Smart cities are considered to be one of the most important applications of the IoT notion. Most smart city applications rely fundamentally on ubiquitous sensing, enabled by Wireless Sensor Network (WSN) technologies. These sensor networks are vulnerable to different challenges that cause failures in some parts of the network, which in turn interfere with the availability of network services and weaken the user experience. In this paper, we introduce a graph-theoretic model of wireless sensor networks used in smart cities. Moreover, we present several challenges, such as natural disasters and random failures and evaluate the system's performance in terms of data delivery, end to end delay, and energy consumption. The evaluation results show that fire is the challenge that causes the most damage among the three challenges examined, while random failure has the least effect on network performance. The results also show that the modeled WSN's can cope well with the challenge of random failures.