6d99ea623943731bf87c628b005be0d2 reseaux oct008version etudiants .ppt

8/13/2019 6d99ea623943731bf87c628b005be0d2 Reseaux Oct008version Etudiants .Ppt

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Rseaux complexes Rels

ENSAT Tanger, Mounir MAOUENE

Intelligence Artificielle; GINF4 , GSEA4 & Master IAB

Automne 2008


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Quest-ce quun graphe G ?

V

E


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relations humaines

hyperliens

sites informatiques relis par

des liens

Individus; ordinateurs; pages

web; aroports ; molcules


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The order of graph

Cardinal of E : eG = e(G) = |E| = |E(G)|

and eG(A,B) is the number of edges of tow nodes.

The degree of v (d(v))

The number of vertices of one node v is called the degree of v.

In-degrees; Out-degrees

The degree of v =1

A graph leave is a node that its degree is 1k-regular graph

The d(v) is the number of edges that contain it.

Gis k-regularif every vertex has degree k.


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PowersetGiven a set S, the powerset of S, (P(S), or 2s) is the set of all S subsets

Hyperedgeis an edge that is allowed to take on any number of vertices,

possibly more than 2.

Hypergraph (H)Is a graph that allows any hyperedge

H = G= (V,E) Eis a set of non-empty subsets of V called hyperedges

or links. Eis a subset of P(V)/{} ,, where P(V) is the powerset of V.A hypergraph is also called a set systemor a family of sets

HA=(A, {eiA/ eiA }), A subset of V, eiEExample: E={HAS_A_MANE , HAS_CLAWS, HAS_TEETH} =

{{HORSE LION ZEBRA} {CAT BEAR TIGER} {ALLIGATOR BEAR LION TIGER}}

NB: Unlike graphs, hypergraphs are difficult to draw on paper


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Isomorphism of a hypergraphH= (V,E) is to a hypergraph G= (V,E), HG if there exists a bijection

: VV and a permutation of IIN such that (ei) = f(i).His strongly isomorphicto Gif the permutation is the identity. One

then writes H G. Note that all strongly isomorphic graphs are

isomorphic, but not vice-versa

Examples .

Consider the hypergraph Hwith edges

H= {e1= {x,y},e2= {y,z},e3= {z,u},e4= {u,x},e5= {y,u},e6= {x,z}} and

G= {f1= {,},f2= {,},f3= {,},f4= {,},f5= {,},f6= {,}}

Then clearly Hand Gare isomorphic (with (x) = , etc.), but they are

not strongly isomorphic, because e1e4e6= {x} and f1f4f6=

A hypergraph automatismis an isomorphism from a vertex set into itself


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The rank r(H) of a hypergraphedge in H, r is the maximum cardinality of the edge

Uniform or k-uniform hypergraphIf all edges have the same cardinality k

Exercise: A graph is a ?-uniform hypergraph

Two vertices (edges) are symmetric- v, w are called symmetric if there exists an automorphism such that(x) = y.

- Two edges eiand ejare said to be symmetricif there exists an

automorphism such that (ei) = ej.

Hypergraph vertex (or edges)-transitive/symmetric)if all of its vertices/edges are symmetric.

hypergraph is transitive.If a hypergraph is both edge- and vertex-symmetric


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V= {v1, v2, v3,, vn} and E= {e1, e2, e3,, em}

Associated to any hypergraph is the incidence matrix A = (aij)

where aij

= 1 if vi

ej

; 0 otherwise

For example, let V = {a, b, c} and E = {{ a }, {a b}, {a c} {a b c}}

1 1 1 1

A = 0 1 0 10 0 1 1


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A transversal (or hitting) set- Let a set T H = (X, E), if [T (e: edge) ] then T is called atransversal

- A transversal Tis called minimal if no proper subset of Tis a

transversal.

The transversal hypergraphThe transversal hypergraph of H is the hypergraph (X, F) whose

edge set Fconsists of all minimal transversals of H.

A transversal hypergraph has applications in machine learning,

game theory indexing of database


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Metric graphseach edge has been associated with an int erval [0, e]

so thatxis the coordinate on the interval, the vertex v1

corresponds tox= 0 and v2to x = eor vice versa. The graph has a

natural metric: for two pointsx,yon the graph, (x,y) is the shortest

distance between them where distance is measured along the edges of

the graph


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The neighborhood of v

denoted NG(v) is a set of vertices adjacent to v not including v itself

Partial graph of G

G = (V, E) and G = (V, E) G is called partial graph of G if V = V

and E E.

Subgraph of G Let G = (V, E) and G = (V, E) G is subgraph of G if : V V and

E E.

Complete graph is a graph in which every node is linked to every other node. A

complete subgraph is called a clic


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Regular graphA graph in which every vertex has the same degree.

Graph = k-regular if every vertex has degree k

Diameter of graph GThe maximum length, among all pairs of vertices in G, of a shortest

path between each pair.

Connect graphThere is a path from any point to any other point in the graph


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Real-World Networks

Network theoryConcerns the study of graphs as a representation of relations

between discrete objects.

Size of a network

The size of a network can be determined in terms of the number ofedges in the network

Density of a networkSize of network/ number of all possible ties

Application:- the speed at which information diffuses among the nodes

- extent to which actors have high levels of social capital and/or social

constraint.

- measure of connectivity of the network

C l t k


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Complex networkIs a network that has certain non-trivial features that do not occur in

simple networks as:

- a heavy-tail in the degree distribution;

- a high clustering coefficient,

- assortativity or disassortativity among vertices;

- community and hierarchical structure at many scales; and


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WALK

connection between two nodes in a graph is called a walk.

A closed walkwhere the beginning and end point of the walk are the same node.

Unrestricted Walksa walk can involve the same node or the same relation multiple times.

Trail between two nodesconnection between two nodes, is any walk that includesconnection at most once


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Eccentricity of tow nodesFor each node, to calculate the distribution of its geodesic distances

to the other nodes.

Diameter of a networkThe diameter of a network is the maximum eccentricity over all the

actors of the network,

NB: Vertices with maximum eccentricity are calledperipheral vertex.

Radius of a networkIs the minimum eccentricity over all the nodes of the network.

NB: Vertices with minimum eccentricity are called the centre


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RANDOM GRAPH

RG is a graph that is generated by some random process The theory

of random graphs is the union between graph theory and probability

theory.

A random graph is obtained by starting with a set of nvertices andadding edges between them at random.

Different random graph modelsproduce different probability

distribution on graphs

P b bili di ib i


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Probability distributiongiven a random variable X: Y between a probabilty space (, F, P),

the sample space, and a mesurable space (Y, ) called the state

space, a probability distribution on (Y, ) is a probability measure P:

[0 1]on the state space whereThe probability distribution Pr of a real-valued random variableXis

completely characterized by its cumultive distribution function:

F(x)=Pr[X x]; x IR

Discrete probability distribution

Discrete distributions are characterized by a probability mass function ,

psuch that Pr[X = x] = p(x)

Continuous probability distribution

These distributions can be characterized by probability density functionfdefined on the real numbers such that

f(x) = Pr[X x] =

x

dttf )(


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Important probability distributions

1- Bernoulli distribution:for 0


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3- Poisson distributionPoisson distributionexpresses the probability of a number

of events occurring in a fixed period of time if these events occur

with a known average rate and independently of the time since the last

event.f(k, ) =

with k ={1, 2, 3,} is the number of occurrences of an event,

is a positive real number, equal to the expected number of

occurrences that occur during the given interval.

The Poisson distribution can be derived as a limiting case of the Binomial

distribution. The Poisson distribution can be applied to systems with a

large number of possible events.

Mean = var =

For sufficiently large values of , (say >>>), the normal distribution with

mean , and variance , is an excellent approximation to the Poisson

distribution.

!k

e k


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4- The normal distributionalso called the Gaussian distribution, is an important family of,

applicable in many fields Each member of the family may be

defined by two parameters, locationand scale: the mean ("average",) and variance (standard deviation squared, 2) respectively.

Que vaut f0,1(x) ?

if = 0 and the max of f is over y; the value of 2 grows when the valueof max of f over y decreases.

)

2

)(exp(

2

1)(

2

2

, 2

xxf


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Stochastic process (random process)

Given a probability space (, F, P), a stochastic process(or randomprocess) with state spaceXis a collection ofX-valued random

variables indexed by a set T("time"). That is, a stochastic process F

is a collection {Ft: t T}where each Ftis anX-valued random

variable. A modificationGof the process Fis a stochastic process

on the same state space, with the same parameter set Tsuch that

P(Ft = Gt)=1 t T.

Familiar examples of time seriesinclude stock market, signals such as

speech, audio and video; medical data, blood pressure or

temperature; and random movement such as brownian motion or

random walks.


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Most commonly studied is the random graph (Erds-Rnyi model),

denoted G(n,p), in which every possible edge occurs independently

with probability p.

A closely related model, denoted G(n,m), assigns equal probability to

all graphs with exactly medges. This model can be viewed as a

snapshot at a particular time of the random graph process, which

is a stochastic process that starts with nvertices and no edges and

at each step adds one new edge chosen uniformlyfrom the set ofmissing edges.

If there are n vertices in a graph, and each is connected to an average

of z edges, then it is trivial to show that p = z/(N 1), which for large

N is usually approximated by z/N. The number of edges connectedto any particular vertex is called the degree k of that vertex, and has

a probability distribution pkgiven by (in the limit where n kz)


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Small world Model

Watts & Strogatz Algorithm

N is the number of nodes the mean degree k such N >> k >> ln(N)>> 1

and 0 1 . This model constructs a undirected graph with Nnodesand Nk/2 edges:

1- Construct a regular ring lattice (circle), a graph with Nnodes each

connected to Kneighbors. That is, if the nodes are labeled n0...nN 1,

there is an edge (ni,nj) if and only if ij k (mod k) ;k(1, k/2) .

2- For every node n0 i N-1take every edge (ni,nj) with i


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4-regular Small world random


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A Real Network: World Wide Web

The nodes of the network are the documents (web pages)

The edges are the hyperlinks (URLs) that point from one documentto another (see fig. www)

The size of this network was close to one billion nodes at the

end of 1999 (Lawrence and Giles, 1998, 1999).

The World Wide Web are directed, the network is characterized by

two degree distributions: the distribution of outgoing edges, Pout(k),

and the distribution of incoming edges, Pin(k),

Pout(k) and Pin(k) have power-law tails:



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The nodes of the network are the documents (web pages)The edges are the hyperlinks (URLs) that point from one document to another

(see fig. www)

The size of this network was close to one billion nodes at the

end of 1999 (Lawrence and Giles, 1998, 1999).

The World Wide Web are directed, the network is characterized by two degree

distributions: the distribution of outgoing edges, Pout(k),

and the distribution of incoming edges, Pin(k),

Pout(k) and Pin(k) have power- law tails:


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The World Wide Web displays the small-world property (Albert,Jeong, and Baraba si (1999))

Network Size (k) llengt

h

path

lrand C:cluster ing

C rand Reference

WWW, site

level, undir.153 127 35.21 3.1 3.35 0.1078 0.00023 Adamic, 1999

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