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Transcript of Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of...
![Page 1: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/1.jpg)
Network Properties
1. Global Network Properties (Chapter 3 of the course textbook “Analysis of
Biological Networks” by Junker and Schreiber)
1) Degree distribution2) Clustering coefficient and spectrum3) Average diameter4) Centralities
![Page 2: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/2.jpg)
1) Degree Distribution
G
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• Cv – Clustering coefficient of node vCA= 1/1 = 1CB = 1/3 = 0.33CC = 0 CD = 2/10 = 0.2 …
• C = Avg. clust. coefficient of the whole network = avg {Cv over all nodes v of G}
• C(k) – Avg. clust. coefficient of all nodesof degree kE.g.: C(2) = (CA + CC)/2 = (1+0)/2 = 0.5
=> Clustering spectrum
E.g. (not for G)
2) Clustering Coefficient and Spectrum
G
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3) Average Diameter
G
u
v
E.g.(not for G)
• Distance between a pair of nodes u and v:
Du,v = min {length of all paths between u and v} = min {3,4,3,2} = 2 = dist(u,v)
• Average diameter of the whole network:
D = avg {Du,v for all pairs of nodes {u,v} in G}
• Spectrum of the shortest path lengths
![Page 5: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/5.jpg)
Network Properties
2. Local Network Properties(Chapter 5 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber)
1) Network motifs2) Graphlets:
2.1) Relative Graphlet Frequence Distance between 2 networks
2.2) Graphlet Degree Distribution Agreement between 2 networks
![Page 6: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/6.jpg)
• Small subgraphs that are overrepresented in a network when compared to randomized networks
• Network motifs:– Reflect the underlying evolutionary processes that generated the network– Carry functional information– Define superfamilies of networks
- Zi is statistical significance of subgraph i, SPi is a vector of numbers in 0-1
• But:– Functionally important but not statistically significant patterns could be
missed– The choice of the appropriate null model is crucial, especially across
“families”
1) Network motifs (Uri Alon’s group, ’02-’04)
![Page 7: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/7.jpg)
• Small subgraphs that are overrepresented in a network when compared to randomized networks
• Network motifs:– Reflect the underlying evolutionary processes that generated the network– Carry functional information– Define superfamilies of networks
- Zi is statistical significance of subgraph i, SPi is a vector of numbers in 0-1
• But:– Functionally important but not statistically significant patterns could be
missed– The choice of the appropriate null model is crucial, especially across
“families”
1) Network motifs (Uri Alon’s group, ’02-’04)
![Page 8: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/8.jpg)
• Small subgraphs that are overrepresented in a network when compared to randomized networks
• Network motifs:– Reflect the underlying evolutionary processes that generated the
network– Carry functional information– Define superfamilies of networks
- Zi is statistical significance of subgraph i, SPi is a vector of numbers in 0-1
• Also – generation of random graphs is an issue:– Random graphs with the same degree in- & out- degree distribution as
data constructed– But this might not be the best network null model
1) Network motifs (Uri Alon’s group, ’02-’04)
![Page 9: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/9.jpg)
1) Network motifs (Uri Alon’s group, ’02-’04)
http://www.weizmann.ac.il/mcb/UriAlon/
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N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
_____
Different from network motifs: Induced subgraphs Of any frequency
2) Graphlets (Przulj, ’04-’09)
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N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
![Page 12: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/12.jpg)
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
![Page 13: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/13.jpg)
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
2.1) Relative Graphlet Frequency (RGF) distance between networks G and H:
![Page 14: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/14.jpg)
Generalize node degree
2.2) Graphlet Degree Distributions
![Page 15: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/15.jpg)
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” ECCB, Bioinformatics, vol. 23, pg. e177-e183, 2007.
![Page 16: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/16.jpg)
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” ECCB, Bioinformatics, vol. 23, pg. e177-e183, 2007.
![Page 17: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/17.jpg)
T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.
Network structure vs. biological function & disease
Graphlet Degree (GD) vectors, or “node signatures”
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Similarity measure between “node signature” vectors
T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.
![Page 19: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/19.jpg)
T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.
Signature Similarity Measure between nodes u and v
![Page 20: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/20.jpg)
Later we will see how to use this and other techniquesto link network structure with biological function.
![Page 21: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/21.jpg)
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.
Generalize Degree Distribution of a network
The degree distribution measures:• the number of nodes “touching” k edges for each value of k.
![Page 22: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/22.jpg)
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.
![Page 23: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/23.jpg)
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.
![Page 24: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/24.jpg)
/ sqrt(2) ( to make it between 0 and 1)
This is called Graphlet Degree Distribution (GDD) Agreementnetween networks G and H.
![Page 25: Network Properties 1.Global Network Properties ( Chapter 3 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber) 1)Degree distribution.](https://reader036.fdocuments.net/reader036/viewer/2022070411/56649caf5503460f94972923/html5/thumbnails/25.jpg)
Software that implements many of these networkproperties and compares networks with respect to them: GraphCrunchhttp://www.ics.uci.edu/~bio-nets/graphcrunch/
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Network models
Degree distribution
Clustering coefficient
Diameter
Real-world (e.g., PPI) networks
Power-law High Small
Erdos-Renyi graphs Poisson Low Small
Random graphs with the same degree distribution as the data
Power-law Low Small
Small-world networks Poisson High Small
Scale-free networks Power-law Low Small
Geometric random graphs Poisson High Small
Stickiness network model Power-law High Small
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Network models
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Network modelsGeometric Gene Duplication and Mutation
Networks
• Intuitive “geometricity” of PPI networks:
• Genes exist in some bio-chemical space• Gene duplications and mutations• Natural selection = “evolutionary
optimization”
N. Przulj, O. Kuchaiev, A. Stevanovic, and W. Hayes “Geometric Evolutionary Dynamics of Protein Interaction Network”, Pacific Symposium on Biocomputing (PSB’10), Hawaii, 2010.
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Network models
Stickiness-index-based model (“STICKY”)
N. Przulj and D. Higham “Modelling protein-protein interaction networks via a stickiness indes”, Journal of the Royal Society Interface 3, pp. 711-716, 2006.