Recursive Structure as a Distinct Dimension of Network Dynamics: A Cycle-Based Metric Reveals Limitations of Integration and Correlation-Based Inference

Quantifying the relationship between network structure and emergent dynamics remains a central challenge in neuroscience and complex systems theory. Existing approaches predominantly emphasize global integration, as formalized in frameworks such as Integrated Information Theory (IIT), or rely on statistical measures derived from observed activity, such as correlation-based connectivity. However, these approaches do not explicitly capture recursive structure, defined as the presence of closed causal pathways enabling feedback. In this study, we introduce a cycle-based metric, Recursive Integration Depth (RIDv4), designed to quantify recursive structure in networked dynamical systems. Using simulated networks across multiple connectivity regimes, we demonstrate that RIDv4 distinguishes structured feedback networks from both weakly connected and randomly dense networks, even when integration is high. Comparisons with baseline cycle-counting and entropy-based metrics show that these approaches fail to reliably capture recursive organization. Furthermore, when applying RIDv4 to connectivity matrices inferred from time-series data, we observe a reversal effect: systems with true underlying feedback structure exhibit lower measured recursion than random systems. This indicates that correlation-based inference does not fully preserve recursive structure and may obscure underlying feedback dynamics. These findings suggest that recursive structure constitutes a meaningful structural property not captured by standard integration or statistical approaches, and that its detection depends critically on how network representations are constructed.