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Random networks

Emergence of Scaling in Random Networks Barabasi & Albert Science, 1999. Random networks. Routing map of the internet http://visualgadgets.blogspot.com/2008/06/graphs-and-networks.html. What is a network?.

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Random networks

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  1. Emergence of Scaling in Random Networks Barabasi & Albert Science, 1999 Random networks Routing map of the internet http://visualgadgets.blogspot.com/2008/06/graphs-and-networks.html

  2. 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

  3. 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

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

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

  6. Sexual / Romantic partners network Bearman, Moody, Stovel. Chains of Affection: The Structure of Adolescent Romantic and Sexual Networks. AJS, 2004 Jefferson High, Columbus, Ohio

  7. Metabolic network of E. Coli

  8. Organization chart

  9. 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?

  10. 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 connectivity • connectivity = number of connections • First model – Erdos & Renyi, 1947

  11. ER random network model • Network model: a random network between n nodes: • Fix the number of vertices to n • For each possible connection between vertices v and u, connect with probability p • P(rank=k) =

  12. ER random network model • Features • Every node has appr. same number of connections • connectivity is scale-dependent! l=l(N) • Tree-like!

  13. Internet-like network evolution http://www.cheswick.com/ches/map/index.htmlhttp://www.cheswick.com/ches/map/movie.mpeg

  14. 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 WWW N=325e3, <k>=5.5, γ=2.1 Power grid N=5e3, <k>=2.7, γ=4 Actor collaboration N=212e3, <k>=29, γ=2.3

  15. 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!

  16. ER model VS. Scale-free network • Adjacency matrix A: • Number the nodes from 1 to N • If vp connected to vq , put 1 in apq 1 2 3 4 5 6 1 2 3 4 5 6

  17. 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

  18. Barabasi model • Goal: generation of random network with “scale-free” property • Number of edges – not fixed • Continuous growth • Preferential attachment • Prob. of a new node to attach to existing one rises with rank of node P(attach to node V) ~ rank(V)

  19. Barabasi Model • Produces scale-free networks • Scale-free distribution – time-invariant. Stays the same as more nodes added

  20. 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

  21. 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!

  22. Scale-free Network features Failure = removal of random node Attack = removal of highly-connected node Network diameter • Robustness to random failure • Susceptibility to deliberate attack % of “damaged” nodes

  23. Scale-free Network features • “Small-world” phenomenon, or: “6 degrees of separation” • Stanley Milgram, 1967, Psychology today

  24. Small-world experiment • Experiment: send a package from Nebraska and Kansas (central US) to Boston, to a person the sender doesn’t know • Motivation: great distance – social and geographical • Only 64 of 296 packages were delivered • For delivered packages: average path length ~ 6

  25. Google search Brin & Page, 1998; Kleinberg, 1999 • Pages are ranked according to incoming links • Incoming 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

  26. Google search

  27. Erdos & Bacon Number • Erdos number: “collaborative distance” of a mathematician from Paul Erdos • Average: ~6 • Kahenman, Auman: 3 • Bacon Number: “collaborative distance” of an actor from Kevin Bacon • http://oracleofbacon.org/ • Average: ~3

  28. Summary • Many real-life, large-scale networks exhibit a scale-free distribution of connectivity • Distribution is power-law • Similar powers for networks of different types • Small-world phenomenon • Key features to enable free-scale property: • Addition of new nodes • Preferential attachment

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