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This work provides an information-theoretic view to better understand the relationships between aggregated vulnerability information viewed by attackers and a class of randomized epidemic scanning algorithms. In particular, this work investigates three aspects: 1) a network vulnerability as the nonuniform vulnerable-host distribution, 2) threats, i.e., intelligent malwares that exploit such a vulnerability, and 3) defense, i.e., challenges for fighting the threats. We first study five large data sets and observe consistent clustered vulnerable-host distributions. We then present a new metric, referred to as the nonuniformity factor, that quantifies the unevenness of a vulnerable-host distribution. This metric is essentially the Renyi information entropy that unifies the nonuniformity of a vulnerable-host distribution with different malware-scanning methods. Next, we draw a relationship between Renyi entropies and randomized epidemic scanning algorithms. We find that the infection rates of malware-scanning methods are characterized by the Renyi entropies that relate to the information bits in a nonunform vulnerable-host distribution extracted by a randomized scanning algorithm. Meanwhile, we show that a representative network-aware malware can increase the spreading speed by exactly or nearly a nonuniformity factor when compared to a random-scanning malware at an early stage of malware propagation. This quantifies that how much more rapidly the Internet can be infected at the early stage when a malware exploits an uneven vulnerable-host distribution as a network-wide vulnerability. Furthermore, we analyze the effectiveness of defense strategies on the spread of network-aware malwares. Our results demonstrate that counteracting network-aware malwares is a significant challenge for the strategies that include host-based defenses and IPv6.