Recent years have witnessed a growing interest in understanding the fundamental principles of how epidemic, ideas or behaviour spread over large networks (e.g. the Internet or online social networks). The conventional approach is to use the susceptible-infected-susceptible (SIS) model or its derivatives. We like to note that these models are often too restrictive and may not be applicable in many realistic situations. In this paper, we propose a 'generalized SIS model' by allowing the existence

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... of intermediate states between susceptible and infected states. To analyse the diffusion process of the generalized SIS model on large graphs, we use the 'mean-field analysis technique' to determine which initial condition leads to or prevents the outbreak of information or virus. For any general connected graphs, we show that the condition which can prevent the spread of contagions depends on two de-coupled effects: the network topology and the parametric values of the generalized SIS model. Experimental results based on both synthetic and real-world datasets show that our methodology can accurately predict the behaviour of the phase-transition process for any general graphs. We also extend our generalized SIS model to analyse the dynamics and behaviour of two competing sources. This is useful if one wants to model competing products in a large network or competition between virus and antidote in a large communication network. We present the analytical derivation and show via experiment how different factors such as initial condition, transmission rates, recovery rates or the number of states can affect the phase transition process and the final equilibrium. Our models and methodology can serve as an essential tool in analysing and understanding the information diffusion process in large networks.