How to Quantify Direct and Indirect Impacts Using Matrix Models

We often talk about the "impact" of A on B, but this includes not only direct influence but also indirect influence through intermediate events or related factors. For example, in a corporate setting, the sales department (A) directly affects customer satisfaction (B), while its performance also indirectly influences the R&D department (C) by increasing investment in new product development, which can improve product quality and further boost customer satisfaction.

Mathematical Model

We consider a system composed of N elements. The possible influence of each element on another can be represented by an N×N matrix M, where M_ij denotes the direct influence strength of element i on element j. If M_ij ≠ 0, element i directly influences element j; otherwise there is no direct influence.

Direct influence can be read directly from the matrix, while indirect influence requires considering multi‑level interactions.

Specifically, the second‑order indirect influence of element i on element j can be expressed by the (i,j) entry of M², because a second‑order effect means the influence passes through one intermediate node. If element i directly influences k and k directly influences j, then the second‑order indirect influence from i to j is the product of those two direct influences summed over all possible intermediate k. Mathematically this is obtained by the matrix product M·M.

Similarly, third‑order indirect influence involves two intermediate nodes and is obtained from M³, representing all paths of length three.

To compute the “total influence” of element i on element j we need to sum all levels of direct and indirect influence. The total influence can be estimated by the infinite series I = M + M² + M³ + …

In practice the series is truncated at a chosen maximum order L, depending on stability and convergence considerations.

Case Study

Assume a simple company model with three departments: Sales (A), R&D (B), and Production (C). Their direct influences are represented by the following matrix:

To calculate the total influence from Sales to Production we include the direct influence (A→C) and the indirect influence through R&D (A→B→C), as well as higher‑order effects.

First, the direct influence A→C is given by the (A,C) entry of the matrix, which equals 0.3.

Next, the second‑order indirect influence via B is obtained by the (A,C) entry of M². The calculation involves multiplying the direct influence A→B by the direct influence B→C.

(Sales to R&D influence)

(R&D to Production influence)

The resulting second‑order indirect influence is … (value omitted). Higher‑order indirect influences, such as third‑order effects, are computed from M³ and capture paths like Sales → R&D → Production → Sales, illustrating feedback loops.

By evaluating the series up to the desired order we obtain the total influence matrix, from which we can read the total impact from Sales to Production (e.g., value 1.371) and observe the complex interplay among departments.

Total impact strength: The total influence from Sales to Production (matrix entry (1,3) = 1.371) combines the direct 0.3 and the accumulated indirect effects, reflecting the comprehensive effect of Sales on Production.

Complexity of inter‑departmental influence: Examining other rows shows that R&D’s total impact on Production (1.422) is particularly high, indicating its pivotal role in the indirect pathways.

Feedback and propagation: Third‑order calculations reveal feedback cycles (e.g., Sales → R&D → Production → Sales), helping understand long‑term strategic effects and potential business loops.

Using matrices to describe and compute direct and indirect influences provides a systematic view of dynamic relationships. The method’s mathematical rigor and computational flexibility make it applicable to many complex systems.

This matrix approach is not limited to economic systems; it can also be applied to ecological, social science, and network science domains, offering researchers a powerful tool to uncover and understand influence propagation.

Finally, the underlying mindset is systems thinking: any single event or action is part of a web of multi‑level interactions, a perspective valuable across enterprises, natural ecosystems, and everyday life.

Appendix: Neumann Series Theorem

In mathematics and engineering, the series Σ M^k is often studied, especially when dealing with direct and indirect influences of matrices. This series is known as the Neumann series of a matrix.

Neumann Series Theorem

For a matrix A, if the spectral radius of A (the absolute values of all eigenvalues) is less than 1, then (I‑A) is invertible and we have:

Here Σ M^k is the sum of the power series of M, and (I‑M)⁻¹ is the inverse of (I‑M). This formula provides a direct way to compute the sum without calculating each power individually.

An important application of the theorem is to assess convergence of the series by examining the eigenvalues of M. If all eigenvalues have magnitude less than 1, the series converges; otherwise it diverges.