How I plan to bother you today

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How I plan to bother you today
? Properties of real-world networks
? Models for those properties
? Discussion on applications (depending on time)
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What do we do in complex network?
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.
Property some individuals are very social
compared to other ones.
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.
Goal spread a saucy rumor!
Biological network
What is the influence of A over C in the social
network?
How strong is the connection from A to C?
Social Network
Idea whatever the network as long as its a
social network, try to target the social
individuals.
How likely is it that if A is infected by a virus
then C will get infected?
Blogs
Facebook
Population
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Transitivity
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1 triangle
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8 connected triples
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C 3/8 0,375
 There are high chances that a husband knows the
family of his wife. 
?Transitivity measures the probability that if A
is connected to B and B is connected to C, then A
is connected to C.
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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.
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).
The significance profile (SP) of the network is
a vector of those frequencies.
4 networks of different micro-organisms are
shown to have very similar TSPs, and in
particular the triad 7 called  feed-forward
loop .
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Lets play the Kevin Bacon Game.
Think of an actor or an actress
? If theyve 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.
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
Hollywoods world is pretty large. What do you
think is the average Bacon Number an american
actor will get?
Only 4 !
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The Small-World property
Through this Kevin Bacons experiment, we know
that although the network of actors is quite big,
the average distance is very small.
Global efficiency of small-world networks.
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.
Efficient to exchange information at a local
scale.
? An e-mail network of 59 812 nodes L 4.95 !
Efficient to exchange information at a global
scale.
? Actor network or 225 226 actors L 3.65 !
It tells you that transmitting information in
small-world networks will be very fast.
At a local level, we have strongly connected
communities.
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. Its a choice
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The Scale-Free property
However, there are things that have an enormous
variation in the distribution.
Many of the things we measure are centered around
a particular value.
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) -ax
c. Here its log-log, so
This value is the typical size.
ln p(x) -a ln x c
apply exponent e
p(x) ecx
c -a
We say that this distribution follows a
power-law, with exponent a.
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, its also called scale-free.
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The Scale-Free property
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
thats neither new nor unique.
Sizes of earthquakes
Wars
Moon craters
Number of citations received / paper
Solar flares
Number of hits on web pages
Computer files
Peoples annual incomes
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 its common, but to understand why
and which other structural parameters can be
there.
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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.
It was said that  the most highly connected
nodes represent an Achilles heel  delete them
and the graph breaks into pieces.
One mechanism was used to build scale-free
networks, called preferential attachment, or
 the rich get richer .
However, it only requires high variability and
not strict scaling
Recent research have shown that complex
networks that claimed to be scale-free have a
power-law but not this Achilles heel.
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.
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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 its 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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Random graphs
The earliest model was made formal by Erdos and
Rényi (although discovered independly by Rapoport
10 years before).
We have N vertices, and two vertices are
connected with probability P.
The goal was not to study properties such as
small-world or scale-free (it wasnt even known).
This theory is interested in the properties that
happen for large graph size, i.e. for n ? 8.
Suppose that a porous stone is immerged in a
bucket of water.
Is there a path from the center to the side?
Above a threshold pc, a cluster containing
vertices in the center and having path to the
side will appear!
What is the probability that the centre of the
stone is moistened?
To illustrate the main property of those
graphs, lets go through an example
Lets model it!
A fluid can flow through channels if they are
wide enough.
The system behaves very differently for p lt pc
and p gt pc its sharp!
A channel has a probability p of being wide
enough, and 1 p too small.
If we model this in two dimensions, we have some
grids (square lattice).
For a sharp transition, think of water in a glass
and the pc 0 degree.
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Modelling the small-world property
We first need to give an idea about a type of
graph called lattice graph.
A lattice is defined in d dimensions. The grid
we saw on the previous slide was a lattice in 2
dimensions, i.e. a 2-lattice.
K 4
K 8
This is a regular 2-lattice.
Each node has edges to its k nearest neighbours.
This is a regular 1-lattice with k 4.
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Modelling the small-world property
In the model by Watts and Strogatz, we begin with
a low-dimension regular lattice.
Does it have to go through a random process?
For each edge, we move randomly one of its ends
to another vertex with probablity p.
No! We also have deterministic constructions
same properties but more control.
The original graph was very clustered we keep
this high clustering.
And by creating shortcuts, we decrease the
average distance, i.e. create a small-world
effect.
Take a circulant graph (a) and add to it a double
step graph (b) you keep the clustering and you
have the shorcuts.
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Scale-free networks
Start with a complete graph of N vertices (a
dense group).
This construction was generalized in the family
of graphs Hn,k.
Make N 1 copies.
Start with a group of n nodes and iterate k steps
(i.e. build k levels).
Link the root (red) to all vertices but the
copies of the root.
It has a scale-free distribution of degrees.
It has a high clustering for a node with k
links, its clustering is C(k) 1/k.
Repeat the process.
Its diameter is 2k 1 (this network is not
small-world).
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Scale-free and small-world
This family Kn,k is defined in a similar way
than the previous one
Start at k 0 with a complete graph Kn.
same
Do k times the following
Add one vertex for each of the existing Kn and
connect it to all vertices of this Kn.
change
There is one K3 so we add one node.
For each of the 4 possible K3, we add one node.
Lets start our example with K3.
No other K3 in the graph next level.
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Preferences of newcomers
Another very popular way to model networks is
to reproduce the growth processes taking place in
the real world new nodes come in!
Im new! Who should I get in touch with?
Only one person knows Tracy, its unlikely that
we get in touch...
Bob and Ted are really cool guys, everybody knows
them so well meet
Herbert Simon showed in 1955 that power laws
are encountered when the rich get richer the
more we already have, the more we get.
So, the most common way of generating a
scale-free network is to use preferential
attachment.
? When a new node arrives, it prefers to link to
the most popular nodes.
In 1965, Derek de Solla Price set up a model
where the probability that a new node links to
another one is proportional to kin 1, where kin
is the incoming degree of the node.
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The Barabasi-Albert (BA) model
Prices model created a directed graph with
variables number of edges added at each node. It
gives the degree distribution pk k .
-(21/m)
Thirty years after, in 1999, Barabasi and
Albert came with their model undirected,
constant number of edges, always gives pk k .
-3
The BA model is the most famous and started
the field. Why?
They gave an important situation where this model
has a strong potential the Web.
You decided to start your website, and its
time to create a link section.
Most likely, you will link to popular websites,
making them even more popular (i.e. it creates a
feedback loop system ? rich get richer).
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Including assortativity in a model
Krapivsky and Redner have considered a directed
version of the BA model with degree-degree
correlations (which we dont have in BA).
In general, Sokolov and Xulvi-Brunet have
proposed a simple algorithm to make a network
assortative with a parameter p
1. Choose randomly two links.
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2. Order their four end-nodes with respect to
their degree.
3. Rewire with probability p to connect small
vertices and high ones together, or 1 p for
random.
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If you repeat, you make the network assortative
and you keep the degree distribution.
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Generalization of the BA model
The Dorogovtsev-Mendes-Samukhin (DMS) model
adds a parameter k0 to the equations of
preferential attachment. If k0 0, we have the
BA.
Krapivsky has shown that if we have a nonlinear
attachment probability, then we dont have a
power law but an exponential a single nodes take
all the newcomers.
The Albert and Barabasi (AB) model uses two
more parameters p and q to rewire changes on the
connections after a node has been attached.
Solé-Pastor Satorras-Smith-Kepler (SPSK) model
uses 3 mechanisms.
Duplicate (copy a randomly selected node with
its connection)
It produces power-law but unrealistic
assortativity and clustering.
Divergence (some connections of a duplicate are
removed)
A more realistic one with duplication and
divergence is the Vazquez Flammini Maritan
Vespignani (VFMV) model.
Mutate (connections are added to the duplicate)
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Triangle-generating protocol
In the same way that many sets of rules can
guide a dynamic process toward to be scale-free,
how can it guide it to be small-world?
Remember the example when you marry somebody,
its quite likely that you will get to know
his/her family (whether you want it or not).
Nodes are dynamically introduced to each other
by a common node
One random node chooses randomly two of its
neighbours and link them. If the node has less
than 2 neighbours, it links to a random node.
With probability p, a random node is removed
with its links, and replaced by a new node with a
randomly chosen neighbour.
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References
Three fundamental reviews on Complex Networks
Complement used for specific parts of this
presentation
For Network Motifs, see Superfamilies of evolved
and designed networks by Ron Milo et al., Science
303 1538 (2004).
The structure and function of complex networks,
M. E. J. Newman, SIAM Review, Vol. 45, No. 2. ,
pp. 167-256. , 2003.
For a number of mechanisms to get a power-law,
see Power laws, Pareto distributions and Zipfs
law, M. E. J. Newman (2006)
Towards a theory of Scale-Free Graphs
Definition, Properties and Implications, Lun Li,
David Alderson, Reiko Tanaka, John C. Doyle,
Walter Willinger, Technical Report
CIT-CDS-04-006, California Tech., Pasadena, 2005.
The story of the porous stone in the bucket of
water is called percolation theory. See Harry
Kesten, What is percolation?, Notices of the
American Mathematical Society Vol. 53 No. 5
(2006)
Complex networks Structure and dynamics, S.
Boccaletti, V. Latora, Y. Moreno, M. Chavez,
D.-U. Hwang, Physics Reports 424, pp. 175-308,
2006.
Francesc Comellas, Complex Networks
Deterministic Models, Physics and Theoretical
Computer Science, IOS Press (2007)