Analysis of complex networks
Philippe Giabbanelli, MOPS&ISNO projects
Intro
When to think about networks
Main course
Properties of complex networks
(small

world, scale

free…)
Side dish
Using a software: Visone
(not the only one!)
Dessert
MoCSSy’s special: tea & cookies
Analysis of complex networks
When to think about networks
Let’s start from something we know…
cellular automata.
Analysis of complex networks
When to think about networks
Here is a neighborhood. Lets say we study some changes in houses.
It’s easy to think of that
space as a nice grid.
Then you can design rules such
«
if most neighbours of X do
barbecues then so does X
».
You can still create a
mapping between this
place and an automaton.
It may be a hassle…
Analysis of complex networks
You want to
study roads?
Sure, nice grid,
get a cellular
automaton
!
or not…
You still can model these roads as cellular automata using a CA with
a dimension greater than 2.
A network, or graph, has a more general
structure
than a CA.
node
edge
The question is not «
can we
model everything using one tool
», but
is it really the most
convenient
? What about the
analysis
of the
model?
Use as a graph when you want to be able to
analyze the structure
. What
is the interchange through the highest number of roads (
i.e.
a
vulnerable
point)? Which roads are the most
central
to travel between two places?
Analysis of complex networks
When to think about networks
Lets look at a second example: populations in epidemiology.
S
I
R
susceptible
infected
removed
α
β
In a compartment model, you consider that people are in three possible
states, and that there are some probabilities to move between these states.
This
ignores the topology
of the population.
People are all connected to each other:
regardless of who you are
, you
have a probability
α
to get infected (even if you don’t see anybody…).
Analysis of complex networks
When to think about networks
Lets look at a second example: populations in epidemiology.
If we want to bring the population in, we represent it as a
graph
: nodes
are people, and edges exist between 2 persons if they know each other.
Who
should we vaccinate first?
We can answer that question
only
if we can analyze the structure (graph).
Analysis of complex networks
Properties of complex networks
slides reused from a presentation in CMPT880
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
2
•
We analyze the structure of (big) networks from the real

world to
understand which properties are underlying them.
• If a
general class of network has a given property
then we can use it to
reason about any
unknown
network of this class.
What do we do in complex network?
Social Network
Biological
network
Blogs
Facebook
Population
Property
: some individuals are very
social compared to other ones.
Goal
: spread a saucy rumor!
Idea
: whatever the network as long as
it’s a social network, try to target the
social individuals.
• There is a tremendous number of applications and since the main two
properties were discovered in 1998, there has been hundreds if not
thousands of papers on complex networks.
How strong is the
connection from A to C?
What is the influence of A over C in
the social network?
How likely is it that if A is infected
by a virus then C will get infected?
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
3
Transitivity
«
There are high chances that a husband knows the family of his wife.
»
1
1
1
1
2
2
2
►
Transitivity measures the probability that if A is connected to B and B
is connected to C, then A is connected to C.
1 triangle
8 connected triples
C = 3/8 = 0,375
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
4
Network Motifs
• When we looked for transitivity, we basically counted the number of
subgraphs of a particular type (triangles and triples).
• We can
generalize this approach to see which patterns are ‘very
frequent’
in the network. Those patterns are called
network motifs
.
• To measure the frequency, we compare with how expected it is to see
such patterns in a random network.
• The
significance profile
(SP) of the network is a vector of those
frequencies.
• For each subgraph, we measure its relative frequency in the network.
• As we are measuring for the 13 possible directed connected graphs of 3
vertices, it is called a
triad
significance profile (TSP).
• 4 networks of
different
micro

organisms are shown to have
very similar
TSPs, and in particular the triad 7 called «
feed

forward loop
».
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
5
Let’s play the
Kevin Bacon Game
.
Think of an actor or an actress…
→
If they’ve been in a film with him, they have Bacon Number one.
→ Otherwise, if they have been in a film with somebody who has Bacon
Number one, then they have Bacon Number two, etc.
Hollywood’s world is pretty large. What do you think is the average
Bacon Number an american actor will get?
Only 4 !
Laurence Fishburne (alias
Morpheus
in
Matrix
)
Played with Kevin Bacon in
Mystic
Rivers
!
Mos Def (in
The Italian Job
)
played with Kevin Bacon in
The
Woodsman
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
6
The Small

World property
• Through this Kevin Bacon’s experiment, we know that although the
network of actors is quite big, the
average distance is very small
.
• A network is said to have the small

world property if the average
shortest path L is
at most logarithmically
on the network size N.
→
An e

mail network of 59 812 nodes… L = 4.95 !
→
Actor network or 225 226 actors… L = 3.65 !
• It tells you that transmitting information in small

world networks will
be very fast.
And so, transmitting viruses will be fast too…
•
Some
authors defined the small

world property with an additional
constraint with the presence of a high clustering. It’s a choice…
At a local level, we have strongly
connected communities.
Efficient to exchange
information at a local scale.
Efficient to exchange
information at a global scale.
Global efficiency of small

world networks.
This value is the typical
size.
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
Many of the things we
measure are centered
around a particular value.
However, there are things that have an
enormous
variation
in the distribution.
If we plot this histogram with logarithmic horizontal
and vertical axis, a pattern will clearly emerge: a
line
.
In a normal histogram, this line is p(x) =

α
x + c. Here it’s log

log, so:
ln p(x) =

α
ln x + c
apply exponent e
p(x) = e
c
x
c

α
We say that this distribution follows a
power

law
, with exponent
α
.
7
The Scale

Free property
A power law is the only distribution that is the same whatever scale we
look at it on,
i.e.
p(bx) = g(b)p(x). So, it’s also called
scale

free
.
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
8
We found that the population has the scale

free property!
In 1955, Herbert Simon already showed that many systems follow a
power law distribution, so that’s neither new nor unique.
• Sizes of earthquakes
• Moon craters
• Solar flares
• Computer files
• Wars
• Number of citations received / paper
• Number of hits on web pages
• People’s annual incomes
The Scale

Free property
It has been found that the distribution of the degree of nodes follows a
power

law in many networks,
i.e.
many networks are scale

free
…
What is important is not so much to find a power

law as it’s common, but
to understand
why
and which
other structural parameters
can be there.
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
9
The Scale

Free property
Myth and reality
• Scaling distributions are a subset of a larger family of heavy

tailed
distributions that exhibit high variability.
• One important claim of the
litterature for scale

free networks
was the presence of highly
connected central hubs.
•
However, it
only requires high
variability and not strict scaling
…
• It was said that «
the most highly
connected nodes represent an
Achilles’ heel
»: delete them and the
graph breaks into pieces.
• Recent research have shown that
complex networks that claimed to be
scale

free
have a power

law but not
this Achilles’ heel
.
• One mechanism was used to build
scale

free networks, called
preferential attachment
, or «
the rich
get richer
».
• It is only one of several, and not less
than 7 other mechanisms give the same
result, so preferential attachment gives
little or no insight in the process
.
Basic Properties
–
Small World
–
Scale Free
–
Other measurements
10
Other measurements
• We have the clustering, distribution of degree, etc.
Are there other
global characteristics relevant to the performances
of the network, in
term of searchability or stability?
• Rozenfeld has proposed in his PhD thesis to
study the cycles
, with
algorithms to approximate their counting (as it’s exponential otherwise).
• Using cycles as a measure for complex networks has received attention:
Inhomogeneous evolution of subgraphs and cycles in complex
networks
(Vazquez, Oliveira, Barabasi. Phys. Rev E71, 2005).
Degree

dependent intervertex separation in complex networks
(Dorogovtsev, Mendes, Oliveira. Phys Rev. E73 2006)
• See also studies on the
correlation of degree
(
i.e.
assortativity).
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