Mathematical Theory of Depth via Depth-Propagation Tensors

This work develops a rigorous mathematical theory of depth as an emergent structural quantity in arbitrary relational systems. Rather than defining depth through combinatorial hierarchy (such as trees or path lengths), the framework derives depth directly from inter-element influence propagation. The paper introduces the Depth-Propagation Tensor (DPT), a recursive tensorial structure that captures hierarchical accumulation across graded propagation levels. Convergence of the depth recursion is established under Lipschitz-type conditions, and the resulting fixed-point formulation places depth within nonlinear operator theory over positive cones. Key contributions include: Formal definition of relational depth as an aggregate functional Recursive propagation framework yielding fixed-point equilibria Construction of the Depth-Propagation Tensor (DPT) Proof of convergence and uniqueness under contraction conditions Operator-theoretic interpretation linking to Perron–Frobenius theory Statistical depth metrics (expectation, variance) Continuous-limit formulation via nonlinear Fredholm integral equations This work forms Part I of the WDS (Weighted Depth Structures) research series, establishing the discrete foundational theory for subsequent nonlinear spectral and phase-transition analysis. The framework is relevant to: Spectral graph theory Nonnegative matrix theory Network centrality analysis Nonlinear dynamical systems Hierarchical structure modeling