Erik Csikos | Searcharxiv

arXiv2026

Mapping the DAG-ness Landscape: Structural Archetypes in Complex Networks

Directed networks arise across biological, social, informational, and engineered systems, yet most analyses treat directedness as a binary property: a network is either a directed acyclic graph (DAG) or it is not. This binary classification obscures the rich spectrum of hierarchical, recurrent, and modular structure present in real systems. In this paper, we empirically evaluate the DAG-ness framework, a four-component measure that quantifies acyclicity, flow alignment, cyclic locality, and pathway complexity across a corpus of 107 networks drawn from twelve structurally diverse domains. Rather than aligning with traditional disciplinary boundaries, our results reveal unexpected cross-domain convergence: diverse systems resolve into four universal structural archetypes. We find that macroscopic acyclicity is pervasive even in feedback-rich systems, and that domains as disparate as neural connectomes and abstract informational networks frequently converge on identical topological constraints. These findings demonstrate that DAG-ness provides a unified, interpretable, and domain-agnostic lens for understanding the hidden laws of directed structure in complex systems.

cs.SI↗

arXiv2026

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

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.

cs.SI↗