Social Position & Social Role Lei Tang 2009/02/13
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.
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
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)
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: Sociamatrix The submatrices corresponding to the ties between and within positions are filled with either all 0’s or all 1’s.
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)
Structural equivalence to Valued Ties For discrete ties, easy to define structural equivalence. (Very strict) For valued ties Euclidean distance Correlation
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
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