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:
A Real Network: World Wide Web
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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 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