MULTILEVEL GRAPHS: A DYNAMIC MATRIX APPROACH

Resumen

This paper presents a dynamic matrix framework for the normalization and simplification of complex
systems using second-order multilevel graphs. Building on the visual alternative proposed by Muñoz
Garro [1], we represent a system as a block adjacency matrix whose entries are integrable functions of
edge weights. We compute the derivative of this matrix with respect to a continuous parameter to quantify the rate of structural change and integrate it to capture the cumulative influence of each connection.
Our mixed methodology combines a theoretical formalization of the correspondence between subgraphs
and submatrices—with conditions for continuity and integrability—and an experimental phase of Python
simulations on three case studies: industrial process networks, corporate information flows, and synthetic
graphs. The results show that the structural derivative identifies critical nodes with high agreement with
established metrics, that a small fraction of edges accounts for most of the system complexity, and that
the method achieves substantial reduction in structural complexity while preserving essential connectivity. Computational costs grow cubically, indicating potential scalability considerations. We conclude
that this model offers a robust quantitative complement to visual techniques and supports automation
and extension to higher-order graphs.