Download Network models - Department of Computing

341: Introduction to Bioinformatics
Dr. Nataša Pržulj
Department of Computing
Imperial College London
[email protected]
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
Protein 3D structure / Network data integration
Software tools for network analysis
Interplay between topology and biology
2
Network properties: summary of last class
Network Comparisons:
 Large network comparison is computationally hard due to NPcompleteness 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:
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.
3
Network properties: summary of last class
1. Global Network Properties
Readings: Chapter 3 of “Analysis of biological networks” by Junker and Schreiber.
 Some Global Network Properties:
1)
2)
3)
4)
5)
6)
Degree distribution
Average clustering coefficient
Clustering spectrum
Average Diameter
Spectrum of shortest path lengths
Centralities
4
Network properties: summary of last class
2. Local Network Properties
Readings: Chapter 5 of “Analysis of Biological Networks” by Junker and Schreiber.
1) Network motifs
2) Graphlets
Two network comparison measures based on graphlets:
2.1) Relative Graphlet Frequency Distance between two networks
2.2) Graphlet Degree Distribution Agreement between two networks
5
What is a network (graph) model?
Does the model network fit the data?
 Use network properties:
 Local
 Global
Why?
 “Hardness” of graph theoretic problems
 E.g., NP-completeness of subgraph isomorphism
 Cannot exactly compare/align networks
•
Use heuristics (approximate solutions)
 Exact comparison inappropriate in biology
•
Due to biological variation
 Noise  revise models as data sets evolve
Why model networks?
 Understand laws  reproduction/predictions
 Network models have already been used in
biological applications:
Network motifs
(Shen-Orr et al., Nature Genetics 2002, Milo et al., Science 2002)
De-noising of PPI network data
(Kuchaiev et al., PLoS Comp. Biology, 2009)
Guiding biological experiments
(Lappe and Holm, Nature Biotechnology, 2004)
 Development of computationally easy algorithms for PPI nets
that are computationally intensive on graphs in general
(Przulj et al., Bioinformatics, 2006)
Network models
We will cover the following network models:
I.
Erdos–Renyi random graphs
II. Generalized random graphs (with the same
degree distribution as the data networks)
III. Small-world networks
IV. Scale-free networks
V. Hierarchical model
VI. Geometric random graphs
VII. Stickiness index-based network model
Erdos–Renyi random graphs (ER)


Model a data network G(V,E) with |V|=n and |E|=m
An ER graph that models G is constructed as follows:



It has n nodes
Edges are added between pairs of nodes uniformly at
random with the same probability p
Two (equivalent) methods for constructing ER graphs:


Gn,p: pick p so that the resulting model network has m
edges
Gn,m: pick randomly m pairs of nodes and add edges
between them with probability 1
Erdos–Renyi random graphs (ER)

Number of edges, |E|=m, in Gn,p is:

Average degree is:
Erdos–Renyi random graphs (ER)

Many properties of ER can be proven theoretically
(See: Bollobas, "Random Graphs," 2002)

Example:
 When m=n/2,suddenly the giant component
emerges, i.e.:
•
•
One connected component of the network has
O(n) nodes
The next largest connected component has
O(log(n)) nodes
Erdos–Renyi random graphs (ER)

The degree distribution is binomial:

For large n, this can be approximated with
Poisson distribution:

where z is the average degree (compute it!)
However, currently available biological
networks have power-law degree distribution
Erdos–Renyi random graphs (ER)

Clustering coefficient, C, of ER is low (for low p)

C=p, since probability p of connecting any two
nodes in an ER graph is the same, regardless of
whether the nodes are neighbors

However, biological networks have high
clustering coefficients
Erdos–Renyi random graphs (ER)

Average diameter of ER graphs is small

It is equal to

Biological networks also have small average
diameters

Summary
Generalized random graphs (ER-DD)

Preserve the degree distribution of data
(“ER-DD”)

Constructed as follows:
 An ER-DD network has n nodes
(so does the data)
 Edges are added between pairs of nodes using
the “stubs method”
Generalized random graphs (ER-DD)

The “stubs method” for constructing ER-DD graphs:




The number of “stubs” (to be filled by edges) is
assigned to each node in the model network
according to the degree distribution of the real
network to be modeled
Edges are created between pairs of nodes with
“available” stubs picked at random
After an edge is created, the number of stubs left
available at the corresponding “end nodes” of the
edges is decreased by one
Multiple edges between the same pair of nodes are
not allowed
Generalized random graphs (ER-DD)

Summary


2 global network properties are matched by ER-DD
How about local network properties (graphlet frequencies)?


Low-density (sparse) graphlets are frequent in ER and ER-DD
However, data networks have lots of dense graphlets, since
data networks have high clustering coefficients
Small-world networks (SW)


Watts and Strogatz, 1998
Created from regular ring lattices by random
rewiring of a small percentage of their edges

E.g.
Small-world networks (SW)

SW networks have:



High clustering coefficients – introduced by “ring
regularity”
Large average diameters of regular lattices – made
small by randomly re-wiring a small percentage of
edges
Summary
Scale-free networks (SF)

Power-law degree distributions: P(k) = k−γ

γ > 0; 2 < γ < 3
Scale-free networks (SF)

Power-law degree distributions: P(k) = k−γ

γ > 0; 2 < γ < 3
Scale-free networks (SF)

Different models exist, e.g.:
 Preferential Attachment Model (SF-BA)
(Barabasi-Albert, 1999)
 Gene Duplication and Mutation Model (SF-GD)
(Vazquez et al., 2003)
Scale-free networks (SF)

Preferential Attachment Model (SF-BA)


“Growth” model: nodes are added to an existing network
New nodes preferentially attach to existing nodes with
probability proportional to the degrees of the existing
nodes; e.g.:

This is repeated until the size of SF network matches
the size of the data
“Rich getting richer”
The starting network strongly influences the properties
of the resulting network (F. Hormozdiari, et al., PLoS Computational


Biology, 3(7):e118, July 2007. )

SF-BA: particularly effective at describing Internet
Scale-free networks (SF)

Gene Duplication and Mutation Model (SF-GD)

Biologically motivated

Attempts to mimic gene duplication and mutation
processes
Scale-free networks (SF)

Gene Duplication and Mutation Model (SF-GD)

At each time step, a node is added to the network as follows:
Scale-free networks (SF)

Summary
Hierarchical model

Preserves network “modularity” via a fractallike generation of the network
Hierarchical model

These graphs do not match any biological data
and are highly unlikely to be found in data sets
Geometric random graphs

“Uniform” geometric random graphs (GEO)
N. Przulj lab, 2004-2010

Geometric gene duplication and mutation model
(GEO-GD)
N. Przulj et al., PSB 2010
Geometric random graphs

“Uniform” geometric random graphs (GEO)





Take any metric space and, using a uniform random
distribution, place nodes within the space
If any nodes are within radius r (calculated via any
chosen distance norm for the space), they will be
connected
Choose r so that the size of the GEO network matches
that of the data
There are many possible metric spaces (e.g.,
Euclidean space)
There are many possible distance norms
(e.g. the Euclidean distance, the Chessboard distance,
and the Manhattan/Taxi Driver distance)
Geometric random graphs
 “Uniform” geometric random graphs (GEO)
 Summary
Geometric random graphs

Geometric gene duplication and mutation model
(GEO-GD)

Gene duplications and mutations can be used to guide the
growth process in a geometric graph
Geometric random graphs

Geometric gene duplication and mutation model
(GEO-GD)

Gene duplications and mutations can be used to guide the
growth process in a geometric graph
Geometric random graphs

Geometric gene duplication and mutation model
(GEO-GD)

Gene duplications and mutations can be used to guide the
growth process in a geometric graph
Geometric random graphs

Geometric gene duplication and mutation model
(GEO-GD)

Gene duplications and mutations can be used to guide the
growth process in a geometric graph
Geometric random graphs

Geometric gene duplication and mutation model
(GEO-GD)

Gene duplications and mutations can be used to guide the
growth process in a geometric graph
Geometric random graphs

Geometric gene duplication and mutation
model (GEO-GD)

This variant also reproduces graphlet properties of
the empirical dataset

Also, these networks have power-law degree
distributions
-GD
Stickiness index-based network model
(N. Przulj and D. Higham, Journal of the Royal Society Interface, vol 3, num 10, pp 711 - 716, 2006.)

Based on the stickiness index:



Assumption: a high degree protein has many binding
domains



A number based on the a protein’s normalized degree in a
PPI network
Used to summarize the abundance and popularity of binding
domains of a protein
However, remember “date” vs. “party” hubs
A pair of proteins is more likely to interact under this
model if both proteins have high stickiness indices
The probability of an edge between two nodes is the
product of their stickiness indices
Stickiness index-based network model
 “Sticky networks” have the expected degree
distribution of the data
 Also, they mimic well the clustering coefficients
and the diameters of real-world networks
 Summary
Software that implements many of these network models and
evaluates their fit to data networks with respect to a variety of
network properties (but there are others):
GraphCrunch: http://bio-nets.doc.ic.ac.uk/graphcrunch/
Software that implements many of these network models and
evaluates their fit to data networks with respect to a variety of
network properties (but there are others):
GraphCrunch: http://bio-nets.doc.ic.ac.uk/graphcrunch2/
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
Protein 3D structure / Network data integration
Software tools for network analysis
Interplay between topology and biology
54