2.
Society Nodes : individuals Links : social relationship (family/work/friendship/etc.) S. Milgram (1967) Social networks: Many individuals with diverse social interactions between them. John Guare Six Degrees of Separation
3.
Communication networks The Earth is developing an electronic nervous system, a network with diverse nodes and links are -computers -routers -satellites -phone lines -TV cables -EM waves Communication networks: Many non-identical components with diverse connections between them.
4.
New York Times Complex systems Made of many non-identical elements connected by diverse interactions . NETWORK
“ the” is the most common, followed by “of”, “to”, etc.
claim: frequency of the n-th most common ~ 1/n (power law, α = 1)
General theme:
rank events by their frequency of occurrence
resulting distribution often is a power law!
Other examples:
North America city sizes
personal income
file sizes
genus sizes (number of species)
People seem to dither over exact form of these distributions (e.g. value of α ), but not heavy tails
10.
Zipf’s Law Linear scales on both axes Logarithmic scales on both axes The same data plotted on linear and logarithmic scales. Both plots show a Zipf distribution with 300 datapoints
Let nbr(u) denote the set of neighbors of u in a graph
all vertices v such that the edge (u,v) is in the graph
The clustering coefficient of u:
let k = |nbr(u)| (i.e., number of neighbors of u)
choose(k,2): max possible # of edges between vertices in nbr(u)
c(u) = ( actual # of edges between vertices in nbr(u))/choose(k,2)
0 <= c(u) <= 1; measure of cliquishness of u’s neighborhood
Clustering coefficient of a graph:
average of c(u) over all vertices u
k = 4 choose(k,2) = 6 c(u) = 4/6 = 0.666…
14.
Clustering : My friends will likely know each other! Probability to be connected C » p C = # of links between 1,2,…n neighbors n(n-1)/2 Networks are clustered [large C(p)] but have a small characteristic path length [small L(p)]. The Clustering Coefficient of a Network
24.
Meta-P(k) Organisms from all three domains of life are scale-free networks! H. Jeong, B. Tombor, R. Albert, Z.N. Oltvai, and A.L. Barabasi, Nature, 407 651 (2000) Archaea Bacteria Eukaryotes Metabolic Network
26.
Protein Network protein-protein interactions PROTEOME
27.
Prot Interaction map Nodes : proteins Links : physical interactions (binding) P. Uetz, et al. Nature 403 , 623-7 (2000). Yeast Protein Network
28.
Prot P(k) H. Jeong, S.P. Mason, A.-L. Barabasi, Z.N. Oltvai, Nature 411, 41-42 (2001) Topology of the Protein Network
29.
P53 … “ One way to understand the p53 network is to compare it to the Internet. The cell, like the Internet, appears to be a ‘ scale-free network ’.” p53 Network Nature 408 307 (2000)
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