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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
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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
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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
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Internet routing map, 1999http://www.cheswick.com/ches/map/
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Power grid, USA, 2001http://www.technologyreview.com/Energy/12474/page2/
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Sexual / Romantic partners networkBearman, Moody, Stovel. Chains of Affection: The Structure of Adolescent
Romantic and Sexual Networks. AJS, 2004
Jefferson High, Columbus, Ohio
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Metabolic network of E. Coli
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Organization chart
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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?
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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
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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) =
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ER random network model
FeaturesEvery node has appr.
same number of connections
connectivity is scale-dependent!
Tree-like!
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Internet-like network evolution
http://www.cheswick.com/ches/map/index.htmlhttp://www.cheswick.com/ches/map/movie.mpeg
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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
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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!
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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
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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
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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)
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Barabasi Model Produces scale-free networks
Scale-free distribution – time-invariant. Stays the same as more nodes added
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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
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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!
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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
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Scale-free Network features
“Small-world” phenomenon, or:
“6 degrees of separation”
Stanley Milgram, 1967, Psychology today
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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
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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
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Google search
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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
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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