Statistical analyses of cyclic and starlike hierarchical dominances in directed graphs

Abstract

The mathematical framework of matrix decomposition implies the possibility to perform statistical analyses of directed graphs focused on the distributions of the independent cyclic and starlike hierarchical components. In this approach the weighted directed graphs with n nodes are built up as a linear combination of starlike graphs with n outgoing edges and a suitable set of three-edge cyclic subgraphs. The applicability of this approach is illustrated by quantifying several general features: e.g., ratio of the cyclic and hierarchical components and asymmetry in the hierarchical components. The applicability of these methods is illustrated by considering the averages over random directed graphs and comparing these values with those characterizing simple directed graphs of tournaments.

General features of Nash equilibria in combinations of elementary interactions in symmetric two-person games

“…The payoff matrix of the other elementary cyclic games can be derived from this matrix by suitably exchanging its rows and columns. Elementary cyclic components are not all independent of each other [28,53,54]. There are, however, many subsets of all C (j,k,l) whose elements span the whole space of cyclic games.…”