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}}}. We show that the correlation functions depend on the spectral dimension and that only for tilde{{boldsymbol{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.
Highlights
Stochastic dynamics on large scale networks has attracted a lot of attention due to its wide occurrence in many disciplines, such as social sciences[1,2,3], physics and biology[4,5,6], communication and control theory[7]
We present the model in the context of porous systems such as brain tissue which can be measured using Magnetic Resonance Imaging (MRI)
Since the behaviour of higher moments and the intermittency property are highly dependent on the second moment[26,27], we focus on the second-moment dynamics for insight into interesting dynamical properties of the model: d⟨mrml⟩ dt
Summary
Stochastic dynamics on large scale networks has attracted a lot of attention due to its wide occurrence in many disciplines, such as social sciences[1,2,3], physics and biology[4,5,6], communication and control theory[7]. Of interest is the population dynamics on networks, which is usually affected by both the topology of the network and some internal stochastic noise in the system. The main question we ask is what kind of information about the topology of the graph can we extract based on the dynamics of measurable functions, such as the correlation function between the sites of the network in a non-equilibrium model. The property mi(t) is linked to a physical measurable quantity in the real world and the graph is the underlying geometry/topology in which the property lives and which usually is a complex network of sites. The model consists of two parts: an interacting part, where diffusion components exchange with strength depending on their location on the graph, and a non-interacting part, where each component follows a stochastic noise with variance σ2. The first causes spreading, while the second pushes towards concentration (a.k.a localization or condensation)
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