Positional Dominance

Positional dominance is a powerful generic concept that unifies the majority of centrality indices and role equivalences. It defines a relation on the nodes of a network, where one node dominates another if it has stronger relationships with comparable others. The result is a new network with the same set of nodes and directed edges $(i,j)$ whenever the position of node $i$ dominates the position of node $j$ in the network.

The position of a node is defined by either the incoming or outgoing links of that node. The strengths of links may vary with a numeric link attribute. When comparing two positions, we are really comparing relationships. A relationship is stronger than another, if its link attribute is at least as large and both source and target are comparable to the source and target of the other relationship. The position of a node dominates that of another, if the relationships defining it are stronger.

The simplest example is the following:

• uniform link strength
• outgoing links
• exclude compared dyad
• source comparability: grouped by uniform (no a-priori distinctions between nodes, all sources created equal)
• target comparability: unique (the relationship to a target can only be matched by a relationship to the very same target)

This results in a network in which a node dominates another positionally if it has outgoing links to the very same other nodes (and, possibly, more). Now include the compared dyad and switch target comparability to grouped by uniform; with these new settings, a node dominates another positionally, if it's outdegree is no less. Finally, by checking single-link dominance, we can have a node dominate any other node simply by having a non-zero degree.

NB: Currently, the provision of positional dominance serves as a preview to future forms of network analysis. A comprehensive explanation of the utility of network positions is in preparation.