1. Concepts of Graph
Theory
Social Networks; Lecture 2

2. Summary
• Graph representation of social networks
• Matrix representation of social networks
• Node degree; average degree; degree distribution
• Graph density
• Walks, trails and paths
• Cutpoits, cutsets and bridges

3. What is a Network?
• A set of dyadic ties, all of the same type,among a
set of actors
• Actors can be persons, organizations ...
• A tie is an instance of a social relation

4. Relations Among Persons
• Kinship
– Mother of, father of, sibling of
• Role-Based
– Boss of, teacher of
– Friend Of
• Affective
– Likes, trusts
• Interactions
– Gives advice to; talks to; sexual interactions
• Affiliations

5. Content and Coding Matter!
• Each relation yields a different structure
and has different effects
• In real data, more then one relation
should be studied.
• Coding:
– What constitutes an edge?
– How to convert interview data into graph data?

6. Example

7. Problem Reformulation

8. Graph Theoretic Concepts
• Consists of a collection of nodes and
lines
G = N, L
N={n1 , n2 , n3 ...ng }
L = {l1 , l2 , l3 ...lL }
• Lines also called “ties” or “edges”
• Nodes occasionally called “agents” or
“actors”

9. Directed and Undirected
Ties
• •Undirected relations
Attended meeting with...
• Communicated with...
• Friend of...
• •Directed flows or subordination
relations
Represent
• “Lends money to”, “teacher Of”
• •Problemshould be symmetric can be measured as non-
-
Ties that
symmetric due to measurement error
• Friendship relations are not always reciprocal

10. Tie Strength
• We can attach values to ties, representing
quantitative attributes
• Strength of relationship
• Frequency of communication
• Information capacity/bandwidth
• Physical distance
• Such graph is called “weighted graph”
4

11. Adjacency Matrices
!quot;#$%&'%()*$+,-%&.
!quot;#$%&'(#)
*#+ *#,, *$% *-$
*#+ . / 0 /
*#,,
*#,, / . / 0
*$% 0 / . / / 5
*$%
*-$ / 0 / . *#+
6
1quot;-2#+#34 /8
5
*#+ *#,, *$% *-$ 7
*#+ . 56 7 *-$
*#,, 5 . / /8
*$% 6 /. 5
*-$ 7 /8 5 .

12. Sparse Matrix
!quot;#$%&'%()*$+,-%&.
!quot;#$%&'(#)
*#+ *#,, *$% *-$
*#+ . Jill
Jen / 0 1/
*#,,
*#,, / . / 0
Jen Joe 3
*$% 0 / . / / 5
Jill Jen 1. *$%
*-$ / 0 / *#+
Jill Jim 3 6
1quot;-2#+#34 Jill 3
Jim
/8
Jim *#,, *$%2*-$
*#+ Joe 5
7
*#+ . Jim 6 27
Joe 5 *-$
*#,, 5 Jen / 3
. /8
Joe
*$% 6 / . 5
*-$ 7 /8 5 .

13. Node Degree
• Degree of a node is a number of lines that
connect it to other nodes
• Degree can be interpreted as
• measure of power or importance of a node
• or
• measure of workload
• In directed graphs:
• indegree: number of incoming edges
• outdegree: number of outgoing edges

14. Marriage Ties Among
!quot;#$%&'
!quot;##$quot;%&'($&)'*+,-%
Leading Florentine Families
()*#*+',-(./
./,#&-0$-&'.quot;+$/$&)
1:#$-%';&-quot;$))quot;-2&
0$+&)
1quot;0quot;'2,+3$/&4'56'7,8-'9quot;4%&00

15. Degree Distribution

16. Graph Density
• Defined as ratio of number of edges in the
graph to the total POSSIBLE number of
edges:
L 2L
∆= =
g(g − 1)/2 g(g − 1)

17. Density and Network Survival:
Help with rice harvest
!quot;#$%&'()%()quot;%*'+quot;%!,-.quot;/
quot;#$%&'()%()quot;%*'+quot;%!,-.quot;/(
0,..quot;1/$2 4#&&'()+5
6'1'+7.-,
!quot;#quot;$%&'($)*#+,-#./$/#$quot;.

18. Components
!quot;#$%&'()quot;&*%+quot;,quot;-'./'#$#%0
BC,&4,)&1,&%,&+,&%C'%&4,)&#'$&+'4&DE&0'$&*%&F4&GGGG9&'$/&+Cquot;&+'4+&AAAH
• Maximal sets of nodes in which every node
can reach every other by some path
!quot;#quot;$%&'#()*+*%*,$
-./quot;0&'#()*+*%*,$+
-0*1*$'.&#,23'$4
5'%'&/0'6$&70,2&80,++9&:,01'%%*&;&<'0=quot;0&>??@A

19. Walks, Trails, Paths
• Walk = a sequence of nodes that can be
visited by following edges
• Trail = walk with no repeated lines
• Path = walk with no repeated node

20. Seven Bridges of
Königsberg

21. Path Length & Distance
• Length of path = number of links
• Length of shortest path between two nodes =
distance or “geodesic”
• Longest geodesic between any two nodes
• = graph diameter

22. !quot;#$%&'(')*+%,#-quot;
Example
0','1,%&'*+ !*
!quot;
0'6*#7+
!!
'4quot;%8quot;quot;#'%8/ (
6quot;#$%&'/0 )
'1,%&':,7,
; quot;
'
# !
&
$ %

23. Cutpoints
!quot;#$%&'#(
• Nodes, if deleted, would disconnect the
network
) *%+,(-./&0/1-&2-+,3,#,+1-.%quot;3+-+&(0%'',0#-',#
• Cutset = set of nodes required to keep a
graph connected
!$**quot;&
!$%
!quot;++
!quot;##
!&'')
!&'()

24. Bridges
!quot;#$%&'()*+,&quot;-&.,+(,
• An edge, if removed, would disconnect the
network 0 1&2),&23$2&#quot;44,#25&4quot;*,5&23$2&6quot;7
quot;23,(6)5,&8,&$2&%,$52&9&52,:5&$:$(
• Local bridge: connects nodes that otherwise
would be far removed
1
'

25. !quot;#$%&'&($')&*+$,-
Centralization
./,*&,'0%#1)12*,/*1$)'#&&*1)(2
• Degree to which network revolves around a
single node
<&/,'! <&/,'=
3/*/'4$5,*&26'$7'8149/&:';1)-

26. Next Time
• Centrality and Power in Social Networks
• Identification of Key Players