Transcript Biological Networks

Biological Networks
Lectures 6-7 : February 02, 2010
Graph Algorithms Review
Global Network Properties
Local Network Properties
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Graph Algorithms Review
Readings: Chapter 2 of “Analysis of biological networks” by Junker and Björn
You will be responsible for knowing the following
about the following 3 algorithms:
• For un-weighted graphs:
– Breadth-First Search (BFS)
• For weighted graphs:
– Dijkstra’s algorithm
– Floyd-Warshal algorithm
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Graph Algorithms Review
For un-weighted graphs:
• Breadth-First Search (BFS)
– Input: un-weighted graph G(V,E), start node s
– Ouput:
• Shortest paths and distances from s to all other nodes
of G
• Connected components of G
– Running time: linear, O(|V|+|E|)
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Graph Algorithms Review
– Order of exploration of G with BFS:
1. Start from the start node s
2. Explore the neighbors of s
3. Explore the neighbors of neighbors of s from the first
explored neighbor to the last one
4. …
– Example :
S:1
2
5
9
6
10
11
3
4
7
8
12
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Graph Algorithms Review
For weighted graphs:
• Dijkstra’s algorithm
– Input: weighted graph G(V,E), start node s
– Output: shortest paths and distances from s to all other
nodes of G
– Running time: O(|V| log|V|+|E|)
• Floyd-Warshal Algorithm
– Input: weighted graph G(V,E)
– Output: Matrix of distances and shortest paths between all
pairs of nodes of G
– Running time: O(|V|3)
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Network Comparisons:
Properties of Large Networks
• Large network comparison is computationally hard due to NPcompleteness of the underlying subgraph isomorphism problem.
• Thus, network comparisons rely on easily computable heuristics
(approximate solutions), called “network properties”
• Network properties can roughly be divided in two categories:
1.
Global network properties: give an overall view of the network, but
might not be detailed enough to capture complex topological
characteristics of large networks
2.
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.
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1. Global Network Properties
Readings: Chapter 3 of “Analysis of biological networks” by Junker and Björn
• Global Network Properties:
1)
2)
3)
4)
5)
6)
Degree distribution
Average clustering coefficient
Clustering spectrum
Average Diameter
Spectrum of shortest path lengths
Centralities
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1. Global Network Properties
1) 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, it might not be representative, since
the distribution of degrees might be skewed.
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1. Global Network Properties
1) Degree Distribution
• 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.
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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.”
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1. Global Network Properties
1) Degree Distribution
• Another Example:
average degree is meaningful
Here P(k) is a Poisson distribution.
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1. Global Network Properties
1) Degree Distribution
• However: degree distribution (and global properties in
general) are weak predictors of network structure.
• Illustration:
G1 and G2 are of the same size (i.e.,|G1|=|G2| -- they have the
same number of nodes and edges) and they have same degree
distribution, but G1 and G2 have very different topologies (i.e.,
graph stucture).
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Examples:
G
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.
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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
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1. Global Network Properties
2) Average Clustering Coefficient
• Definition:
average clustering coefficient of a network is
the average Cv over all the nodes v∈ V.
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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:
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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
…
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)
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.
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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:
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4) and 5) Average Diameter and Spectrum of Shortest Path Lengths
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)
1. Global Network Properties
6) Node Centralities
(Readings: Chapter 3 of “Analysis of biological
networks”-Junker,Björn)
• Definitions:
– Centrality quantifies the topological importance of a node
(edge) in a network.
There are many different types of centralities:
1. degree centrality Cd: 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.
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1. Global Network Properties
6) Node Centralities
• Definitions:
– Centrality quantifies the topological importance of a node (edge) in a network.
There are many different types of centralities:
1.
Degree centrality, Cd(v): nodes with a large number of neighbors (i.e., edges)
2.
Closeness centrality, Cc(v): nodes with short paths to all other
nodes in the network have high closeness centrality
have high centrality. Therefore, we have Cd(v)=deg(v).
Cc(v)=
1
 dist(u,v)
uV

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1. Global Network Properties
6) Node Centralities
• Definitions:
– Centrality quantifies the topological importance of a node (edge) in a network.
There are many different types of centralities:
3.
Betweenness centrality, Cb(v): Nodes (or edges) which occur in many of
the shortest paths have high betweeness centrality
.
Cb(v)=  st(v)
st
st
sv
v t
The above summation means that there is a sum on the top and on the
bottom of the fraction.
Above:

σst = the number of shortest paths from s to t (they may or not pass
through node v)
σst(v) = the number of shortest paths from s to t that pass through v.
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1. Global Network Properties
6) Node Centralities
• Definitions:
– Centrality quantifies the topological importance of a node (edge) in a network.
There are many different types of centralities:
4.
Eccentricity centrality, Ce(v):
Eccentricity of a node v is defined as ecc(v) =
maxdist(u,v)
v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:

Ce(v) = 1/Ecc(v)
Thus, central nodes have higher Ce since they have lower ecc.
There exist many other definitions of node centralities.
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1. Global Network Properties
6) Node Centralities
• Example:
Degree
From highest
to
lowest
Closeness
Betweeness
D
F, G
H
F, G
D, H
F, G
A, B
A, B
I
C, E, H
C, E
D
I
I
A, B
J
J
C, D, J
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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.
– http://centibin.ipk-gatersleben.de/
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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
Two network comparison measures based on them:
2.1) Relative Graphlet Frequence Distance between two
networks
2.2) Graphlet Degree Distribution Agreement between two
networks
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2. Local Network Properties
1) Network Motifs
• Definition: A network motif is a small overrepresented partial subgraph of real network.
Here, over-represented means that it is overrepresented 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?
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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?
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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.
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2. Local Network Properties
1) Network Motifs
• Motifs:
– May provide insight into both the structure and function of the
whole network.
– Can potentially define universal classes of networks.
 Networks of similar type share the same motifs (e.g., all networks that
tranmit information, but in different domains) – see examples in next
class
 Motifs could reflect the evolutionary processes that generated these
network classes
• Issue: network null model used to define motifs
• Another issue: partial versus induced subgraphs
Motifs are partial subgraphs!
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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
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2. Local Network Properties
2) Graphlets
• Definition: Graphlets are small connected induced nonisomorphic subgraphs of a large network.
They do not need to be over-represented  no issues with the
null model.
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2. Local Network Properties
2) Graphlets
• Graphlet frequencies: count the occurrences
of all small (2 to 5 node) graphlets in a
network.
• Thus, we can compare these frequencies
between two networks – this is Relative
Graphlet Frequency Distance (RGF-distance)
measure of structural similarity between two
networks.
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2. Local Network Properties
2) Graphlets
• Graphlet Degree Distribution Agreement (GDD-agreement):
• Generalization of the degree distribution to a spectrum
of GDD distributions
• Degree distribution measures: the number of nodes
touching k edges for each value of k
• An edge is the only 2-node graphlet (graphlet denoted
by G0 in the examples below)
• There is nothing special about an edge
• Why not count how many triangles, squares,... a node
touches?
•“GDD signature” of a node – how many times a node
touches each of the graphlets at a given orbit
(see examples in next class)
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