How Scale-Free are Biological Networks. Raya Khanin and Ernst Wit Department of Statistics University of Glasgow Biological Networks Protein interaction network : proteins that are (or might be) connected by physical interactions
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How Scale-Free are Biological Networks.
Raya Khanin and Ernst Wit
Department of Statistics
University of Glasgow
Protein interaction network: proteins that are (or might be) connected by physical interactions
Metabolic network: metabolic products and substrates that participate in one reaction
Gene regulatory network: two genes are connected if the expression of one gene modulates expression of another one by either activation or inhibition
Degree (or connectivity)
of a node, k, is the number
of links (edges) this node has.
The degree distribution,
P(k), is the probability that
a selected node has exactly
k links. Networks are classified by
their degree distributions.
(Barabasi and Oltvai, Nature, 2004)
Network of citations between scientific papers
Network of collaborations, movie actors
Electrical power grids, airline traffic routes, railway networks
World Wide Web and other communication systems
"These laws, applying equally well to the cell and the ecosystem, demonstrate how unavoidable nature's laws are and how deeply self-organization shapes the world around us.“(A.Barabasi, 2002).
“Scale-free networks are everywhere. They can be seen in … terrorist networks. What this means for counter-terrorism experts?”
Global guerrillas (network organization, infrastructure disruption, and the emerging marketplace of violence)
Scale|free : http://www.scalefree.info (Social Networking, Communities of Practice and Knowledge Management: publish articles such as “love and knowledge”, “what lovers tell us about persuasion”, “how ‘social contagion’ affects consumer behaviour)
Properties of scale-free systems are invariant to changes in scale. The ratio of two connectivities in a scale-free network is invariant under rescaling:
This implies that scale-free networks are self-similar, i.e. any part of the network is statistically similar to the whole network and parameters are assumed to be independent of the system size.
Scale-free (self-similarity) properties of a common cauliflower plant: it is virtually impossible to determine whether one is looking at a photograph of a complete vegetable or its part, unless an additional scale-dependent object (a match) is added. (A) Complete vegetable; (B) a small segment of the same vegetable; (C) small part of the segment shown in B. The same match was used in all three photographs to provide a sense of scale for an otherwise scale-free structure (Gomez et al, 2001).
Note: vegetable was purchased in Sloan supermarket in Manhattan’s Upper West Side.
Protein interaction networks
Protein domain networks
Gene co-expression networks
Distribution of gene expression and spot intensities on microarrays
Frequency of occurrence of generalized parts in genomes of different organisms
the data is usually graphically fitted
by a straight line on a log-log scale
(approximately) under H0: network is scale-free
t is calculated (observed) values of T with estimated
p-value is the probability a network has such connectivities if they were drawn from the power-law distribution.
Uetz 25 2.050.00004
Schwikowski 26 1.8650
Ito 27 2.020
Li 28 2.30
Rain 29 2.10
Giot 21 1.530
Tong 9 1.440
Lee 20 1.990
Guelzim 17 1.490
Spellman/Cho 1.27(1.06) 0
is the cut-off, s. t. the number of connections is less than expected for pure scale-free networks for
and the behaviour is approximately scale-free within the range
Uetz 25 1.6 8.67 0.37
Schwikowski 26 1.26 6.2 0.105
Ito 27 1.79 26 0
Li 28 2.1 19.5 0.0178
Rain 1.12 11.5 0.2
Giot 21 1.09 20 0.0013
Tong9 0.96 23.7 0
Lee 20 1.96 294 0
Guelzim17 1.18 15 0.00001
Spellman/Cho 1.07(0.78) 73(99) 0.7(0.1)
conclude that connectivity distributions
of gene transcriptional network for yeast
and E-coli are scale-free, and thus
“bacterial and fungal genetic networks are
free of characteristic scale with respect to
the distributions of both regulating and
Number of regulated genes per regulating protein
We found, however, that the scale-free property does not even hold and therefore such biological conclusions are invalid