Wasserman and Faust(1994) Study group of Social Network Analysis @ Social Computing Lab

1. Chapter 6. Structural Balance
and Transitivity
Chapter 7. Cohesive subgroup
박건우
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2. 6.1 Structural Balance
• individual’s attitudes or opinions coincided with the attitudes or
opinions of other “entities” of people : cognitive balance
• Structural balance : when two people are like each other, then they
are consistent in their evaluation of all other people.
– 내가 어떤 사람을 좋아하면, 나와 서로 좋아하는 다른 누군가 또한 의견
을 같이 한다
– structurally balanced group은 두 개의 subset으로 나눌 수 있다.
• only consider signed graph / signed digraph
• Generalization : structural balance < clusterability < transitivity
• (-)(-) = (+), (+)(+) = (+), (-)(+) = (-)
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3. 6.1.1 Signed Nondirectional Relations
• Type of lines
– positive attitude : (+), negative attitude : (-)
• The eight possible P-O-X triples
– sign of a cycle : product of the sign the lines
– First row : balanced, Second row : unbalanced
• Balance : the cycle has a positive sign
• A graph is balanced if all cycles are balanced
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4. 6.1.1 Signed Nondirectional Relations
• Balanced vs Unbalanced
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5. 6.1.2 Signed Directional Relations
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6. 6.2 Clusterability
• signed graph가 balanced 라면, 두 개의 set으로 나눌 수 있고 두 set
은 negative line으로 연결되어 있다. 하지만, 실제로는 네트워크에 두
개만의 cluster가 존재하지는 않는다. cluster 개념을 더 일반화 하자.
• Definition of Clusterability
– 즉, unbalanced graph도 clusterable할 수 있다.
• 어떤 조건에서, graph가 clusterable 할 것인가?
– 1) signed graph가 clusterable할 조건
– 2) complete signed graph가 clustering 되는 조건
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7. 6.2.1 The Clustering Theorem
• Theorem 6.1
– A signed graph has a clustering if and only if the graph contains no
cycles which have exactly one negative line
• Theorem 6.2
– The following four statements are equivalent for any complete signed
graph
• The graph is clusterable
• The graph has a unique clustering
• The graph has no cycle with exactly one negative line
• The graph has no cycle of length 3 with exactly one negative line
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8. 6.2.2 Summary(Clusterability)
• all balanced signed (di)graphs are clusterable, but clusterable
signed (di)graphs may or may not be balanced
• (-)(-)(-) triples
– not balanced
– but clusterable
• balance는 cycle의 sign이 (+)가 되어야 하므로 balance 하지는 않지만(전체
노드를 2개로 cluster하는 것이 불가능), clusterability는 각 3개의 노드가 다른
cluster에 속한다고 볼 수 있으므로 성립.
• balance : only two subsets are allowed
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9. 6.3 Generalization of Clusterablity
• 실제 네트워크의 전체적인 분포가 어떻게 되는지를 보면, 통계적 기
법으로 사용할 수 있다.(14장에서 다룸)
• 실제 네트워크들을 보고, 어떤 경향성을 나타내는지 보았다.
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10. 6.3.1 Empirical Evidence
• 800개의 sociomatrices의 특징들을 봄
1) many relations are directional
2) asymmetric dyads are very common
3) signed relations are rather rare
• some new theories are revised to unsigned relations
4) In some digraphs, one subset of actors chose a second, while actors in
this second subset chose members of a subset
• clusters appeared to be ranked -> ranked clusterablity
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11. 6.3.2 Ranked Clusterability
• for complete signed directed graphs
• [++], [--]는 같은 level을 나타내지만, [+-]는 서로 다른 level의 cluster
관계를 나타낸다.
– [++] : 같은 level의 같은 cluster
– [ -- ] : 같은 level의 다른 cluster
– [+- ] : 다른 level의 다른 cluster. +를 받은 쪽이 higher rank
• Complete signed directed graph에서 정의되는 ranked clusterability
를 unsigned graph에서 생각한다면?
=> Transitivity
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12. 6.4 Transitivity
• Definition 6.4
– The triad involving actors i, j, and k is transitive if whenever i->j and j->k
then i->k
– either of the two conditions is not met, triple is vacuously transitive
• Theorem 6.3
– A digraph is transitive if every triad it contains is transitive
– transitive digraph가 asymmetric dyad를 가지지 않는다면(all choices are
reciprocated) clusterable 하다.
– 즉, clusterability는 transitivity의 special case이고, 가장 general한
concept이다.
• Structural balance < Clusterability < Transitivity
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13. 6.4 Transitivity
1) Transitive
6,7,8,9
2) Vacuously transitive
1,2,3,4,5
3) Intransitive
10,11,12,13,14,15,16
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14. 7. Cohesive Subgroups
• Cohesive subgroup
– 서로 상대적으로 강하게 연결된 subgroup
• 7.1 Background
– social cohesion model
• The more tightly that individuals are tied into a network, the more they are
affected by group standards
• Two factors are operating : how many ties an individual has to the group and
how closed the entire group is to outsiders
– 어떤식으로 정량화 할 것인가?
• cohesion 개념은 다양한 네트워크 특성을 이용해 정량화가 가능하고, 다른 방
식으로 formalize 가능하다
– The mutuality of ties
– The closeness of reachability of subgroup members
– The frequency of ties among members
– The relevant frequency of ties among subgroup members compared to non-
members
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15. 7.2 Subgroups Based on Complete Mutuality
• Definition of a clique : a maximal complete subgraph of three or
more nodes
– mutual dyads are not considered
• 특징
– 한 노드가 여러 clique에 속할 수 있다
– 노드는 어느 clique에도 속하지 못할 수 있다
– 다른 clique에 완전히 속하는 clique는 없다. maximal이니까
– What are cliques here?
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16. 7.2 Subgroups Based on Complete Mutuality
• Clique의 문제점.
– clique 내의 actor들의 차이를 줄 수 없다
• Clique는 너무 엄격한(“stingy”) 개념이므로, 두 가지 방향으로 clique
를 완화시킨 cohesive subgroup을 보자
1) reachability, path distance, diameter
2) nodal degree
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17. 7.3 Subgroups based on reachability and
diameter
• important social processes occur through intermediaries
• Subgroup members might be adjacent, but if they are not adjacent,
then the path connecting them should be relatively short
• n-clique, n-clan, n-club
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18. 7.3 Subgroups based on reachability and
diameter
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19. 7.3 Subgroups based on reachability and
diameter
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20. 7.4 Subgroups Based on Nodal Degree
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21. 7.5 Comparing Within to Outside Subgroup
Ties
• 다른 subgroup으로 나가는 tie보다 상대적으로 subgroup 내부로 보내
는 tie가 많다면 cohesive subgroup이라는 idea
• LS set : comparing ties within the subgroup to ties outside the
subgroup by focusing on the greater frequency of the ties among
subgroup members compared to the ties from subgroup members to
outsiders
• Lambda Sets : cohesive subset은 subgroup에서 line을 없애는 것으
로부터 robust해야 한다는 idea에서 출발
– subset 내의 모든 pair의 line connectivity가 subset 내와 subset 밖의 어
떤 node의 pair connectivity보다도 크다
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22. 7.6 Measure of Subgroup Cohesion
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23. 7.7 Directional Relations /
7.8 Valued Relations
• Directed relation 과 valued relation에서 cohesive subgroup을 어떻게
정의하는가.
• 조건을 주어 undirected network와 같은 형태로 바꾸고, 앞에서 나왔
던 방법을 그대로 사용한다.
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24. 7.9 Interpretation of Cohesive Subgroups
• 해석은 크게 3가지 관점에서 볼 수 있다.
– Individual actor
– Subset of actors
– Whole group
• Individual Actor
– 다른 actor characteristic들을 보면서, member들과 non-member들 간에
어떤 차이가 있는지 볼 수 있고, 해석 할 수 있다.
• Subset of actors
– actor들 간 유사한 특성들을 가질 수 있다. (homophilly)
• Network as a whole
– 전체 네트워크에 그룹이 몇 개 있는지? 큰 하나의 그룹인지 아니면 나누
어져 있는지 등으로 해석 가능
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