Network Theory:
Computational Phenomena and Processes
Social Network Analysis
Dr. Henry Hexmoor
Department of Computer Science
Southern Illinois University Carbondale
Degree, Indegree, Outdegree
Centrality
Degree Centrality:
C D (i ) 
n
x
ij
j 1
Indegree Centrality:
n
C I (i ) 
x
ji
j 1
Outdegree Centrality:
C O (i ) 
n
x
i 1
ij
C D ( i )  normalized
Centrality

C D (i )
n 1
 Percentage
Eigenvector Centrality=CE (i)=I’th entry of eigenvector e

(     )V  
  [ 1 ,  2 ,...,  n ]
e = largest eigenvalue of adjacency matrix
Betweenness Centrality
C B (k ) 

 ikj
 ij
,i  j  k
 ikj= number of geodesic linking across i and j has pass through node k.
Normalized betweenness=
C B (k )
( n  1)( n  2 ) 2
Closeness Centrality
n
C c (i ) 
d
ij
j 1
 Centrality
n
: C  (i ) 

j 1
 
 
Scaling factor
Adjustment
ij
(   C  ( j ) )
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
v5
v1
v2
v3
v4
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 network
Actors are tied to ----Organization/Attributes;
e.g. Affinity network, Homophily network
Attribute m1………………..Attribute mn
Org1……………………..…….Org n
1
2
.
.
n
1
2
.
.
n
People
Sociogram ≡
{
}
Points-------------individuals
Lines --------------relationship
Attributes
Centrality
Star graph
A has higher degrees. A is central to all.
Centrality: Quantifying a network node.
𝐶𝐷 (i)=
𝑛
𝑗=1 𝑥ij
Normalized Centrality
E
F
Centrality:
𝐶𝐷 (A)=4
A
D
C
B
Normalized Centrality:
𝐶𝐷 (A) = 4 = 80%
‘
𝐶𝐷 (A)=
6-1
5
𝐶𝐷 (B)=3, 𝐶𝐷 (C)=2, 𝐶𝐷 (D)=2, 𝐶𝐷 (E)=2 , 𝐶𝐷 (F)=1
A is more central than F
Directed network centrality:
E
F
𝐶𝑜 (A)=3
𝐶𝑜 (F)=1
A
D
𝐶𝐼 (A)=2
𝐶𝐼 (F)=∅
C
B
𝐶𝐼 (B)=2
𝐶𝐼 (D)=2
𝐶𝐼 (E)=2
𝐶
(A)
2
𝐷
𝐶𝐷‘(A)=
= = 40% (Normalize centrality)
5
6-1
Prestigue of A=B=C=E=2 (Indegree)
Eigen vectors
Vector X is a matrix with n rows and column, linear
operator A, maps the vector X to matrix product AX
𝑥1
𝑥2
.
𝐴1,1
𝐴2,1
.
. → .
.
.
.
.
𝑥𝑛
𝐴𝑛,1
𝐴1,2 … … .
𝐴2,2 … … .
.
.
.
.
𝐴𝑛,2
→
A𝑋 = ƛ. A
𝐴1,𝑛 𝑥1
𝐴2,𝑛 𝑥2
.
.
.
.
.
.
.
.
𝑎𝑛,𝑛 𝑥𝑛
=
𝑦1
𝑦2
.
.
.
.
𝑦𝑛
Eigen Value
3
−3
ƛ to the equations |𝐴 − ƛ. I| ≠ ∅
Second degree centrality
Node
A
Value
.534
B
.275
C
.363
D
.199
E
.199
F
.199
G
.199
H
.102
I
.102
Eigen value centrality
J
.164
𝐶𝐷 (A)= 𝐶𝐷 (O)=6 higher Eigen values
K
.164
L
.164
M
.164
N
.164
O
.441
Consider this 16 degree graph network:
I
H
B
M
N
O
C
G
A
D
E
L
K
J
F
Betweenness Centrality
Betweennes centrality measures the extent to
which a vertex lies on paths between other
vertices.
𝛿𝑖𝑘𝑗 =Number of paths from i to j
passing through k
𝐶𝐵 (K)=
𝛿𝑖𝑘𝑗
𝛿𝑖𝑗
𝛿𝑖𝑗 = Number of shortest distance path
from i to j
K= Geodesic distance
Normalized centrality:
E
A
F
D
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%
C
B
𝐶𝐵 (𝐾)
𝐶′𝐵 =
[(𝑛−1)(𝑛−2)/2]
Closeness Centrality
Closeness centrality is the mean distance from a
vertex to other vertices.
E
F
A
D
C
Node
(i)
𝐶𝑓
𝐶𝐶
𝐶𝐶 (%)
B
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%
𝑛
𝑗=1 𝑑𝑖𝑗
𝐶𝑐 (i)=[𝐶𝑓 (i)]−1
𝐶𝑓 (i)=
f= farness
c= closeness
d= distance between i & j
n= total number of nodes
x (𝑛 − 1)
Eigen vector
Eigen vector for N(i) = Neighbours of i
= {J ∈ 𝑉 | 𝑖, 𝑗 ∈ E}
where N=(𝑉, ∈)
Eigen vector centrality:
𝐶𝐸 (i)=
J∈𝑁(𝑖) 𝐶𝐸 (J)
Therefore,
ƛ. E = AT . E
Page Rank Cetrality
The 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:
𝐶𝑃𝑅 (i)=
𝐶𝑃𝑅 (J)
𝐽∈𝑁(i) 𝑑𝑒𝑔
𝑂𝑢𝑡 (J)
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:
𝑪𝑩𝑪 =
𝑱∈𝑵(𝒊)(𝜶
+ [𝜷 x 𝑪𝑩𝑪 (𝐉)])
=𝜶 𝐥𝐨𝐠 𝒏 𝐢 + 𝜷.
𝑱∈𝑵(𝐢) 𝑪𝑩𝑪 (𝑱)
Here,
𝛼 = Importance of the node i′ s degree
(local importance)
𝛽 = Imporatance of neighbour ′ s centrality
(global imporatance)
Density
Density: 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
Ego Density
L = number of links in
network
n = number of nodes in
the network
Structural Hole (Ron Burt)
Let’s consider this,
1
gap
2
Structural Hole
• 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
Brokering
Brokering is bridging different group of individuals.
1.Coordinator (local brokers; Intragroup brokering)
C
e.g. manger, mediating employees B
A
B as
Coordinator/
Broker
2. Consultant (Intergroup brokering by an outsider
e.g. middle man in business between buyers &seller,
stock agent )
Consultant
Seller
Buyer
Brokering
3. Representation (represents A when negotiating
with C) e.g. hiring a mechanic to buy car for you
A
C
B
4. Gate Keeper(e.g a butler, chief of staff)
Actor Producer Agent
Actor
Producer
Dyadic Relation
Dyads:
A
B
A
B
A
B
A
B
Triads: when a triad
consists of many ties,
an open triad (triangle)
is forbidden.
A
C
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.
Bonacich
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.
Diameter
Reverse distance:
 Max ( d ij ), i , j  V
RD ij  d ij  (1  Diameter )
Integration:
 RD
I (k ) 
Reverse distance
jk
jk
n 1
Network distance
I (k )
I ( k ) 
max
jk
( RD jk )
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.
 RD
R (k ) 
kj
jk
n 1
R (k )
R ( k ) 
max
jk
( RD
jk
)
Edge between-ness
Number of shortest path from s to t that
pass through edge e
C EB ( e ) 

 st ( e )
 st
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

 [( x ik  x jk )  ( x ki  x kj )
2
2
i j k
E.g.,
A
A 

B 0
0
C

D 0

E 0
B
C
D
E
0
1
1

1
1
0

0
0
1

0

0

0

0


0
d
1
AB
0
0
E
B
A
C
D
A  B have distance zero
d DE 
2  1 . 41