Epsilon Equitable Partition: A positional analysis method for large social networks

Positional analysis is considered an important tool in the analysis of social networks. It involves partitioning of the set of actors into subsets such that actors in a subset are similar in their structural relationships with other actors. Traditional methods of positional analysis such as structural equivalence, regular equivalence, and equitable partitions are either too strict or too generic. For real world large graphs, most of these methods result into almost trivial partitions. We propose a useful relaxation to the concept of equitable partition called an epsilon equitable partition. A variant of epsilon equitable partition called maximal epsilon equitable partition is also proposed and formulated as an optimization problem. A fast algorithm for computing epsilon equitable partitions is proposed. We also present the results of performing positional analysis on a number of networks and demonstrate empirically that positional analysis with our notion gives raise to non-trivial and meaningful partitions. Along with the static network analysis with respect to positions, we study the impact of positional analysis on the evolution of networks. Our results show that positional analysis indeed plays an important role in the evolution of networks.