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This work investigates three aspects: (a) a network vulnerability as the non-uniform vulnerable-host distribution, (b) threats, i.e., intelligent worms that exploit such a vulnerability, and (c) defense, i.e., challenges for fighting the threats. We first study five data sets and observe consistent clustered vulnerable-host distributions. We then present a new metric, referred to as the non-uniformity factor, which quantifies the unevenness of a vulnerable-host distribution. This metric is essentially the Renyi information entropy and better characterizes the non-uniformity of a distribution than the Shannon entropy. We then analytically and empirically measure the infection rate and the propagation speed of network-aware worms. We show that a representative network-aware worm can increase the spreading speed by exactly or nearly a non-uniformity factor when compared to a random-scanning worm at the early stage of worm propagation. This implies that when a worm exploits an uneven vulnerable-host distribution as a network-wide vulnerability, the Internet can be infected much more rapidly. Furthermore, we analyze the effectiveness of defense strategies on the spread of network-aware worms. Our results demonstrate that counteracting network-aware worms is a significant challenge for the strategies that include host-based defense and IPv6.