Localization phase transition in stochastic dynamics on networks with hub topology

Highlights

• Derivation of the full phase diagram, predicting the transition between localization and balancing.

• The average mass ratio satisfies a logistic growth equation.

• Extension to multiple hubs dynamics.

• Applicability of the model to analyze MR measurements of porous structure is discussed.

Abstract

Dynamics among central sources (hubs) providing a resource and large number of components enjoying and contributing to this resource describes many real life situations. Modeling, controlling, and balancing this dynamics is a challenging problem that arises in many scientific disciplines. We analyze a stochastic dynamical system exhibiting this dynamics with a multiplicative noise. We show that this model can be solved exactly by passing to variables that describe the mass ratio between the components and the hub. We derive a deterministic equation for the average mass ratio in the absence of noise on the hub. This equation describes logistic growth. We derive the phase diagram of the model with and without noise on the hub. We show that when there in no noise on the hub there is no localization phase. In the presence of noise on the hub, we identify two regimes by deriving the equilibrium distribution of the process. The first regime describes balance between the non-hub components and the hub, in the second regime the resource is concentrated mainly on the hub. We generalize the results to a system with multiple hubs. We show that there is less concentration on the hubs as the number of hubs increases, and in the limit of infinite hubs the average mass ratio grows or decays exponentially. Surprisingly, in the limit of large number of components the transition values do not depend on the amount of resource given by the non-hub nodes. We propose an interesting application of this model in the context of porous media using Magnetic Resonance (MR) techniques.

Introduction

Population 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], physics [3] and biology, communication and control theory [4]. This dynamics is mainly affected by the topology of the network, as well as some internal stochastic noise. In many applications there are only a few nodes playing a major role in the dynamical process, distributing and carrying most of the resources [5], [6]. For example, this can be the case in models describing population dynamics, economic growth [7], and distributed control systems [4]. An additional application of this problem is in the context of diffusion measurements of porous systems, such as brain tissue, using Magnetic Resonance (MR) techniques. In this last case, the sensitivity of the MR signal to self-diffusion of water molecules can be utilized to extract information about the network of cells (neurons) in the brain. The concept of self-diffusion of molecules in a network of pores was already introduced in Refs. [8], [9], [10], [11]. The main challenge is how to determine the topology of the network based on the MR measurements [9].

Our model consists of a system of interacting sites on a graph $\mathcal{G}$ with $N$ vertices and $E$ edges between them. We are interested in the stochastic dynamics of some characteristic property ${\left\{m_i\left(t\right)\right\}}_{i\in \mathcal{G},t\ge 0}$. The property $m_i\left(t\right)$ is linked to a physical measurable quantity in the real world and the graph is the underlying geometry/topology in which the property lives. The topology is a complex network of sites. The model is described by the following family of stochastic differential equations in the Stratonovich form on the graph $\mathcal{G}$:

$$\frac{d{m}_i\left(t\right)}{dt}=\sum _j{J}_{ij}{m}_j\left(t\right)-\sum _j{J}_{ij}{m}_i\left(t\right)+g_i\left(t\right)m_i\left(t\right),$$

with the initial conditions $m_i\left(0\right)=m_{i,0}$. The term $g_i\left(t\right)$ is a multiplicative white noise, such that $〈g_i\left(t\right)〉=f_i$, and $〈g_i\left(t\right)g_j\left(t^{\prime }\right)〉={\sigma }_i^2{\delta }_{ij}\delta (t-t^{\prime })$. We choose the Stratonovich form, since its solution is a limit of a physical system involving white noise with short memory [12]. The topology of the network is encoded in the adjacency matrix $J$ of the graph. The model consists of two parts: an interacting part, where the interaction strength depends on the location on the graph, and a non-interacting part, where each component follows a stochastic noise with different variance ${\sigma }_i^2$. The first part causes spreading, while the second pushes towards concentration (a.k.a localization or condensation). The model was already analyzed in the mean field topology, i.e., when all the nodes are connected and interact at the same rate. In this case, the equilibrium distribution is a Pareto power-law [1]. It was also analyzed on trees [13], [14], and random graphs assuming separable probability distribution on the nodes [15]. The model on the lattice is known in the mathematical literature as the time-dependent Parabolic Anderson Model (PAM) [16], [17]. The phase diagram of the model in this case depends on the dimension of the lattice. On a general network, phase transitions depend on the spectral dimension of the network [9].

Here, we present and analyze a specific topology in which the model is shown to be solvable. Namely, we consider a directed graph with $N+1$ nodes, one hub node interacting with $N$ independent non-hub nodes. In the context of MR measurements of diffusion in a porous structure, the MR signal measured is assumed to be composed of two contributions: one coming from hindered diffusion in the extracellular space and the other from restricted diffusion in the intracellular space [18], [19], [20], [21]. The hub node represents the magnetization in the extracellular space (e.g., water), $h_0\left(t\right)$, and the non-hub nodes represents $N$ independent intracellular pores with magnetization, $m_i\left(t\right)$. The motion of molecules between these regions changes the value of the magnetization as a function of time and is represented by the interaction term between nodes, $J_{ij}$. The effect of the magnetic field gradient can be incorporated in the stochastic noise, for example, in $f_i$, and/or its variance ${\sigma }_i^2$. In the economic context, the system describes the dynamics of the money hold by the hub, which represents the state/bank, and the money of each agent $m_i\left(t\right)$. In this case, the agents deposit money in the bank and the bank pays interest on it. The stochastic noise represents the bank/state and the agent’s investments in the stock market and housing [1], [7]. Analysis of the dynamics of the sums of the money held by the agents and the bank/state was curried out in Ref. [7].

Here, we analyze the dynamics of the mass ratio between each agent and the bank/state (hub). First, in the absence of noise on the hub and then in the presence of noise on the hub. We show that in the absence of noise, there is no localization phase transition. We show this in two ways by calculating the Lyapunov exponents of the process and by deriving the equilibrium distribution for the mass ratio. On the other hand, in the presence of noise on the hub, there exists localization phase transition. We show this by deriving the equilibrium distribution of the process. Finally, we generalize our analysis to multiple number of hubs.

Our main result is a full phase-diagram of the model. We show that this model can be described by a stochastic equation for the mass ratio between each of the non-hub nodes and the hub, and a decoupled non-linear equation for the average relative mass of all the non-hub nodes with respect to the hub. This equation describes logistic grows. It is deterministic in the absence of noise on the hub. The phase transition is characterized by only one parameter. This parameter accounts for the exchange rate between the non-hub nodes and the hub and the variances of the multiplicative noises.