1. Network analysis Sushmita Roy BMI/CS 576 www.biostat.wisc.edu/bmi576
Dec 3rd, 2013

2. Key concepts Network measures Network models Degree
Degree distribution
Average path length and shortest path length
Clustering coefficient
Modularity
Network motifs
Centrality measures
Network models
Random networks
Scale free networks

3. Directed and undirected networks
Vertex/Node A E A E F D F D
Edge
Directed Edge B B C C
Undirected network
Directed network

4. Node degree Undirected network Directed network
Degree, k: Number of neighbors of a node
Directed network
Indegree, kin: Number of incoming edges
Out degree, kout: Number of outgoing edges
Average degree (undirected network) A E
Indegree of F is 4
Outdegree of E is 1 F D
Directed Edge B C

5. Average degree Consider an undirected network with N nodes and L edges
Let ki denote the degree of node i
Average degree is
Average degree is equivalently defined as

6. Degree distribution
P(k) gives the probability that a selected node has k edges
Different networks can have different degree distributions
A fundamental property that can be used to characterize a network

7. Different degree distributions
Poisson distribution
The mean is a good representation of ki of all nodes
Exhibited in Erdos Renyi networks
Power law distribution
Also called scale free
There is no “typical” node that captures the degree of nodes.

8. Poisson distribution A discrete distribution
The Poisson is parameterized by which can be easily estimated by maximum likelihood
P(X=k) k

9. Power law distribution
Used to capture the degree distribution of most biological/real networks
Typical value of is between 2 and 3.
MLE exists for but is more complicated
See Power-Law Distributions in Empirical Data. Clauset, Shalizi and Newman, 2009 for details
P(k)

10. Erdos Renyi random graphs
Dates back to 1960 due to two mathematicians Paul Erdos and Alfred Renyi.
Provides a probabilistic model to generate a graph
Starts with N nodes and connects two nodes with probability p
Node degrees follow a Poisson distribution
Tail falls off exponentially, suggesting that nodes with degrees different from the mean are very rare

11. Generating a graph using the ER model
Input
p: probability of an edge
N: number of nodes in the network
Output: An ER network of N nodes with on p*N(N-1)/2 edges on average
For each possible edge add with probability p

12. Scale free networks
Degree distribution is captured by a power law distribution
Such networks are ubiquitous in nature
Scale-free networks can be generated by the preferential attachment model from Barabasi-Albert
A “rich gets richer” model

13. Generating a Scale free network with the preferential attachment model
Input:
N: number of nodes
m: number of existing nodes to connect
Output: a scale-free network
At each iteration
Add a node with m connections
Select a node i as one of the m neighbors with probability

14. Poisson versus Scale free
Barabasi & Oltvai

15. Path lengths The shortest path length between two nodes A and B:
The smallest number of edges that need to be traversed to get from A to B
Mean path length is the average of all shortest path lengths
Diameter of a graph is the longest of all shortest paths in the network

16. Scale-free networks are ultra-small
Average path length is log log N
In a random network (Erdos Renyi network) the average path length is log N

17. Clustering coefficient
Measure of transitivity in the network
If A is connect to B, and B is connected to C, how often is A connected to C
Clustering coefficient Ci for each node i is
ni is the number of edges among neighbors of i
The ratio of the number of edges connecting i’s neighbors to the max possible
Average clustering coefficient gives a measure of nodes to form clusters B C A ?

18. Clustering coefficient exampleB A C D

19. Let’s look at some large networks
We will consider networks of nodes
One is generated using the Preferential attachment model
One is generated using the ER model

20. Networks generated from the different models
ER random network
Preferential attachment

21. Degree distributions of the two networks
Preferential attachment
ER random network

22. Comparing other properties of the networks

23. Relationship between clustering coefficient and degree
Define C(k) as the average clustering coefficient of all nodes with degree k
In some networks
If this is true, the networks are said to have a hierarchical organization
Smaller node sets are linked together to form larger modules.

24. Hierarchical network
A hierarchical network generated by replicating the current set of nodes
Scale-free distribution of degrees
Inverse relationship between C(k) and degree
Barabasi & Oltvai, 2004

25. Hierarchical organization is seen also among nodes
Regulators are hierarchically organized with different roles per level
Top: Master regulators influence many genes
Middle: Bottle necks directly targeting most genes
Bottom: Essential regulators
Hierarchical structure of S. cerevisiae regulatory network
Yu & Gerstein 2006, Jothi et al. 2009

26. Given a network how can we test what degree distribution it follows?
Compute the empirical degree distribution
Degree distribution can Poisson or Power law
Estimate parameters of the distribution from the data
Pick the distribution that fits the data better.

27. Properties of scale free networks
Degree distribution is best captured by a power law distribution
Average clustering coefficient is higher than expected from a random network
Average path length is smaller than expected from a random network

28. Centrality measures in networks
A measure of how important network node is
Four types of centrality measures defined for each node
Degree centrality
The degree of a node
Betweenness centrality
The number of shortest paths between two nodes that passes through the node of interest
Closeness centrality
Sum of a distances from other nodes
Eigenvector centrality
Given by the largest eigen vector of the adjacency matrix

29. Eigenvector centrality
Based on the idea that nodes with high score should influence the importance of a node more
Given by
The centrality measures are given by the entries of the first eigen vector
Google’s page rank algorithm makes use of a type of Eigen vector centrality
Largest eigen value
Neighbors of v

30. Degree centrality of a node is correlated to functional importance of a node
Yeast protein-protein interaction network
Red nodes on deletion cause the organism to die
Red nodes also among the most degree central

31. Network motifs
Degree distributions capture important global properties of the network
Can we say something about more local properties of the network?
Network motifs are defined as small recurring subnetworks that occur much more than a randomized network
A subgraph is called a network motif of a network if its occurrence in randomized networks is significantly less than the original network.
Some motifs are associated to explain specific network dynamics
Milo Science 2002

32. Network motifs of size three in a directed network

33. Finding network motifs
Enumerating motifs
Subgraph enumeration
Calculating the number of occurrences in randomized networks
Milo 2002

34. Network motifs found in many complex networks
The occurrence of the feedforward loop in both networks suggests a fundamental similarity in the design on these networks

35. Structural common motifs seen in the yeast regulatory network
Auto-regulation
Multi-component
Feed-forward loop
Single Input
Multi Input
Regulatory Chain
Feed-forward loops involved in speeding up in response of target gene
Lee et.al. 2002, Mangan & Alon, 2003

36. Modularity in networks
Modularity “refers to a group of physically or functionally linked nodes that work together to achieve a distinct function”
-- Barabasi & Oltvai
Similar idea is captured by the “community structure” in networks
Two questions
Given a network is it modular?
Given a network what are the modules in the network?

37. A modular network
Module 2
Module 3
Module 1

38. Assessing the modularity of a network
Modularity of a network can be assessed in two ways:
Recall the average clustering coefficient
A modular network is one that has a significantly higher clustering coefficient than a network with equivalent number of nodes and degree distribution
If we know an existing grouping of nodes, we can compute modularity (Q) as
difference between within group (community) connections and expected connections within a group
Q defined as in: Finding and evaluating community structure in networks,

39. Finding modules in a graph
Given a graph find the densely connected subgraphs
Graph clustering algorithms are applicable here
Hierarchical clustering using the edge weight as a distance
How to define weight?
Markov clustering algorithm
Girvan-Newman algorithm

40. Girvan-Newman algorithm
Initialize
Compute betweennees for all edges
Repeat until convergence criteria
Remove the node with the highest betweennees
Recompute betweenness of affected edges
Convergence criteria can be
No more edges
Desired modularity.

41. Zachary’s karate club study
Node grouping based on betweenness
Each node is an individual and edges represent social interactions among individuals. The shape and colors represent different groups.

42. Summary of network analysis
Given a network, its topology can be characterized using different measures
Degree distribution
Average path length
Clustering coefficient
Centrality measures
Allow us to assess the importance of different nodes
Network motifs
Overrepresentation of subgraphs of specific types
Network modularity