The main investigation in this thesis is the research on a future quantum internet. The focus is laid on the network structure of this quantum network. The network is rst examined by the investigation of the Kuramoto model for classical oscillators. The main property of the network is the ability to synchronise all nodes such that all oscillate with the same frequency. Several parameters of the oscillators are varied to verify why and when synchronisation takes place. After modelling the calculated equations of motion, an interesting conclusion arises. For a symmetric ring network, a stable conguration is not always found but by introducing asymmetric oscillators in the network, the synchronous state can be found. This conclusion leads to the idea that a quantum network requires certain dierences in its structure in order to guarantee transmission to take place. Furthermore, the existence of synchronisation heavily depends on the parameters used for the oscillators. The analysis is then continued in the quantum domain where optomechanical systems are introduced. At rst, two systems are connected to each other by a gaseous interaction and an electric interaction via a Dung circuit. The main investigation is again into the synchronisation, which implies that the operators belonging to both systems behave the same as a function of time. Then several synchronisation measures are introduced to measure the ability of the systems to synchronise. As predicted, both systems synchronise in terms of the operators for each system. Then a quantum network is introduced, where a complex yet ecient network is created. This quantum network is a small world network, where a transmitter node is able to connect to the receiver node by only a few links. Furthermore, multiple transmissions can take place at the same time and links which are not connected do not synchronise with the transmitter node. After modelling this network, the results are in compliance with the theory available. Challenges for a practical quantum network include the experimental basis of being able to utilise optomechanical systems outside laboratory circumstances. Also, the network should be tested for practical use and expanded in a much larger size both in theoretical analysis as well as in experiments.

Modelling the spread of contagious diseases among people has been a research topic for over a hundred years. However, the increase in computation power in the recent years allows for more advanced scientific models. The well-known susceptible-infected-susceptible model is used to describe the spreading of a disease among a group of people. This is modelled as a network, where persons are represented by nodes and their connections are links in the network. In this thesis, instead of a typical static network, the network itself changes structure based on the disease states of the nodes. In other words, there are two independent processes; the spreading of the disease over the network (state of the nodes) and the adaption of the network to the disease (state of the links). As a spreading model, the Markovian adaptive susceptible-infected-susceptible model (ASIS model for short) is introduced. It is shown that the model has one steady state, named the trivial steady state, in which all individuals are healthy. When the infection rate is sufficiently high, the system undergoes a phase transition from the disease-free state to an endemic state, where most nodes are infected. The state where most nodes are infected is named the metastable state. After being in the metastable state for a long time, the system collapses to the trivial steady state. The spreading of contagious processes is not just limited to disease spreading. Other relevant examples, such as opinion, gossip, fake news, neuron transmittance in the brain, etc. can be modelled using adaptive models as well. In this thesis, the ASIS model is extended by allowing different rules for the link-breaking and link-creation processes in what we call the Generalised ASIS framework. In total 36 models have been analysed simultaneously. Out of the 36 models, 9 showed a partially unstable metastable state. This resulted in rapid oscillations of the number of infected nodes just above the epidemic threshold. The relation between the epidemic threshold and the effective link-breaking rate was also determined. For 5 cases, the epidemic threshold is independent of the effective link-breaking rate. In 18 cases, the epidemic threshold scales linearly in the link-breaking rate. The remaining 13 cases are bounded between a constant and a linear link-breaking rate, however, its exact dependence remains unclear. In the G-ASIS framework, it was conjectured that the metastable state of the Markov process can be accurately approximated by the steady state of the mean-field approximation. It was shown this is not true for every model. However in the ASIS model, the mean-field approximation showed close resemblance to the averaged stochastic results. This may be caused by a positive correlation between the nodes. For the other models, the deviation from the mean field approximation can probably be contributed to the fact that nodes are not positively correlated. Other higher order mean field approximations should be examined to approximate the averaged behaviour of the Markov process.