District heating networks (DHNs) are critical infrastructures that ensure production and residents' living. Analyzing vulnerability characteristics of DHNs is of great significance in heating safety decision-making. In this study, we analyze the structure, function, operation, and failure characteristics of DHNs and propose vulnerability analysis methods. We develop failure simulation models under fully-controllable (FC) and uncontrollable (UC) conditions for six failure scenarios based on component importance. To verify the proposed methods' effectiveness, we applied them to a DHN in a Chinese city. Upon analyzing the appositeness of various vulnerability assessment indicators, we find that the heat source connectivity efficiency loss rate, established based on hydraulic distribution in DHNs, can effectively characterize both topological and functional vulnerability. Our analysis also reveals that controllable DHNs, which evenly distribute heat supply among connected users for fairness, can result in lower flow and greater functional vulnerability compared to UC DHNs. In particular, under large area failure (AF) scenarios, the average functional vulnerability of UC DHNs increases by 27.59% to 38.30% compared to small-AF scenarios, while that of controllable DHNs increases by 103.78% to 120.58%. The proposed vulnerability assessment framework considering topology and function from both quantitative and qualitative perspectives, can grasp heating vulnerabilities multi-dimensionall