Emergence of Scaling in Random Networks

Albert-Laszlo Barabsi & Reka Albert

Barabasi• Director of Northeastern University’s Center for Complex

Network Research

• Diseaseome- linking of diseases through shared genes

• Human Dynamics – understanding human behavior using statistical physics

http://en.wikipedia.org/wiki/Albert-L%C3%A1szl%C3%B3_Barab%C3%A1si

Albert

• Professor of Physics and Adjunct Professor of Biology at Pennsylvania State University

• Boolean modeling of biological systems

• Network Theory

http://en.wikipedia.org/wiki/R%C3%A9ka_Albert

Traditional Theories – Random Graphs

• Erdos – Renyi : edges are independent, all are equally likely

• Watts – Strogatz : Small world – small average shortest path length, large clustering coefficient

Observations

• Data sets show random graphs are not good approximations

• Hubs are followed by smaller hubs followed by smaller hubs

• Evidence of a scale invariance

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Proposed Model to Represent Scale Invariance (Scale Free)

• 2 Requirements

• Incorporation of growth

• Preferential Attachment

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Growth

• Assumptions of ER and WS – fixed N

• More realistic to assume an increasing N

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Preferential Attachment (Yule/Simon Process)

• ER and WS assume equal probability of connection

• Data suggests “rich get richer” – nodes have greater chance of connecting to nodes that have more neighbors

• Positive Feedback

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Proposed Model

• Π() = probability of connecting to vertex i (k neighbors) =

• Start with small number of nodes that are fully connected (

• For ( t = 0, t = , t += 1)

• Add a new vertex with m edges

• Connect to m other nodes with probability Π()

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Model

Conclusions of the Model

• Scale free – independent of

• Robust- removing 1 node does not change much

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Growth and Preferential Attachment:Are both needed?

• No growth

• Π() =

• N is constant, number of edges are increasing

• fully connected network

• No Preferential Attachment

• New nodes are connected with

• Π() = = constant probability

• - Similar to random networks

• Not scale free

Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

Both Are Necessary

Discussion Questions

• Can power laws be assumed by ?

• Clauset, Shazili, Newman

• http://en.wikipedia.org/wiki/Albert-L%C3%A1szl%C3%B3_Barab%C3%A1si

• http://en.wikipedia.org/wiki/R%C3%A9ka_Albert

• Emergence of Scaling in Random Networks. Barabasi and Albert. Science 286, 509 (1999)

• Power-Law Distributions in empirical data. Clauset, Shalizi, and Newman. Siam Review 51, 661-703 (209)