Emergence of Scaling in Random Networks

Barabasi & Albert

Science, 1999

Routing map of the internet

http://visualgadgets.blogspot.com/2008/06/graphs-and-networks.html

What is a network?

A graph is : an ordered pair G = (V,E) comprising a set V of vertices or nodes together with a set E of edges or lines, which are 2-element subsets of V

A set of elements together with interactions between them

Representation: a set of dots connected with (directed) lines

Where networks arise?

Computer networks Internet, LAN, Token-ring, 1553

Biology Gene regulation, food chain, metabolic networks

Data storage structures: WWW, data-base trees

Power transmition Electric power grid, hydraulic transmition

Social interaction Citation patterns, friendships, professional hierarchy

Computation Flow field computation, stress field computation

Internet routing map, 1999http://www.cheswick.com/ches/map/

Power grid, USA, 2001http://www.technologyreview.com/Energy/12474/page2/

Sexual / Romantic partners networkBearman, Moody, Stovel. Chains of Affection: The Structure of Adolescent

Romantic and Sexual Networks. AJS, 2004

Jefferson High, Columbus, Ohio

Metabolic network of E. Coli

Organization chart

Large-scale, “natural” networks

How “random” are “natural” networks (WWW, internet, gene regulation, …)“natural” ~ no apriori structure defined

What are the key characteristics of natural networks?

What is “Random Network”? Random network – ensemble of many

possible networks:Fixed or unfixed number of vertices (dots)Fixed or unfixed number of edges (lines)Any two vertices have some probability of being

connected

Key notion: node connectivityconnectivity = number of connections

First model – Erdos & Renyi, 1947

ER random network model

Network model: a random network between n nodes:Fix the number of vertices to nFor each possible connection between vertices v

and u, connect with probability p

P(rank=k) =

ER random network model

FeaturesEvery node has appr.

same number of connections

connectivity is scale-dependent!

Tree-like!

Internet-like network evolution

http://www.cheswick.com/ches/map/index.htmlhttp://www.cheswick.com/ches/map/movie.mpeg

ER model and real life Real-life networks are scale-free:

Connectivity follows power-law: P(k) ~ kγ

γ = 2.1…4○ very low connection numbers are possible

Actor collaboration

N=212e3, <k>=29, γ=2.3

WWW

N=325e3, <k>=5.5, γ=2.1

Power grid

N=5e3, <k>=2.7, γ=4

ER model VS. Scale-free network ER: same average number of connections per node – tree-

like SF: hubs present – few nodes with large number of

connections – hierarchy!

ER model VS. Scale-free network Adjacency matrix A:

Number the nodes from 1 to NIf vp connected to vq , put 1 in apq

1 2 3 4 5 6

1 2 3 4 5 6

ER model VS. Scale-free network Adjacency matrix of ER: ~ uniform

distribution of 1’s Adjacency matrix of SF: 1’s lumped in

columns & rows for few nodes

ER

SF

Barabasi model

Goal: generation of random network with “scale-free” property

1. Number of edges – not fixedContinuous growth

2. Preferential attachmentProb. of a new node to attach to existing one

rises with rank of node

P(attach to node V) ~ rank(V)

Barabasi Model Produces scale-free networks

Scale-free distribution – time-invariant. Stays the same as more nodes added

Barabasi Model

Removal of either assumptions destroys scale-free property:

Without node addition with time → fully connected network after enough time

Without preferential attachment → exponential connectivity

ER Vs. Barabasi

Graph diameter:the average length of shortest distance

between any two vertices

For same number of connections and nodes, ER has larger diameter than scale-free networks

No small-world in ER!

Scale-free Network featuresN

etw

ork

diam

eter

% of “damaged” nodes

Robustness to random failure Susceptibility to deliberate attack

Failure = removal of random node

Attack = removal of highly-connected node

Scale-free Network features

“Small-world” phenomenon, or:

“6 degrees of separation”

Stanley Milgram, 1967, Psychology today

Small-world experiment

Experiment: send a package from Nebraska and Kansas (central US) to Boston, to a person the sender doesn’t knowMotivation: great distance – social and

geographical

Only 64 of 296 packages were delivered

For delivered packages: average path length ~ 6

Google search

Brin & Page, 1998; Kleinberg, 1999

Pages are ranked according to incoming linksIncoming link from a high-score page is more

valuable

Meaning: after random clicks, a user will be on high-ranked page

Prefers old, well-connected pages

Google search

Erdos & Bacon Number Erdos number: “collaborative distance”

of a mathematician from Paul ErdosAverage: ~6Kahenman, Auman: 3

Bacon Number: “collaborative distance” of an actor from Kevin Baconhttp://oracleofbacon.org/Average: ~3

Summary

Many real-life, large-scale networks exhibit a scale-free distribution of connectivity

Distribution is power-lawSimilar powers for networks of different typesSmall-world phenomenon

Key features to enable free-scale property:Addition of new nodesPreferential attachment