1. 341: Introduction to Bioinformatics
Dr. Nataša Pržulj
Department of Computing
Imperial College London
Winter 2011

2. Topics Introduction to biology (cell, DNA, RNA, genes, proteins)
Sequencing and genomics (sequencing technology, sequence alignment algorithms)
Functional genomics and microarray analysis (array technology, statistics, clustering and classification)
Introduction to biological networks
Introduction to graph theory
Network properties
Network/node centralities
Network motifs
Network models
Network/node clustering
Network comparison/alignment
Software tools for network analysis
Interplay between topology and biology 2 2

3. Network Comparisons: Properties of Large Networks
Large network comparison is computationally hard due to NP-completeness of the underlying subgraph isomorphism problem:
Given 2 graphs G and H as input, determine whether G contains a subgraph that is isomorphic to H.
Thus, network comparisons rely on easily computable heuristics (approximate solutions), called “network properties”
Network properties can roughly & historically be divided in two categories:
Global network properties: give an overall view of the network, but might not be detailed enough to capture complex topological characteristics of large networks.
Local network properties: more detailed network descriptors which usually encompass larger number of constraints, thus reducing degrees of freedom in which the networks being compared can vary. 3

4. 1. Global Network Properties
Readings: Chapter 3 of “Analysis of biological networks” by Junker and Björn
Global Network Properties:
Degree distribution
Average clustering coefficient
Clustering spectrum
Average Diameter
Spectrum of shortest path lengths
Centralities

5. 1. Global Network Properties
Degree Distribution
Definitions:
degree of a node is the number of edges incident to the node.
Average degree of a network:
average of the degrees over all nodes in the network.
However, avg. deg might not be representative, since the distribution of degrees might be skewed. x
deg(x)=5

6. 1. Global Network Properties 1) Degree Distribution
Let P(k) be the percentage of nodes of degree k in the network. The degree distribution is the distribution of P(k) over all k.
P(k) can be understood as the probability that a node has degree k.

7. 1. Global Network Properties 1) Degree Distribution
Example:
(log-log plot)
Here P(k) ~ k-γ , where often 2 ≤ γ < 3. This is a power-law, heavy-tailed distribution.
Networks with power-law degree distributions are called scale-free networks. In them, most of the nodes are of low degree, but there is a small number of highly-linked nodes (nodes of high degree) called “hubs.”

8. 1. Global Network Properties 1) Degree Distribution
Another Example:
average degree is meaningful
Here P(k) is a Poisson distribution.

9. 1. Global Network Properties 1) Degree Distribution
However: degree distribution (and global properties in general) are weak predictors of network structure.
Illustration:
G and H are of the same size (i.e.,|G|=|H| -- they have the same number of nodes and edges) and they have same degree distribution, but G and H have very different topologies (i.e., graph stucture).

10. Examples: G

11. Research debates… Assortative vs. disassortative mixing of degrees:
Do high-degree nodes interact with high-degree nodes?
Done by:
Pearson corr. coefficient between degrees of adjacent vertices
Average neighbor degree; then average over all nodes of degree k
Structural robustness and attack tolerance:
“Robust, yet fragile”
Scale-free degree distribution:
“Party” vs. “date” hubs
J.D. Han et al., Nature, 430:88-93, 2004
Bias in the data collection – sampling?
M. Stumpf et al., PNAS, 102: , 2005
J. Han et al., Nature Biotechnology, 23: , 2005
High degree nodes:
Essential genes
H. Jeong at al., Nature 411, 2001.
Disease/cancer genes
Jonsson and Bates, Bioinformatics, 22(18), 2006
Goh et al., PNAS, 104(21), 2007 11

12. 1. Global Network Properties 2) Average Clustering Coefficient
Definition:
clustering coefficient Cv of a node v:
Cv = |E(N(v))|/(max possible number of edges in N(v))
Where N(v) the neighborhood of v, i.e., all nodes adjacent to v
Cv can be viewed as the probability that two neighbors of v are connected.
Thus 0 ≤ Cv ≤ 1.
By definition: For vertex v of degree 0 or 1, by definition Cv=0.

13. 1. Global Network Properties 2) Average Clustering Coefficient
Example:
|N(v)|= 4, since there are 4 nodes in N(v), i.e., N(v)= {1, 2, 3, 4}
|E(N(v))|= 3, since there are 3 edges between nodes in N(v)
Max possible number of edges between nodes in N(v) is: choose(4,2) = 6.
Therefore Cv= 3/6 = 1/2

14. 1. Global Network Properties 2) Average Clustering Coefficient
Definition:
average clustering coefficient, C, of a network is the average Cv over all the nodes v∈ V.

15. 1. Global Network Properties 3) Clustering Spectrum
Definition:
clustering spectrum, C(k), is the distribution of the average clustering coefficients of all nodes of degree k in the network, over all k.
Example:

16. of degree k E.g. 2) And 3) 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 = (1+0)/2 = 0.5
=> Clustering spectrum
E.g.
(not for G) G
Need to evaluate whether the value of C (or any other property) is statistically significant.

17. 1. Global Network Properties 4) Average Diameter
Definition: the distance between two nodes is the smallest number of links that have to be traversed to get from one node to the other.
Definition: the shortest path is the path that achieves that distance.
Definition: the average network diameter is the average of shortest path lengths over all pairs of nodes in a network.

18. 1. Global Network Properties 5) Spectrum of shortest path lengths
Definition:
Let S(d) be the percentage of node pairs that are at distance d. The spectrum of shortest path lengths is the distribution of S(d) over d.
Example:

19. 4) and 5) Average Diameter and Spectrum of Shortest Path Lengths
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)

20. 1. Global Network Properties 6) Node Centralities
(Readings: Chapter 3 of “Analysis of biological networks”-Junker,Björn)
Rank nodes according to their “topological importance”
Definition:
Centrality quantifies the topological importance of a node (edge) in a network.
There are many different types of centralities.
There are many different types of centralities:
Degree centrality
Closeness centrality
Eccentricity centrality
Betweenness centrality
Subgraph centrality
Eigenvector centrality
Software tools: Visone (social nets) and CentiBiN (biological nets)

21. 1. Global Network Properties 6) Node Centralities
Definitions:
Degree centrality, Cd(v): nodes with a large number of neighbors (i.e., edges) have high centrality. Therefore, we have Cd(v)=deg(v).
Example of a use of degree centrality:
In PPI networks, nodes with high degree centrality are considered to be “biologically important.” We will learn later in the course what this means.
2. Closeness centrality, Cc(v): nodes with short paths to all other nodes in the network have high closeness centrality
Cc(v)=

22. 1. Global Network Properties 6) Node Centralities
Definitions:
3. Betweenness centrality, Cb(v): Nodes (or edges) which occur in many of the shortest paths have high betweeness centrality.
Cb(v)=
Above:
The above summation means that there is a sum on the top and on the bottom of the fraction.
σst(v) = the number of shortest paths from s to t that pass through v
σst = the number of all shortest paths from s to t (they may or not pass through node v) 22

23. 1. Global Network Properties 6) Node Centralities
Definitions:
4. Eccentricity centrality, Ce(v): nodes with short paths to any other node have high eccentricity centrality
Eccentricity of a node v is defined as ecc(v) =
So it is the maximum shortest path length from node u to all other nodes v in V.
Eccentricity centrality of a node v:
Thus, central nodes have higher Ce since they have lower ecc.
There exist many other definitions of node centralities. 23 23

24. 1. Global Network Properties 6) Node Centralities
Example:
Degree
Closeness
Betweeness
From highest D
F, G H
D, H to
A, B I
C, E, H
C, E
lowest J
C, D, J

25. 1. Global Network Properties 6) Node Centralities
You need to know how to compute these centralities (and all other network properties) by hand on small networks.
For large real-world networks, you could use software, e.g., CentiBiN.

26. Network Properties 2. Local Network Properties
(Chapter 5 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber)
They encompass a larger number of constraints, thus reducing degrees of freedom in which networks being compared can vary
How do we show that two networks are different?
How do we show that they are the same?
How do we quantify the level of similarity?

27. Network Properties 2. Local Network Properties
(Chapter 5 of the course textbook “Analysis of Biological Networks” by Junker and Schreiber)
Network motifs
Graphlets
Two network comparison measures based on graphlets:
2.1) Relative Graphlet Frequence Distance between two networks
2.2) Graphlet Degree Distribution Agreement between two networks 27

28. 2. Local Network Properties 1) Network Motifs
(Uri Alon’s group, )
Definition: A network motif is a small over-represented partial subgraph of real network.
Here, over-represented means that it is over-represented when compared to networks coming from a random graph model.
Problem: What is expected at random, i.e., which network “null model” to use to identify motifs?

29. 2. Local Network Properties 1) Network Motifs
Example of a random graph model:
Erdos-Renyi (ER) random graphs – Definition:
A graph on n nodes (for some positive integer n)
Edges are added between pairs of nodes uniformly at random with same probability p
ER graphs usually have a small number of dense (in term of number of edges) subgraphs
There will be no regions in the network that have large density of edges. Why?

30. 2. Local Network Properties 1) Network Motifs
Example:
If motifs are identified when comparing the data with ER model networks, every dense subgraph would come up as a motif because they do not exist in our ER model networks.

31. 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”
Feed-forward loop

32. 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”
Random graphs with the same in- and out- degree distribution as data might not be the best network null model
Motifs are partial subgraphs, while we use induced ones to understand network structure

33. 2. Local Network Properties 1) Network Motifs
Example:
Feed-forward loop
Shen-Orr, Milo, Mangan, and Alon, “Network motifs in the transcriptional
regulation network of Escherichia coli,” Nature Genetics, 2002

34. 1) Network motifs (Uri Alon’s group, ’02-’04)
Also, see Pajek, MAVisto, and FANMOD

35. 2) Graphlets (Przulj group, ’04-’10)
_____
Different from network motifs:
Induced subgraphs
Of any frequency (don’t need to be over-represented)
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free or Geometric?,” Bioinformatics, vol. 20, num. 18, pg , 2004.

36. N. Przulj, D. G. Corneil, and I
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg , 2004.

37. N. Przulj, D. G. Corneil, and I
N. Przulj, D. G. Corneil, and I. Jurisica, “Modeling Interactome: Scale Free
or Geometric?,” Bioinformatics, vol. 20, num. 18, pg , 2004.

38. 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 , 2004.

39. 2.2) Graphlet Degree Distributions
Generalize node degree
2.2) Graphlet Degree Distributions

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

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

42. 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 , 2008.

43. Similarity measure between “node signature” vectors
T. Milenkovic and N. Przulj, “Uncovering Biological Network Function via Graphlet Degree Signatures”, Cancer Informatics, vol. 4, pg , 2008.

44. 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 , 2008.

45. T. Milenković and N. Pržulj, “Uncovering Biological Network Function via Graphlet Degree Signatures,” Cancer Informatics, 2008: , 2008 (Highly Visible).

46. SMD1
YBR095C
40%
PMA1
T. Milenković and N. Pržulj, “Uncovering Biological Network Function via Graphlet Degree Signatures,” Cancer Informatics, 2008: , 2008 (Highly Visible).

47. T. Milenković and N. Pržulj, “Uncovering Biological Network Function via Graphlet Degree Signatures,” Cancer Informatics, 2008: , 2008 (Highly Visible).

48. 90%* *Statistically significant threshold at ~85% SMD1 RPO26 SMB1
T. Milenković and N. Pržulj, “Uncovering Biological Network Function via Graphlet Degree Signatures,” Cancer Informatics, 2008: , 2008 (Highly Visible).

49. Later we will see how to use this and other techniques
to link network structure with biological function

50. 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.

51. N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.

52. N. Przulj, “Biological Network Comparison Using Graphlet Degree Distribution,” Bioinformatics, vol. 23, pg. e177-e183, 2007.

53. This is called Graphlet Degree Distribution (GDD) Agreement
/ sqrt(2) ( to make it between 0 and 1)
This is called Graphlet Degree Distribution (GDD) Agreement
between networks G and H.

54. Software that implements many of these network
properties and compares networks with respect to them:
GraphCrunch

55. Software that implements many of these network
properties and compares networks with respect to them:
GraphCrunch

56. Topics Introduction to biology (cell, DNA, RNA, genes, proteins)
Sequencing and genomics (sequencing technology, sequence alignment algorithms)
Functional genomics and microarray analysis (array technology, statistics, clustering and classification)
Introduction to biological networks
Introduction to graph theory
Network properties
Network/node centralities
Network motifs
Network models
Network/node clustering
Network comparison/alignment
Software tools for network analysis
Interplay between topology and biology 56 56