Dynamical Mean-Field Theory of Complex Systems on Sparse Directed Networks

Abstract

Although real-world complex systems typically interact through sparse and heterogeneous networks, analytic solutions of their dynamics are limited to models with all-to-all interactions. Here, we solve the dynamics of a broad range of nonlinear models of complex systems on sparse directed networks with a random structure. By generalizing dynamical mean-field theory to sparse systems, we derive an exact equation for the path-probability describing the effective dynamics of a single degree of freedom. Our general solution applies to key models in the study of neural networks, ecosystems, epidemic spreading, and synchronization. Using the population dynamics algorithm, we solve the path-probability equation to determine the phase diagram of a seminal neural network model in the sparse regime, showing that this model undergoes a transition from a fixed-point phase to chaos as a function of the network topology.

21 pages, 8 figures

References in corpus (14)

• Spectra of Sparse Random Matrices
• Mapping the Structure of Directed Networks: Beyond the "Bow-tie" Diagram
• Dynamical TAP equations for non-equilibrium Ising spin glasses
• Emergence of the giant weak-component in directed random graphs with arbitrary degree distributions
• Eigenvalue Outliers of non-Hermitian Random Matrices with a Local Tree Structure
• Parallel dynamics of disordered Ising spin systems on finitely connected directed random graphs with arbitrary degree distributions
• Inference of kinetic Ising model on sparse graphs
• High prevalence regimes in the pair quenched mean-field theory for the susceptible-infected-susceptible model on networks
• Generating Simple Directed Social Network Graphs for Information Spreading
• Nature of the Volcano Transition in the Fully Disordered Kuramoto Model
• Dynamically selected steady states and criticality in non-reciprocal networks
• Incorporating Heterogeneous Interactions for Ecological Biodiversity
• Dynamical Phase Transitions in Graph Cellular Automata
• Dynamical Theory for Adaptive Systems

Cited by in corpus (7)

• Irreversibility in Non-reciprocal Chaotic Systems
• Epidemic threshold and localization of the SIS model on directed complex networks
• Random matrix ensemble for the covariance matrix of Ornstein-Uhlenbeck processes with heterogeneous temperatures
• Starvation suppression in dense scale-free metabolic networks: Dynamical mean-field analysis of catalytic reaction networks
• Latent geometry emerging from network-driven processes
• Superstable Geometry in Triadic Percolation
• Zero-temperature dynamics of the spherical model with non-reciprocal interactions