Reliable electrical power supply is a prerequisite for the modern society, and if it fails, it can cause severe consequences in terms of economic losses and even fatalities. It is thus important to analyse the vulnerability of the electric power system. Network analysis has previously been used to analyse the vulnerability of electric transmission systems. Recent events in Sweden, however, have shown that perturbations in distribution systems can also cause severe societal consequences. Thus, we argue that vulnerability analysis at the distribution level is equally important. Furthermore, previous work has focused on the technical aspects of the system, and in this paper we take a step towards incorporating the societal aspects of vulnerability by suggesting new network analytic measures. We analyse the distribution systems in two Swedish municipalities using the proposed measures. We conclude that the proposed measures can increase the value of using network analysis when analysing societal vulnerability to perturbations in electric distribution systems and that such analysis also can be useful in emergency mitigation and preparedness planning.