Here we developed a new conceptual, stochastic Heterogeneous Opinion-Status model (HOpS model), which is adaptive network model. The HOpS model admits to identify the main attributes of dynamics on networks and to study analytically the relation between topological network properties and processes taking place on a network. Another key point of the HOpS model is the possibility to study network dynamics via the novel parameter of heterogeneity. We show that not only clear topological network properties, such as node degree, but also, the nodes' status distribution (the factor of network heterogeneity) play an important role in so-called opinion spreading and information diffusion on a network. This model can be potentially used for studying the co-evolution of globally aggregated or averaged key observables of the earth system. These include natural variables such as atmospheric, oceanic and land carbon stocks, as well as socio-economic quantities such as global human population, economic production or wellbeing.