definations graph theory

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


2/50

Types of graphsTypes of graphs Weighted graphsWeighted graphs

MultigraphsMultigraphs PseudographsPseudographs

DigraphsDigraphs

Simple graphsSimple graphs


3/50

Weighted graphis a graph for which each edge has an associatedweight, usually given by a weight function

w: Ep R, generally positive


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


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


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.


7/50

Adjacency Matrix of Multigraphs

02140

20100

11030

40301

00012

E

D

C

B

A

EDCBA


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


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


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?


11/50

Degree Centrality

B

ED

C

A

14

3

1

1

degree

00010

00010

00011

1110100110

E

D

C

B

A

EDCBA


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


13/50


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


14/50


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


15/50

Closeness centrality

j

jid

N

iCC ,

1

The normalised inverse of the sum of

topological distances in the graph.


16/50

B

ED

C

A

02212

20212

22011

11101

22110

E

D

C

B

A

EDCBA

!

n

j

jid1

,

64

6

7

7



17/50


B

ED

C

A Closeness

0.67

1.00

0.67

0.570.57


18/50

Node B is the most central one in spreading

information from it to the other nodes in the

network.



19/50

B

ED

C

A

Local metrics

Node B is the most central oneaccording to the degree,betweenness and closenesscentralities.


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


21/50

Degree: Difficulties


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

!


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


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.


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


26/50

Eigenvector Centrality:

Difficulties

In regular graphsall the

nodes have exactly thesame value of theeigenvector centrality,which is equal to

n

1


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)


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


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


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


31/50

Subgraph Centrality

Nodes EE(i)1,2,8 3.9024,6 3.7053,5,7 3.638


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

!!

!

,),()(,),()(

,),()(


33/50

Subgraph Centrality:

Comparisons

Nodes EE(i)

45.696

45.651

VjVijECiECVjVijBCiBC

VjVijCCiCC

VjVijDCiDC

!!

!

!

,),()(,),()(

,),()(

,),()(


34/50

Path of length 6 Walk of length 8

Shortest path

Communicability


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


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


37/50

1!j 01 "pJ Vp

Communicability


38/50

Communicability

2uj

qp jj JJ sgnsgn ! qp jj JJ sgnsgn {

pp

q

q


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


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


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


42/50

G5

communicability

graph

!G !5 G

pqG ? A1,0



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.



44/50

Social (Friendship) Network

Communities: Example


45/50

Communities: Example

The Network

Its CommunicabilityGraph


46/50

Communities

Social Networks


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.


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.


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?


50/50

Exercise 2. Illustration.

definations graph theory

Documents