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Network Properties
1. Global Network Properties
(Chapter 3 of the course textbook “Analysis of Biological
Networks” by Junker and Schreiber)
1)
2)
3)
4)
Degree distribution
Clustering coefficient and spectrum
Average diameter
Centralities
1) Degree Distribution
G
2) Clustering Coefficient and Spectrum
• Cv – Clustering coefficient of node v
CA= 1/1 = 1
CB = 1/3 = 0.33
CC = 0
CD = 2/10 = 0.2
…
G
• C = Avg. clust. coefficient of the whole network
= avg {Cv over all nodes v of G}
• C(k) – Avg. clust. coefficient of all nodes
of degree k
E.g.: C(2) = (CA + CC)/2 = (1+0)/2 = 0.5
=> Clustering spectrum
E.g.
(not for G)
3) Average Diameter
u
• 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)
G
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
E.g.
(not for G)
Network Properties
2. Local Network Properties
(Chapter 5 of the course textbook “Analysis of Biological
Networks” by Junker and Schreiber)
1) Network motifs
2) Graphlets:
2.1) Relative Graphlet Frequence Distance between 2 networks
2.2) Graphlet Degree Distribution Agreement between 2 networks
1) Network motifs (Uri Alon’s group, ’02-’04)
• 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)
• 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)
• 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)
http://www.weizmann.ac.il/mcb/UriAlon/
2) Graphlets (Przulj, ’04-’09)
_____
Different from network motifs:
 Induced subgraphs
 Of any frequency
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,”
Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
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:
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg. 3508-3515, 2004.
2.2) Graphlet Degree Distributions
Generalize node degree
N. Przulj, “Biological Network Comparison Using Graphlet Degree
Distribution,” ECCB, Bioinformatics, vol. 23, pg. e177-e183, 2007.
N. Przulj, “Biological Network Comparison Using Graphlet Degree
Distribution,” ECCB, Bioinformatics, vol. 23, pg. e177-e183, 2007.
Network structure vs. biological function & disease
Graphlet Degree (GD) vectors, or “node signatures”
T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet
Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.
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.
Signature Similarity Measure between nodes u and v
T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet
Degree Signatures”, Cancer Informatics, vol. 4, pg. 257-273, 2008.
Later we will see how to use this and other techniques
to link network structure with biological function.
Generalize Degree Distribution of a network
The degree distribution measures:
• the number of nodes “touching” k edges for each value of k.
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,”
Bioinformatics, vol. 23, pg. e177-e183, 2007.
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,”
Bioinformatics, vol. 23, pg. e177-e183, 2007.
N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,”
Bioinformatics, vol. 23, pg. e177-e183, 2007.
/ sqrt(2) ( to make it between 0 and 1)
This is called Graphlet Degree Distribution (GDD) Agreement
netween networks G and H.
Software that implements many of these network
properties and compares networks with respect to them:
GraphCrunch
http://www.ics.uci.edu/~bio-nets/graphcrunch/
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
Network models
Network models
Geometric 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.
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.