Hong Zhang

Continuous-time random walks (CTRWs) are highly appreciated as a model of anomalous diffusion (AD) in complex media such as structured and heterogeneous environments. These walks are used to describe AD phenomena characterized by a nonlinear functional dependence of the mean-squared displacement (MSD) of particles with time. The introduction of asymmetric waiting-time distributions (WTDs) depending on the diffusion directions leads to biased CTRWs and to AD phenomena. Here, we examine the general formulation of such CTRWs with asymmetric WTDs in continuous space. From the balance equations, we derive the corresponding generalized Montroll-Weiss equation in Fourier-Laplace space as well as the generalized master equations with asymmetric waiting times. For random jump lengths obeying an asymmetric modified Gaussian distribution (AMGD), we analytically obtain the probability-density function of the particles as well as the first two moments in Laplace space. We unveil that CTRWs with exponential WTDs and jump lengths following the AMGD exhibit a ballistic diffusion, whereas in a symmetric case for the jump lengths both the MSD and variance at long time have a linear scaling with time. The AD properties of the MSD and of the variance at long times depend on a exponent of the power-law WTDs, while the AMGD-distributed jump lengths the AD scaling exponents for such CTRWs. In this continuous-space model, the possibility of length-distributed jumps—in contrast to the case of only neighboring-site jumps allowed in the discrete version of a similar model we proposed recently [G. Li , ]—leads to different resulting scaling properties of the MSD. The analytical results are in close agreement with the results of computer simulations, both for the case of asymmetric exponential and of power-law WTDs. The current theoretical results are extendable to more elaborate CTRW models. The latter are applicable for the description of more complicated AD-related phenomena taking place in the presence of external fields.

To describe the heterogeneity of the neural network, we propose an anomalous random neural network (ARNN) with arbitrary distributed waiting times, which is a generalization of the random neural network. We investigate the signal flow process in ARNN based on the renewal process and obtain the generalized master equations for the time evolution of the probability of the state vector of neurons. From the obtained master equations, we obtain the generalized rate equations for the time evolution of the average potential of each neuron in both closed and open ARNN systems. It is proved that when the distribution of waiting time is exponential, the generalized rate equations for open ARNN systems can reduce to Hopfield neural networks; when the distribution of waiting time is power-law, the corresponding rate equations become the generalized fractional-order Hopfield neural networks. Particularly, for a single neuron, we derive a power-law firing rate that matches the experiment [Lundstrom et al., Nat. Neurosci. 11, 1335 (2008)].

We propose a continuous-time unidirectional random walk model for a heterogeneous population of particles involving subdiffusive trapping effects. In this model, after escaping from the trap, each particle either jumps forward with a random probability or remains in the same place. The population heterogeneity is captured by modeling the jump probability as a beta-distributed random variable. The randomness in this transition parameter generates an effective jump probability with the ensemble self-reinforcement. We derive the limiting probability for the ensemble average of the particle position using an integral subordination formula. We show that the average particle position can be represented by a time-changed Pólya process involving an inverse stable subordinator.

Epidemic spreading in heterogeneous networks has attracted great interest in recent years. To capture the significant effect of residence of individuals on epidemic spreading, we consider herein a simple susceptible-infected-susceptible model with random waiting time in heterogeneous networks. We provide the analytical dynamical expressions for the time evolution for infected individuals and find a fractional memory effect of power-law waiting time on anomalous epidemic spreading. This work provides new quantitative insights in describing contagion processes and could help model other spreading phenomena in social and technological networks.

The subject of aging continuous time random walks (CTRWs) has attracted increasing attention in recent years. To describe the aging behaviors of random particles whose jumps are biased by a nonhomogeneous velocity field, we propose herein a generalized scheme of aging CTRWs in flows and obtain the corresponding generalized master equation in Fourier–Laplace space for probability density functions. Moreover, we derive the generalized aging advection diffusion equation for particles with a power law waiting time and Gaussian jump length densities, investigate the corresponding ensemble and time mean square displacements, and show how anomalous diffusion depends on the age of the process and on the moving fluids.