A reduced Mason’s rule for solving linear systems of equations recursively: an application to the integrated ownership problem in a network of companies

This paper considers the study of integrated ownership in a universe of companies represented by a corporate network. The difficulty is that the system of relations is characterized by a number of cross-shareholdings (i.e., cycles). Given a source node s and a target node t, the problem is to establish whether s owns t, either directly or indirectly. In the literature, the problem is tackled by computing an ownership matrix and solving a linear system of equations. Solving the system for checking whether s owns t may cause an unnecessary computational effort which can be avoided. We propose a recursive algorithm based on the Mason’s gain formula, which is usually applied in network flow-propagation problems for analyzing the feedback effect. The number of operations needed to compute the formula grow up factorially as the number of nodes in the network increases. In fact, the solution process is hard when the network contains Strongly Connected Components of large size. Yet, in real applications corporate networks are sparse and have few Strongly Connected Components of small size. We propose a recursive algorithm based on a new simplified version of the Mason’s formula that is not computed on the whole network but only on groups of firms which form a Strongly Connected Component. This reduces the computational burden for determining the integrated ownership of a source company s in all the possible target nodes.