Analysis of complex networks

rucksackbulgeAI and Robotics

Dec 1, 2013 (3 years and 9 months ago)

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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).