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After the groundbreaking work of Erd$\ddot{o}$s-R$\acute{e}$nyi random graph, the random networks has made great progress in recent years. One of the eye-catching modeling is time-varying random network model capable of encoding the instantaneous time description of the network dynamics. To further describe the random duration time for the nodes to be inactive, we herein propose a dinner party anomalous random networks model, and derive the analytical solution of the probability density function for the node being active at a given time. Moreover, we investigate the gift delivery and viral transmission in dinner party random networks. This work provides new quantitative insights in describing random networks, and could help model other uncertainty phenomena in real networks.

In this paper we propose an open anomalous semi-Markovian random neural networks model with negative and positive signals with arbitrary random waiting times. We investigate the signal flow process in the anomalous random neural networks based on renewal process, and obtain the corresponding master equation for time evolution of the probability of the potential of the neurons. As examples, we discuss the special cases of exponential waiting times and power law ones, and find the fractional memory effect of the probability of the system state on its history evolution. Besides, the closed random neural networks model is introduced and the corresponding rate equation is given.

Chemical master equation plays an important role to describe the time evolution of homogeneous chemical system. In addition to the reaction process, it is also accompanied by physical diffusion of the reactants in complex system that is generally not homogeneous, which will result in non-exponential waiting times for particle reactions and diffusion. In this paper we shall introduce a chemical continuous time random walk under anomalous diffusion model based on renewal process to describe the general reaction-diffusion process in the heterogeneous system, where the waiting times are arbitrary distributed. According to this model, we will develop the systematic stochastic theory including generalizing the chemical diffusion master equation, deriving the corresponding mass action law, and extending the Gillespie algorithm. As an example, we analyze the monomolecular $A\leftrightarrow B$ reaction-diffusion system for exponential and power-law waiting times respectively, and show the strong fractional memory effect of the concentration of the reactants on the history of the concentration in power-law case.

To capture the dynamic behaviors of reaction-subdiffusion in flow fields, in the present paper we analyze a simple monomolecular conversion A $\rightarrow$ B. We derive the corresponding master equations for the distribution of A and B particles in continuous time random walks scheme. The new results are then used to obtain the generalizations of advection-diffusion reaction equation, in which the diffusion and advection operators both depend on the reaction rate.