Characteristic function of connectivity with obstacles

Previous works usually simplify the network area as a disk, a square or a torus without obstacles. However, geometry in practical environment is much more intricate. Rectilinear communication between nodes is often blocked by external surroundings, such as buildings and mountains. Complicated shapes of the obstacles may cause the computation intractable. In this paper, we propose a characteristic function of connectivity to capture the geometric features of an ad-hoc network. The function can be expressed as power series, of which the orders and coefficients is based on the characteristics of the network. We formulate the shapes with fundamental factors and investigate their algebraic impact on connectivity. The critical transmission range is also calculated to mitigate the influence of obstacles.