Transcript Introduction to Bioinformatics.

Modeling Nature
February 2009
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Modeling Nature
LECTURE 3: Network models
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Overview
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Some definitions
Basic characteristics of networks
Special network topologies
Examples from nature and sociology
Relation to the tasks
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Definition of a Network
A network is a system of N similar entities
called nodes (a.k.a. edges), where each
node interacts with (i.e. ‘has a relation to’)
certain other nodes in the system.
This interaction is visualized through a
connection (a.k.a. vertex).
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Some examples
Undirected
network
Directed
network
Selfconnection
and multiple
edges
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A more complex example
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A large network
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A gene
regulatory
network
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Characteristics of Networks
• Characteristic Path Length (L)
– The average number of associative links
between a pair of concepts
• Clustering Coefficient (C)
– The fraction of associated neighbors of a
concept that are also connected
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Q
A possible path
between
nodes P and Q
P
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Picture pathlengths and
clustercoefficients in these networks
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Characteristics of Networks
• The coordination number z is the average
number of links per vertices,i.e. there are a total
of Nz/2 connections in an undirected network.
• The network diameter D is the maximum
degree of separation between all pairs of
vertices.
For a network with N vertices and coordination
number z we thus have a diameter D of:
zD ≈ N
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Characteristics of Networks
• Branching Factor a.k.a. Degree (k)
– The number of other nodes connected to
this node i.e. the number of vertices of a
node
• Degree distribution
– The relation between the degree and
number of nodes in the network that have
exactly this degree.
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Connection probability p
The probability that a given edge occurs is
called the connection probability p.
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Characteristics of Networks
Clique
A clique is a set of vertices for which:
(i) every node of the same clique is connected by
an edge to every other member of the clique
and
(ii) no node outside the clique is connected to all
members of the clique.
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Cliques
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Special Network Topologies
In many situations networks can have a
special structure (topology) or properties.
We will consider the following cases.
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1. Regular network
A regular network is a network where each
node has an identical connection scheme .
? YES
? NO
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2. Fully connected network
A fully connected network is a network
where each node is connected to all other
nodes.
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3. A sparse network
A sparse network is a network that exhibits
a (very) small amount of connections.
(opposite: dense)
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dense network
sparse network
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4. Random network
A random network is a network that is
generated by some random process.
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Random network
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Small-World (SW) network
A SW network is a property of the network
rather than a specific topology – though the
SW-property has implications for the
network architecture.
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Small-world networks
In such a network there are numerous clusters of
richly interconnected elements and a small
number of connections between the clusters.
This type of network has been offered as a model of
the "six-degrees-of-separation" concept.
A small world network falls between a regular and
random network in its properties, as depicted in
this figure:
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Small-world networks
p is the probability that a randomly chosed
connection will be randomly redirected elsewhere
(i.e., p=0 means nothing is changed, leaving the
network regular; p=1 means every connection is
changed and randomly reconnected, yielding
complete randomness).
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Small-world networks
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Four network types
a
fully connected
c
regular
b
d
random
“small world
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Network Evaluation
Type of network
k
C
L
Fully-connected
N-1
Large
Small
Random
<<N
Small
Small
Regular
<<N
Large
Large
Small-world
<<N
Large
Small
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Varying the rewiring probability p:
from regular to random networks
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C(p)/C(0)
L(p)/L(0)
0
0.0 0.00001 0.0001 0.001 0.01 0.1 1.0
p
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First an example …
the accumulation of knowledge
and the growth of the ‘semantic
network’ in children
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Example : Semantic Network
lemon
gravitation
pear
apple
orange
Newto
n
Einstein
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Growth of knowledge
semantic networks
lemon
gravitation
pear
apple
orange
Newto
n
Einstein
• Average separation should be small
• Local clustering should be large
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Strongest links of/with
APPLE
PIE
PEAR
ORANGE
TREE
CORE
FRUIT
(20)
(17)
(13)
( 8)
( 7)
( 4)
NEWTON
APPLE
ISAAC
LAW
ABBOT
PHYSICS
SCIENCE
(22)
(15)
( 8)
( 6)
( 4)
( 3)
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Semantic net at age 3
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Semantic net at age 4
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Semantic net at age 5
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The growth of semantic
networks obeys a logistic law
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
97
91
85
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43
37
31
25
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0
1
# concepts/max. # concepts
1
age
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L as a function of age (× 100)
2.8
2.3
1.8
200
300
400
500
600
700
= semantic network
= random network
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C as a function of age (× 100)
0.2
0.15
0.1
0.05
0
200
300
400
500
600
700
= semantic network
= random network
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Small-worldliness
Walsh (1999)
• Measure of how well small path length is
combined with large clustering
• Small-worldliness = (C/L)/(Crand/Lrand)
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Small-worldliness
as a function of age
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adult
3.5
small-worldliness
3
2.5
2
1.5
1
0.5
0
200
300
400
500
600
700
age
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Small-Worldliness
Some comparisons
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4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Semantic
Network
Cerebral
Cortex
Caenorhabditis
Elegans
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What causes the smallworldliness in the semantic
net?
Optimal efficient organization
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Strongest links in semantic net
of adult males [Shields, 2001]
• TOP 40 of concepts
• Ranked according to their k-value
(number of associations with other
concepts)
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Semantic top 40
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3
4
5
6
7
8
9
10
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15
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Concept
ME
MAN
GOOD
SEX
NO
MONEY
YES
NOTHING
WORK
PEOPLE
FOOD
WATER
TIME
LIFE
WHAT
LOVE
BAD
GIRL
UP
CAR
k
1088
1057
871
851
793
763
720
690
690
684
683
676
639
633
623
621
615
575
569
557
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Concept
BLACK
HOUSE
OUT
DEATH
HOME
NICE
RED
HARD
WHITE
NOW
OF
SCHOOL
WOMAN
HELP
SEA
BED
PAIN
DOG
NEVER
DOWN
k
548
546
537
528
509
502
488
462
457
456
451
450
450
446
443
442
429
424
418
410
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Scale-Free (SF) networks
A SF network is a network where the degree
distribution has a very specific structure
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Scale-Free Networks
(Barabasi et al, 1998)
• In the late 1990s: Analysis of large data sets became possible
• Finding: the degree distribution often follows a power law: many
lowly connected nodes, very few highly connected nodes:
• Examples
– Biological networks: metabolic, protein-protein interaction
– Technological networks: Internet, WWW
– Social networks: citation, actor collaboration
– Other: earthquakes, human language
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Scale-Free Networks
Some small-world networks are also what are called
scale-free.
In a scall-free networks the characteristic clustering
is maintained even as the networks themselves
grow arbitrarily large.
The mathematical properties and methods of
analysis of such scale-free networks allow broad
types of analysis, modeling and simulation.
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Scale-Free Networks
* Here is a picture of a part of such a scale-free network :
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Scale-Free Networks
(Barabasi et al, 1998)
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Scale-Free Networks
In any real network some nodes are more highly connected
than others.
P(k) is the proportion of nodes that have k-links.
For large, random graphs only a few nodes have a very small
k and only a few have a very
large k, leading to a bell-shaped
distribution, such as this one:
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Scale-Free Networks
Scale-free networks fall off more
slowly than random ones.
Such networks are governed by a
power law of the form
Because of this power law relationship,
a log-log plot of P(k) versus k gives a
straight line.
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Scale-Free Networks
(Barabasi et al, 1998)
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Random versus scale-free
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Random (□) and scale free (○)
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Linear axes
Logarithmic axes
In scale-free networks, some nodes act as "highly
connected hubs" (high degree), although most nodes
are of low degree.
Scale-free networks' structure and dynamics are
independent of the system's size N, the number of
nodes the system has. In other words, a network that is
scale-free will have the same properties no matter what
the number of its nodes is.
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Scale-free networks can grow by the process of
preferential attachment : new links are made preferably
to hubs: the probability of a new link is proportional to the
links of a node.
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Some examples…
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Nodes: email-addresses, links:
emails
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Nodes: people, links: # of sexual
partners
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Protein network C.elegans
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The WWWeb is scale free
Web pages :
Inlinks and outlinks (red and blue)
Network nodes (green)
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100 000 Internet routers and the
physical connections between them
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Mycoplasma genitalium Metabolic Network
Degree distribution
Horizontally log of degree (=
number of connections),
vertically log of number of
genes with this degree
Mycoplasma genitalium
500 nm
580 Kbp
477 genes
74% coding DNA
Obligatory parasitic
endosymbiont
Metabolic Network
Nodes are genes, edges
are gene co-expressions
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Degree distributions in human gene coexpression network. Coexpressed genes
are linked for different values of the correlation r, King et al, Molecular Biology
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and Evolution, 2004
Social Networks
A social network is a social structure made of
nodes (which are generally individuals or
organizations) that are tied by one or more
specific types of interdependency, such as
values, visions, ideas, financial exchange,
friendship, kinship, dislike, conflict or trade.
The resulting graph-based structures are
often very complex.
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Many more examples…
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Relation to the Tasks
Task 3a. Global Social Networks
Task 3b. The neural network
structure of consciousness.
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END of LECTURE 3
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