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

Area of Science:

• Complex systems science
• Network science
• Theoretical physics

Background:

• Real-world complex systems often feature sparse and heterogeneous interactions, unlike simplified models assuming all-to-all connections.
• Analytical solutions for dynamics in complex systems are typically restricted to models with full connectivity.

Purpose of the Study:

• To develop an analytical method for solving nonlinear dynamics in complex systems with sparse, directed, and random network structures.
• To generalize existing dynamical mean-field theory to sparse network regimes.

Main Methods:

• Generalized dynamical mean-field theory to accommodate sparse network structures.
• Derived an exact path-probability equation for the effective dynamics of a single degree of freedom.
• Employed the population dynamics algorithm to solve the derived equation.

Main Results:

• Developed a general solution applicable to diverse nonlinear models, including those in neural networks, ecosystems, epidemic spreading, and synchronization.
• Determined the phase diagram for a key neural network model in the sparse regime.
• Observed a transition from a fixed-point phase to chaotic dynamics as a function of network topology.

Conclusions:

• The generalized dynamical mean-field theory provides an effective framework for analyzing complex systems on sparse networks.
• Network topology plays a crucial role in determining the dynamical behavior of complex systems, including transitions to chaos.
• The findings offer new analytical tools for understanding phenomena across various scientific domains.