On Power-Law Relationships of the Internet Topology Michalis Faloutsos Petros Faloutsos Christos Faloutsos

Evolution of Network Models Erdos-Renyi random graph model (1959)  Problem: 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 pages 90%: ≤10 incoming links 3 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 characteristic node, no intrinsin scale  scale-free network y ∝ x α 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) d v : outdegree of a node v r v : the rank of a node v (index in the order of decreasing outdegree) R: constant (-0.81/-0.82/- 0.74/-0.48  rank exponent can distinguish graphs of different nature) d v ∝ r v R

Power-law 2 (outdegree exponent) f d : the frequency of outdegree d. the # of nodes w/ outdegree d O: constant(-2.15/-2.16/- 2.2/-2.48  fundamental property of the network) f d ∝ d O

Using this fact… A novel perspective of the structure of the internet Estimate important parameters Design and performance analysis of protocols Generate realistic topologies for simulation purposes

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 random process… 1. The first customer always choose the first table 2. 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 permutations

The chinese restaurant process F(k,N,T): the prob. that in at time T, when N tables are full, a random table is occupied by k guests F(k,N,T) ∝ (1/k) 1+α 1+α = γ ∈ (1,2]

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