Network analysis

Sushmita RoyBMI/CS 576

www.biostat.wisc.edu/bmi576sroy@biostat.wisc.edu

Dec 3rd, 2013

Key concepts

• Network measures– Degree– Degree distribution– Average path length and shortest path length– Clustering coefficient– Modularity– Network motifs– Centrality measures

• Network models– Random networks– Scale free networks

Directed and undirected networks

Undirected network

Vertex/Node

Edge Directed Edge

Directed network

A

B C

D

E

F

A

B C

D

E

F

Node degree

• Undirected 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)

Directed Edge

A

B C

D

E

FIndegree of F is 4Outdegree of E is 1

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

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

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.

Poisson distribution

• A discrete distribution

• The Poisson is parameterized by which can be easily estimated by maximum likelihood

k

P(X=k)

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)

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

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

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

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

Poisson versus Scale free

Barabasi & Oltvai

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

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

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

A

BC

?

Clustering coefficient example

A

C

BG

D

Let’s look at some large networks

• We will consider networks of 800-1000 nodes• One is generated using the Preferential attachment

model• One is generated using the ER model

Networks generated from the different models

Preferential attachment ER random network

Degree distributions of the two networks

Preferential attachment ER random network

Comparing other properties of the networks

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.

Hierarchical network

Barabasi & Oltvai, 2004

A hierarchical network generated by replicating the current set of nodes

Scale-free distribution of degrees

Inverse relationship between C(k) and degree

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

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.

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

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

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

Neighbors of v

Largest eigen value

Degree centrality of a node is correlated to functional importance of a node

Red nodes on deletion cause the organism to dieRed nodes also among the most degree central

Yeast protein-protein interaction network

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

Network motifs of size three in a directed network

Finding network motifs

• Enumerating motifs– Subgraph enumeration

• Calculating the number of occurrences in randomized networks

Milo 2002

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

Structural common motifs seen in the yeast regulatory network

Lee et.al. 2002, Mangan & Alon, 2003

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

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?

A modular network

Module 1

Module 2

Module 3

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, http://arxiv.org/abs/cond-mat/0308217v1

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

Girvan-Newman algorithm

• Initialize– Compute betweennees for all edges

• Repeat until convergence criteria1. Remove the node with the highest betweennees2. Recompute betweenness of affected edges

• Convergence criteria can be– No more edges– Desired modularity.

Zachary’s karate club study

Each node is an individual and edges represent social interactions among individuals. The shape and colors represent different groups.

Node grouping based on betweenness

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