Opinion dynamics models study how the interaction among people influences the opinion formation process. In most opinion dynamics models, only one opinion could exist in the steady state, which is different from the real-life opinion formation process. In 2009, Shao \emph{et al.} introduced a non-consensus opinion (NCO) model, which allows different opinions to coexist in the steady state. This thesis extends the NCO model by introducing a special type of nodes, Byzantine nodes, to play the role of dishonest people. The Byzantine NCO model is more in line with the real-world opinion formation process because it considers that people who express opinions are not always honest. I build an NCO model simulation algorithm and use this algorithm to perform simulations on three different network models: small-scall graphs, the Erdős–Rényi random graph and the scale-free network. In Byzantine node selection, three different strategies are proposed, according to the degree of the selected nodes. I find a new steady state for the NCO model: the cyclic steady state. The cyclic behaviour of the NCO and Byzantine NCO model is discussed, and some networks with a long cycle period are given. I also introduced a general method to generate networks with extremely long cycle periods. The other properties of the Byzantine NCO model, such as the probability of cyclic behavior, the final opinion distribution and the convergence time are researched. By performing simulations on the network models, I find that the introduction of Byzantine nodes could help the system to reach a steady state with a more balanced opinion ratio. The introduction of Byzantine nodes could decrease the critical threshold of the NCO model and promote the coexistence steady state. A mechanism in which Byzantine nodes influences the convergence time by influencing the steady state is suggested.

The spreading process of diseases has been an important research topic for many years. It has profound effects on the development of human social behaviors. The underlying social network structure may change when individuals change their connection with other individuals in response to the epidemic. The classic susceptible-infected-susceptible (SIS) model is used to model the spread of an epidemic on a network, where all individuals are defined as nodes and the connections between the individuals are regarded as links. Besides the typical static network, the structure of the network can be related to the state of nodes (infected or susceptible) by link breaking and link creation processes. So the extended network, adaptive susceptible-infected-susceptible model (ASIS model) can be derived.

To study the spreading process on static networks and adaptive networks, we use stochas- tic simulations and mean field approximations. We assume that the spreading process over the network is a continuous-time discrete-state Markov process. But most recent works use the discrete-time simulator, which is actually an approximation of the process. In this report, we extend a existing continuous-time simulator towards adaptive networks. This existing simulator is based on the Gillespie algorithm. We perform the simulations using both discrete-time Markov chain and continuous-time Markov process. And based on the simulation results, we demonstrate that the continuous-time simulator has a better performance than the discrete-time simulator on modeling both static SIS network and ASIS network with high accuracy.

The second part of this work aims to study the characteristics of the ASIS network in the metastable state. We observe three possible states of the ASIS network: the endemic state, disease-free state and bistable state. The degree distribution of the graph follows a binomial distribution in some cases. By plenty of simulations with different parameters, we illustrate under what circumstances the degree distribution follows the binomial distribution.