Social network dominance based on analysis of asymmetry

We focus on analysis of dominance, power, influence - that by definition asymmetric - between pairs of individuals in social networks. We conduct dominance analysis based on the canonical analysis of asymmetry that decomposes a square asymmetric matrix into two parts, a symmetric one and a skew-symmetric one, and then applies the singular value decomposition (SVD) on the skew-symmetric part. Each individual node can be projected as one 2-dimensional point based on its row values at each pair of successive singular vectors. The asymmetric relationship between two individuals can then be captured by areas of triangles formed from the two points and the origin in each 2-dimensional space. We quantify node dominance (submissive) score based on the relative position of the node's coordinate from coordinates of all other nodes it dominates (subdues) in the projected singular vector spaces. We conduct dominance/submissiveness analysis for several representative networks including perfect linear orderings, networks with tree structure, and networks with random graphs and examine the departures of a real social network from those representative graphs. Empirical evaluations demonstrate the effectiveness of the proposed approach.