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# 2 Graph Theory

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### 2 Graph Theory

1. 1. Concepts of Graph Theory Social Networks; Lecture 2
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. 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. 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. 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. 6. Example
7. 7. Problem Reformulation
8. 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. 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. 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. 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. 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. 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. 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. 15. Degree Distribution
16. 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. 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. 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. 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. 20. Seven Bridges of Königsberg
21. 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. 22. !quot;#\$%&'(')*+%,#-quot; Example 0','1,%&'*+ !* !quot; 0'6*#7+ !! '4quot;%8quot;quot;#'%8/ ( 6quot;#\$%&'/0 ) '1,%&':,7, ; quot; ' # ! & \$ %
23. 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. 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. 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. 26. Next Time • Centrality and Power in Social Networks • Identiﬁcation of Key Players