1. Social Position & Social Role Lei Tang 2009/02/13

2. Social Postion  Position: A collection of individuals who are similarly embedded in networks of relations.  Position is different from cluster (or cohesive subgroup)  Group is formed based on adjacency, proximity or reachability  This is typically adopted in current data mining.  Position is based on the similarity of ties among subsets of actors.  Actors occupying the same position need not be in direct, or even indirect contact with each other.

3. Social Role  Roles: the patterns of relations which obtain between actors of between positions  Position focuses on actors while roles focus on relations  E.g. Kinship role defined as combination of marriage and descent.  Can be modeled in three levels:  Actors  Subsets of actors  The whole network  Based on multiple relations and the combinations of these relations

4. Overview of Positional & Role Analysis Multirelational Data Usual Role Analysis Usual Positional Analysis Roles and Positions Group actors Group Relations Group actors (Individual level) (group level)

5. Structural Equivalence  Actor I and J are structurally equivalent:  For all the other actors k (!=I or J), actor I has tie to k iff actor J has tie to k.  Example: 1 2 3 4 5 Sociamatrix 12345 1-0110 20-110 300-01 4000-1 50000- The submatrices corresponding to the ties between and within positions are filled with either all 0’s or all 1’s.

6. Positional Analysis  Major objective: simplify the information in a network data set  Tasks:  A formal definition of equivalence  A measure of equivalence  A representation of equivalence  Density matrix  Image matrix  Reduced graph  Asses adequacy (Goodness of fit)

7. Structural equivalence to Valued Ties  For discrete ties, easy to define structural equivalence. (Very strict)  For valued ties  Euclidean distance  Correlation

8. Partition Actors (Clustering)  Consider each row as one data instance.  Agglomerative Hierarchical clustering  CONCOR (convergence of iterated correlations)  Based on multirelaitonal network A,  Calculate pairwise correlation matrix C1  Compute pairwise correlation matrix C2 based on C1.  Continue until we get the block of +1/-1  CONCOR is connected to PCA. (the top eigenvector)  Can only split into two positions, like divisive clustering +1 +1

9. Role Analysis  Consider different combination of relations  E.g. if aRb denotes a is mother of b, then  aRRb represent a new relation (grandmother)  They assume the relations between actors are already known.  But for us, this is seldom known.  The book is focusing more on analysis rather than methodology