Understanding the relationships between players through network structures plays an important role in analyzing modern social and economic systems. For example, in shareholding networks, it is essential to quantitatively evaluate how each player influences others to analyze power dynamics and influence structures. Although the Shapley-Shubik power index is a well-known measure used to evaluate each player’s influence in games where voting power is distributed among players, this index do not explicitly consider network structures, and they fail to fully capture the relationships and influence propagation between players. Previous studies have extended the Shapley-Shubik power index to networks, introducing the Network Power Index (NPI) to represent the power exerted by one company i over another company j in shareholding networks. In this study, we give a graph-theoretic interpretation and description of the idea and mechanism of NPI and discuss it from a purely mathematical approach.