Spectral Analysis of a Non-Equilibrium Stochastic Dynamics on a General Network

Abstract

Abstract Unravelling underlying complex structures from limited resolution measurements is a known problem arising in many scientific disciplines. We study a stochastic dynamical model with a multiplicative noise. It consists of a stochastic differential equation living on a graph, similar to approaches used in population dynamics or directed polymers in random media. We develop a new tool for approximation of correlation functions based on spectral analysis that does not require translation invariance. This enables us to go beyond lattices and analyse general networks. We show, analytically, that this general model has different phases depending on the topology of the network. One of the main parameters which describe the network topology is the spectral dimension $$\tilde{{\boldsymbol{d}}}$$ d˜ . We show that the correlation functions depend on the spectral dimension and that only for $$\tilde{{\boldsymbol{d}}}$$ d˜ > 2 a dynamical phase transition occurs. We show by simulation how the system behaves for different network topologies, by defining and calculating the Lyapunov exponents on the graph. We present an application of this model in the context of Magnetic Resonance (MR) measurements of porous structure such as brain tissue. This model can also be interpreted as a KPZ equation on a graph.

Keywords

• Moment Lyapunov Exponent
• Limited Measurement Resolution
• Lattice Topology
• Vertex Function
• Transient Graph