Dr. Henry HexmoorDepartment of Computer Science

Southern Illinois University Carbondale

Network Theory:Computational Phenomena and Processes

Social Network Analysis

Degree, Indegree, Outdegree Centrality

Degree Centrality:

Indegree Centrality:

Outdegree Centrality:

n

jijD xiC

1

)(

n

jjiI xiC

1

)(

n

iijO xiC

1

)(

Eigenvector Centrality=CE (i)=I’th entry of eigenvector e

e = largest eigenvalue of adjacency matrix

Percentagen

iCCentralitynormalizediC DD

1)()(

V

)( ],...,,[ 21 n

Betweenness Centrality

kjiijikjkCB ,)(

Normalized betweenness=

= number of geodesic linking across i and j has pass through node k.

2)2)(1()(

nnkCB

ikj

Closeness Centrality

n

jijc diC

1

)(

))(()(:1

n

jij jCiCCentrality

Scaling factor

Adjustment

One-node network

1-Connection 0-No connection One-mode network: Actors are tied to one another considering one type of relationship; i.e Binary Adjacency matrix

v1 v2 v3 v4 v5

v1 0 1 0 1 1

v2 0 0 0 1 0

v3 0 0 0 0 1

v4 0 0 0 0 0

v5 0 1 0 0 0

Two-node network

Two-node network: Actors are tied to events.• Incident network • Bipartite graph

e.g. Student attending classes

Affiliation networkActors are tied to ----Organization/Attributes;e.g. Affinity network, Homophily network

Sociogram ≡

Org1……………………..…….Org n Attribute m1………………..Attribute mn

12..n

12..n

{ }Points-------------individualsLines --------------relationship

People Attributes

Centrality Star graph

A has higher degrees. A is central to all.

Centrality: Quantifying a network node.(i)=

ij

Normalized Centrality

Centrality: Normalized Centrality:

A is more central than F

A

F D B

CE

(A)=4 (A)= ‘ (A) 6-1

45

= = 80%

(B)=3 (C)=2 (D)=2 (E)=2 (F)=1 , , , ,

Directed network centrality:

Prestigue of A=B=C=E=2 (Indegree)

A

F D

C

B

(A)=3 (F)=1

(A)=2 (F)=

(B)=2 (D)=2 (E)=2

(A)= ‘ (A) 6-1

25= = 40% (Normalize centrality)

E

Eigen vectorsVector X is a matrix with n rows and column, linear operator A, maps the vector X to matrix product AX

𝑥1𝑥2....𝑥𝑛

→

𝑥1𝑥2....𝑥𝑛

¿

𝑦1𝑦 2

.

.

.

.𝑦𝑛

𝑋→

A ¿ƛ . A

Eigen Value

•   3−3

to the equations ¿ 𝐴− ƛ . I∨≠∅

Second degree centrality Consider this 16 degree graph network:

NodeA

Value.534

B .275

C .363

D .199

E .199

F .199

G .199

H .102

I .102

J .164

K .164

L .164

M .164

N .164

O .441

Eigen value centrality(A)= (O)=6 higher Eigen values

M

N

L

KJ

C A

B

DG

H

I

E

F

O

Betweenness Centrality

Betweennes centrality measures the extent to which a vertex lies on paths between other vertices.

(K)=

=Number of paths from i to j passing through k

= Number of shortest distance path from i to j

K= Geodesic distance

Normalized centrality:

A

D

CE

F B

Node No of distinct path from the node

Normalized(C’)

E 4.0 40%

A 3.5 35%

D 1.0 10%

B 0.5 5%

C 0.0 0%

F 0.0 0%

=

Closeness CentralityCloseness centrality is the mean distance from a vertex to other vertices.

A

D

CE

F B

Node(i)

(%)

A 6 5/6 83%

E 7 5/7 72%

D 7 5/7 72%

B 8 5/8 63 %

C 9 5/9 55%

F 11 5/11 46%

(i)=

(i)=f= farnessc= closenessd= distance between i & jn= total number of nodes

Eigen vector

Eigen vector for N(i) = Neighbours of i = {J }

where N=(, )Eigen vector centrality:

Therefore,

(i)=

. ¿

Page Rank CetralityThe numerical weight that it assigns to any given element E is referred to as the PageRank of E and denoted by PR(E).Page Rank Centrality:

(i)=

Bonacich/Beta Centrality • Both centrality and power were a function of

the connections of the actors in one's neighborhood.

• The more connections the actors in your neighborhood have the more central you are.

• The fewer the connections the actors in your neighborhood, the more powerful you are.

• It is the weighted centrality

Bonacich/Beta Centrality:

=

Here,

(local importance)

DensityDensity: It is the level of ties/connectedness in a network; It is a measure of a network’s distance from a complete graph.

Complete graph: Every node is connected to every node in the network

L = number of links in networkn = number of nodes in the network

Ego Density

Structural Hole (Ron Burt)Let’s consider this,

• The gap between connected components is the hole

• Structural hole provides diversity of information for nodes that bridge them

• Without structural hole information becomes redundant and less available

1 2

Structural Hole

gap

Brokering

Brokering is bridging different group of individuals.1.Coordinator (local brokers; Intragroup brokering) e.g. manger, mediating employees

2. Consultant (Intergroup brokering by an outsider e.g. middle man in business between buyers &seller, stock agent )

A

B C

B as Coordinator/ Broker

Seller

Consultant Buyer

Brokering3. Representation (represents A when negotiating with C) e.g. hiring a mechanic to buy car for you

4. Gate Keeper(e.g a butler, chief of staff)

Actor Producer Agent

Actor

Producer

A

B

C

Dyadic Relation

Dyads: Triads: when a triad consists of many ties, an open triad (triangle) is forbidden.

A

A

A

A

B

B

B

B

C

A B0

Triad Relations (census)

Components Component is a group where all individuals are connected to one another by at least one path.

• Weak Component: A component ingoing direction of ties.

• Strong Component: A component with directional ties.

• Clique: A subgroup with mutual ties of three or more. who are directly connected to one another

by mutual ties

Bonacich Centrality

o CBC = Degree Centrality

High Degree + Low Betweenness : Ego Connection are redundant

Low Closeness+ High Betweenness : Rare node but pivotal to many

In triads, there is a structural force toward transitivity.

Reverse distance: )1( DiameterdRD ijij

VjidMaxDiameter ij ,),(

Principle of strength of weak tie.(Granovetter, 1973):There is a social force that suggests transitivity. If A has ties to B and B to C, then there is tie from A to C.

Bonacich

Integration:

1)(

n

RDkI kj

jkReverse distance

Network distance

)(max)()(

jkjk RDkIkI

Degree to which a node’s inward ties integrate it into the network.

Radiality

Degree to which a node’s outward ties connects the node with novel nodes.

1)(

n

RDkR kj

kj

)(max)()(

jkjk RDkRkR

Edge between-ness

st

stEB

eeC )()(

Number of shortest path from s to t that pass through edge e

Number of shortest path from s to t

This is important in diffusion studies like epidemics

Social Capital

The network closure argument: Social Capital is created by a strongly interconnected network.

The structural hole argument:Social Capital is created by a network of nodes who broker connections among disparate group.

Structural Equivalence= similarity of position in a network

Euclidean Distance

E.g.,

22 )()[( kjkijkik xxxxkji

01000100000001100110EDCBA

EDCBA

E A

C

B

D

0ABd

BA have distance zero

41.12 DEd