Emergence of Scaling in Random Networks Barabasi Albert

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? 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, 1999 http: //www. cheswick. com/ches/map/

Power grid, USA, 2001 http: //www. technologyreview. com/Energy/12474/page 2/

Sexual / Romantic partners network Bearman, 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 connectivity = 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 n For each possible connection between vertices v and u, connect with probability p P(rank=k) =

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

Internet-like network evolution http: //www. cheswick. com/ches/map/index. html http: //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 WWW Power grid N=212 e 3, <k>=29, γ=2. 3 N=325 e 3, <k>=5. 5, γ=2. 1 N=5 e 3, <k>=2. 7, γ=4

ER model VS. Scale-free network ER: same average number of connections per node – treelike 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 N If vp connected to vq , put 1 in apq 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 SF ER

Barabasi model Goal: generation of random network with “scale-free” property 1. Number of edges – not fixed Continuous growth 2. Preferential attachment Prob. 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 features Network diameter Failure = removal of random node Attack = removal of highlyconnected node % of “damaged” nodes Robustness to random failure Susceptibility to deliberate attack

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 know Motivation: 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 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

Google search

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

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