Thermoacoustic instabilities that arise in the context of lean premixed combustion are a major challenge in the development of gas turbines and aero engines. This thesis contributes to the research on modal analysis of thermoacoustic instabilities by applying linear stability analysis and sensitivity analysis of longitudinal thermoacoustic modes within a low order network model framework. The first part of the thesis is dedicated to the better understanding of different types of modes, particularly the interplay between classical acoustic modes and intrinsic thermoacoustic (ITA) modes in a simple thermoacoustic system. Anticorrelated modal sensitivities are found to arise due to a pairwise interplay between acoustic and ITA modes. The magnitude of the sensitivities increases as the interplay between the modes grows stronger. The results show a global behaviour of the modes linked to the presence of exceptional points in the spectrum. Non-normal behaviour and its consequences are also investigated in this setup. The use of sensitivity information is shown to be capable of quantitatively assess the degree of coupling and decoupling of the thermoacoustic modes \francol{from} the acoustic components of a simplified geometry. The investigation shows the effect of an area expansion on the modes of a thermoacoustic tube. The last part of the thesis proposes an optimisation procedure based on the sensitivity information to identify the optimal volume and placement of dampers, which are used to damp thermoacoustic oscillations.