definations graph theory
Post on 08-Apr-2018
220 Views
Preview:
TRANSCRIPT
-
8/7/2019 Definations graph theory
1/50
Introduction to NetworkIntroduction to Network
Theory:Theory:Modern Concepts, AlgorithmsModern Concepts, Algorithms
and Applicationsand ApplicationsErnesto EstradaErnesto Estrada
Department of Mathematics, Department of PhysicsDepartment of Mathematics, Department of Physics
Institute of ComplexInstitute of Complex SystemsSystems at Strathclydeat Strathclyde
University of StrathclydeUniversity of Strathclydewww.estradalab.orgwww.estradalab.org
-
8/7/2019 Definations graph theory
2/50
Types of graphsTypes of graphs Weighted graphsWeighted graphs
MultigraphsMultigraphs PseudographsPseudographs
DigraphsDigraphs
Simple graphsSimple graphs
-
8/7/2019 Definations graph theory
3/50
Weighted graphis a graph for which each edge has an associatedweight, usually given by a weight function
w: Ep R, generally positive
-
8/7/2019 Definations graph theory
4/50
07.05.01.20
7.001.200
5.01.204.30
004.305.1
0005.10
E
D
C
B
A
EDCBA
Adjacency Matrix of Weighted
graphs
-
8/7/2019 Definations graph theory
5/50
Degree of Weighted graphsThe sum of the weights associated to every edgeincident to the corresponding node
The sum of the corresponding row or column ofthe adjacency matrix
07.05.01.20
7.001.200
5.01.204.30004.305.1
0005.10
E
D
C
B
A
EDCBA Degree1.5
4.96
2.83.3
-
8/7/2019 Definations graph theory
6/50
Multigraph or pseudographis a graph which is permitted to have multipleedges. Is an ordered pair G:=(V,E) withV a set of nodesE a multiset of unordered pairs of vertices.
-
8/7/2019 Definations graph theory
7/50
Adjacency Matrix of Multigraphs
02140
20100
11030
40301
00012
E
D
C
B
A
EDCBA
-
8/7/2019 Definations graph theory
8/50
Directed Graph (digraph)Directed Graph (digraph) Edges have directionsEdges have directions
The adjacency matrix is not symmetricThe adjacency matrix is not symmetric
01000
10100
1001020010
00010
E
D
C
B
A
EDCBA
-
8/7/2019 Definations graph theory
9/50
Simple GraphsSimple graphs are graphs without
multiple edges or self-loops. They areweighted graphs with all edge weightsequal to one.
B
ED
C
A
-
8/7/2019 Definations graph theory
10/50
Local metrics
Local metrics provide a measurement of a
structural property of a single node Designed to characterise
Functional role what part does this node
play in system dynamics? Structural importance how important is this
node to the structural characteristics of thesystem?
-
8/7/2019 Definations graph theory
11/50
Degree Centrality
B
ED
C
A
14
3
1
1
degree
00010
00010
00011
1110100110
E
D
C
B
A
EDCBA
-
8/7/2019 Definations graph theory
12/50
Betweenness centrality
The number of shortest paths in the graph
that pass through the node divided by thetotal number of shortest paths.
kjiji jkikBC i j {{! ,, ,,VV
-
8/7/2019 Definations graph theory
13/50
Betweenness centrality
B
Shortest paths are:
AB, AC, ABD, ABE, BC, BD,
BE, CBD, CBE, DBE
B has a BC of 5
A
C
D E
1,;1,,
1,;1,,1,;1,,
1,;1,,
1,;1,,
EDEBD
ECEBC
DBDBC
EAEBA
DADBA
VV
VVVV
VV
VV
-
8/7/2019 Definations graph theory
14/50
Betweenness centrality
Nodes with a high betweenness centrality
are interesting because they control information flow in a network
may be required to carry more information
And therefore, such nodes
may be the subject of targeted attack
-
8/7/2019 Definations graph theory
15/50
Closeness centrality
j
jid
N
iCC ,
1
The normalised inverse of the sum of
topological distances in the graph.
-
8/7/2019 Definations graph theory
16/50
B
ED
C
A
02212
20212
22011
11101
22110
E
D
C
B
A
EDCBA
!
n
j
jid1
,
64
6
7
7
Closeness centrality
-
8/7/2019 Definations graph theory
17/50
Closeness centrality
B
ED
C
A Closeness
0.67
1.00
0.67
0.570.57
-
8/7/2019 Definations graph theory
18/50
Node B is the most central one in spreading
information from it to the other nodes in the
network.
Closeness centrality
-
8/7/2019 Definations graph theory
19/50
B
ED
C
A
Local metrics
Node B is the most central oneaccording to the degree,betweenness and closenesscentralities.
-
8/7/2019 Definations graph theory
20/50
Which is the most central node?
A
B
and the winner is
A is the most centralaccording to thedegree
B is the most centralaccording to closeness
and betweenness
-
8/7/2019 Definations graph theory
21/50
Degree: Difficulties
-
8/7/2019 Definations graph theory
22/50
Extending the
Concept of DegreeMake xi proportional to the average of the centralitiesof its is network neighbors
where P is a constant. In matrix-vector notation we
can write
In order to make the centralities non-negative we selectthe eigenvectorcorresponding to the principal eigenvalue(Perron-Frobenius theorem).
j
n
j
iji xAx !
!1
1P
AxxP1
!
-
8/7/2019 Definations graph theory
23/50
Eigenvalues and Eigenvectors
The value is an eigenvalue of matrix A ifthere exists a non-zero vector x, such that
Ax=x. Vector x is an eigenvector ofmatrix A The largest eigenvalue is called the principal
eigenvalue
The corresponding eigenvector is the principaleigenvector
Corresponds to the direction of maximumchange
-
8/7/2019 Definations graph theory
24/50
Eigenvector Centrality
The corresponding entry of the principal
eigenvector of the adjacency matrix of thenetwork.
It assigns relative scores to all nodes in thenetwork based on the principle thatconnections to high-scoring nodescontribute more.
-
8/7/2019 Definations graph theory
25/50
Node EC
1 0.500
2 0.2383 0.2384 0.5755 0.354
6 0.3547 0.1688 0.168
Eigenvector Centrality
-
8/7/2019 Definations graph theory
26/50
Eigenvector Centrality:
Difficulties
In regular graphsall the
nodes have exactly thesame value of theeigenvector centrality,which is equal to
n
1
-
8/7/2019 Definations graph theory
27/50
Subgraph Centrality
Aclosed walk of lengthkin a graph is a succession
of k (not necessarilydifferent) edges startingand ending at the samenode, e.g.
1,2,8,1 (length 3)
4,5,6,7,4 (length 4)
2,8,7,6,3,2 (length 5)
-
8/7/2019 Definations graph theory
28/50
!
1//
.
.
01
10
A
!
1
2
1
3
3
3 Q
Q
A
!
1
2
1
2
2
2 Q
Q
A
Subgraph Centrality
The number ofclosedwalk of length kstarting
at the same node i is givenby the ii-entry of the kthpower of the adjacencymatrix
iikki A!Q
-
8/7/2019 Definations graph theory
29/50
ic
AcAcAcAcAciEE
l
l
l
iiiiiiiiii
Qg
!
!
!
0
4
4
3
3
2
21
0
0
.
Subgraph Centrality
We are interested in giving weights in decreasingorder of the length of the closed walks. Then,
visiting the closest neighbors receive more weightthat visiting very distant ones.
The subgraph centrality is then defined as the
following weighted sum
-
8/7/2019 Definations graph theory
30/50
Subgraph Centrality
By selecting cl=1/l! we obtain
where eA is the exponential of the adjacency
matrix.For simple graphs we have
g
!!
0 !ll
liiEE Q iieiEE
A!
? A!
!n
j
jjeixiEE
1
2 P
-
8/7/2019 Definations graph theory
31/50
Subgraph Centrality
Nodes EE(i)1,2,8 3.9024,6 3.7053,5,7 3.638
-
8/7/2019 Definations graph theory
32/50
Subgraph Centrality:
Comparsions
Nodes EE(i)
1,2,8 3.9024,6 3.73,5,7 3.638
Nodes BC(i)
1,2,8 9.5284,6 7.1433,5,7 11.111
VjVijECiEC
Vj
Vij
CCi
CC
VjVijDCiDC
!!
!
,),()(,),()(
,),()(
-
8/7/2019 Definations graph theory
33/50
Subgraph Centrality:
Comparisons
Nodes EE(i)
45.696
45.651
VjVijECiECVjVijBCiBC
VjVijCCiCC
VjVijDCiDC
!!
!
!
,),()(,),()(
,),()(
,),()(
-
8/7/2019 Definations graph theory
34/50
Path of length 6 Walk of length 8
Shortest path
Communicability
-
8/7/2019 Definations graph theory
35/50
k
pq
sk
k
s
pqspq WcPbG g
"
!
s
pqP
Let be the number ofwalks of length k>sbetween p and q.
Let be the number ofshortest paths of length s
between p and q.
kpqW
DEFINITION (Communicability):
sb and must be selected such as the communicability converges.kc
Communicability
-
8/7/2019 Definations graph theory
36/50
pqk
pq
k
pq ek
G AA !! g!0 !
jeqxpx jn
j
jpq
P
!!
1
Communicability
By selecting bl=1/l! and cl=1/l! we obtain
where eA is the exponential of the adjacency
matrix.For simple graphs we have
-
8/7/2019 Definations graph theory
37/50
1!j 01 "pJ Vp
Communicability
-
8/7/2019 Definations graph theory
38/50
Communicability
2uj
qp jj JJ sgnsgn ! qp jj JJ sgnsgn {
pp
q
q
-
8/7/2019 Definations graph theory
39/50
u
u
u
u
!
22
2211
1
j
jj
j
jj
jjj
jjjpq
jj
jj
eqpeqp
eqpeqpeqpG
PP
PPP
JJJJ
JJJJJJ
intracluster
intercluster
Communicability
jj eqeqp jj
j
j
jpq
PP NNNN
!
!
!(clusterinter
2
j
clusterintra
2
p
-
8/7/2019 Definations graph theory
40/50
Communicability &
CommunitiesA community is a group of nodes for wich theintra-cluster communicability is larger than the
inter-cluster one
These nodes communicates better among themthan with the rest of extra-community nodes.
jj eqeqp jj
j
j
j
PPNNNN
!
!
"clusterinter
2
clusterintra
2
p
-
8/7/2019 Definations graph theory
41/50
e
"!5
0if0
0if1
x
xxLet
The communicability graph 5(G) is the graphwhose adjacency matrix is given by 5((G)) results
from the elementwise application of the function5(G) to the matrix ((G).
Communicability Graph
-
8/7/2019 Definations graph theory
42/50
G5
communicability
graph
!G !5 G
pqG ? A1,0
Communicability Graph
-
8/7/2019 Definations graph theory
43/50
Acommunity is defined as a clique
in the communicability graph.
Identifying communities is reduced
to the all cliques problem in the
communicability graph.
Communicability Graph
-
8/7/2019 Definations graph theory
44/50
Social (Friendship) Network
Communities: Example
-
8/7/2019 Definations graph theory
45/50
Communities: Example
The Network
Its CommunicabilityGraph
-
8/7/2019 Definations graph theory
46/50
Communities
Social Networks
-
8/7/2019 Definations graph theory
47/50
ReferencesAldous & Wilson, Graphs and Applications. AnIntroductory Approach, Springer, 2000.
Wasserman & Faust, Social Network Analysis,Cambridge University Press, 2008.
Estrada & Rodrguez-Velzquez, Phys. Rev. E
2005, 71, 056103.
Estrada & Hatano, Phys. Rev. E. 2008, 77,036111.
-
8/7/2019 Definations graph theory
48/50
Exercise 1Identify the most central node according to the followingcriteria:
(a) the largest chance of receiving information from closestneighbors;(b) spreading information to the rest of nodes in thenetwork;(c) passing information from some nodes to others.
-
8/7/2019 Definations graph theory
49/50
Exercise 2T.M.Y. Chan collaborates with 9 scientists incomputational geometry. S.L. Abrams also collaborates with
other 9 (different) scientists in the same network. However,Chan has a subgraph centrality of 109, while Abrams has 103.The eigenvector centrality also shows the same trend,EC(Chan) = 10-2; EC(Abrams) = 10-8.
(a) Which scientist has more chances of being informed aboutthe new trends in computational geometry?(b) What are the possible causes of the observed differencesin the subgraph centrality and eigenvector centrality?
-
8/7/2019 Definations graph theory
50/50
Exercise 2. Illustration.
top related