Monitoring Stealthy Diffusion

Starting with the seminal work by Kempe et al., a broad variety of problems, such as targeted marketing and the spread of viruses and malware, have been modeled as selecting a subset of nodes to maximize diffusion through a network. In cyber-security applications, however, a key consideration largely ignored in this literature is stealth. In particular, an attacker often has a specific target in mind, but succeeds only if the target is reached (e.g., by malware) before the malicious payload is detected and corresponding countermeasures deployed. The dual side of this problem is deployment of a limited number of monitoring units, such as cyber-forensics specialists, so as to limit the likelihood of such targeted and stealthy diffusion processes reaching their intended targets. We investigate the problem of optimal monitoring of targeted stealthy diffusion processes, and show that a number of natural variants of this problem are NP-hard to approximate. On the positive side, we show that if stealthy diffusion starts from randomly selected nodes, the defender's objective is submodular, and a fast greedy algorithm has provable approximation guarantees. In addition, we present approximation algorithms for the setting in which an attacker optimally responds to the placement of monitoring nodes by adaptively selecting the starting nodes for the diffusion process. Our experimental results show that the proposed algorithms are highly effective and scalable.