Integrating sparse and heterogeneous network structure into DMFT for Eq. (uds) and deriving phase diagrams

Develop a dynamical mean-field theory that rigorously incorporates sparse and heterogeneous network structures into the analysis of nonlinear systems governed by the differential equation dot{x}_i(t) = - f(x_i) + sum_{j=1}^N A_{ij} g(x_i, x_j), and determine the phase diagrams of such sparse complex systems in the limit N → ∞.

Background

Dynamical mean-field theory (DMFT) has yielded exact solutions for the effective single-variable dynamics of many complex systems in the fully connected (dense) regime. However, real-world interaction networks are typically sparse (finite mean degree) and heterogeneous (fluctuating local topology), and incorporating these features into DMFT has posed a longstanding challenge.

The paper focuses on sparse directed networks and presents an exact path-probability equation solvable via population dynamics for several models (e.g., neural networks, ecosystems, SIS, Kuramoto). The cited sentence delineates the broader unresolved task of integrating sparse and heterogeneous structures within DMFT and of deriving phase diagrams for sparse complex systems—objectives that have remained out of reach in general formulations.