Network Properties

1
Network Properties

• Global Network Properties
• (Chapter 3 of the course textbook Analysis
  of Biological Networks by Junker and Schreiber)
• Degree distribution
• Clustering coefficient and spectrum
• Average diameter
• Centralities

2
1) Degree Distribution
G
3
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
• 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 (10)/2 0.5
• gt Clustering spectrum
• E.g.
• (not for G)

G
4
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)
• 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

G
v
E.g. (not for G)
5
Network Properties

• 2. Local Network Properties
• (Chapter 5 of the course textbook Analysis of
  Biological Networks by Junker and Schreiber)
• Network motifs
• Graphlets
• 2.1) Relative Graphlet Frequence Distance between
  2 networks
• 2.2) Graphlet Degree Distribution Agreement
  between 2 networks

6
1) Network motifs (Uri Alons 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

7
1) Network motifs (Uri Alons 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

8
1) Network motifs (Uri Alons 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

9
1) Network motifs (Uri Alons group, 02-04)
http//www.weizmann.ac.il/mcb/UriAlon/
10
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.
11
N. Przulj, D. G. Corneil, and I. Jurisica,
Modeling Interactome Scale Free or
Geometric?, Bioinformatics, vol. 20, num. 18,
pg. 3508-3515, 2004.
12
N. Przulj, D. G. Corneil, and I. Jurisica,
Modeling Interactome Scale Free or
Geometric?, Bioinformatics, vol. 20, num. 18,
pg. 3508-3515, 2004.
13
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.
14

• Generalize node degree

2.2) Graphlet Degree Distributions
15

N. Przulj, Biological Network Comparison Using
Graphlet Degree Distribution, ECCB,
Bioinformatics, vol. 23, pg. e177-e183, 2007.
16

N. Przulj, Biological Network Comparison Using
Graphlet Degree Distribution, ECCB,
Bioinformatics, vol. 23, pg. e177-e183, 2007.
17
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.
18
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.
19
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.
20
Later we will see how to use this and other
techniques to link network structure with
biological function.
21
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.
22

N. Przulj, Biological Network Comparison Using
Graphlet Degree Distribution, Bioinformatics,
vol. 23, pg. e177-e183, 2007.
23
N. Przulj, Biological Network Comparison Using
Graphlet Degree Distribution, Bioinformatics,
vol. 23, pg. e177-e183, 2007.
24
/ sqrt(2) (? to make it between 0 and 1)
This is called Graphlet Degree Distribution (GDD)
Agreement netween networks G and H.
25
Software that implements many of these
network properties and compares networks with
respect to them GraphCrunch http//www.ics.uci.e
du/bio-nets/graphcrunch/
26
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
27
Network models
28
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 (PSB10), Hawaii, 2010.
29
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.