Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale
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Transcript of Dr. Henry Hexmoor Department of Computer Science Southern Illinois University Carbondale
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