A Continuous Multi-Component Measure of Directed Acyclicity (DAG-ness)

Abstract: Directed acyclic graphs (DAGs) are fundamental to the study of causal structures, hierarchical systems, and information flow. While directedness and acyclicity are defined as binary properties, real-world networks often exhibit continuous degrees of "DAG-ness" due to structural noise, back-edges, or localized feedback loops. Our previous attempt to quantify DAG-ness as a continuous measure suffered from topological redundancy, where overlapping cyclic penalties artificially deflated scores for networks with minor feedback. In this paper, we resolve these limitations by introducing a strictly orthogonal, 4-dimensional continuous DAG-ness framework. By independently measuring the volume of feedback $A(G)$, the alignment of flow $F(G)$, the macroscopic locality of feedback $M(G)$, and dynamical pathway complexity $S(G)$, the proposed measure eliminates collinearity and the "Dilution Trap." Empirical evaluation on synthetic diagnostic graphs demonstrates enhanced mathematical stability, while deterministic application to classical number-theoretic systems (the Kaprekar and Collatz graphs) confirms the framework's ability to rigorously isolate topological flow from dynamical entrapment. The resulting composite score $D(G)$ provides a highly scalable, interpretable, and mathematically sound metric for structural network analysis.