• Int-11-97, Int-04-98, Int-12-98(45% growth)• Rout-95• Observation of Log-log plot: linear regression(least-

square method) correlation coeff. of ≥ 96%

Power-law 1

• (rank exponent)• dv: outdegree of a node v

• rv : the rank of a node v (index in the order of de-creasing outdegree)

• R: constant (-0.81/-0.82/-0.74/-0.48 rank expo-nent can distinguish graphs of different nature)

• dv ∝ rv R

Power-law 2

• (outdegree exponent)• fd: the frequency of

outdegree d. the # of nodes w/ outdegree d

• O: constant(-2.15/-2.16/-2.2/-2.48 fun-damental property of the network)

• fd ∝ dO

Using this fact…

• A novel perspective of the structure of the inter-net

• Estimate important parameters• Design and performance analysis of protocols• Generate realistic topologies for simulation pur-

poses

The chinese restaurant process

• A restaurant w/ countably many tables, labelled 1,2, …

• Customers walk in and sit down at some table• Tables are chosen according to the following ran-

dom process…

1. The first customer always choose the first table2. The nth customer chooses the first unoccupied table w/ prob. α/(n-1+ α), and an occupied table w/ prob. c/(n-1+α)C: # of people sitting at that table

a probability distribution

The chinese restaurant process

• The prob. Of a seating is invariant under permu-tations

The chinese restaurant process

• F(k,N,T): the prob. that in at time T, when N ta-bles are full, a random table is occupied by k guests

• F(k,N,T) ∝ (1/k)1+α

• 1+α = γ∈(1,2]

END