Studies on the propagation of malware in mobile networks have revealed that the spread of malware can be highly inhomogeneous. Platform diversity, contact list utilization by the malware, clustering in the network structure, etc. can also lead to differing spreading rates. In this paper, a general formal framework is proposed for leveraging such heterogeneity to derive optimal patching policies that attain the minimum aggregate cost due to the spread of malware and the surcharge of patching. Using Pontryagin's Maximum Principle for a stratified epidemic model, it is analytically proven that in the mean-field deterministic regime, optimal patch disseminations are simple single-threshold policies. Through numerical simulations, the behavior of optimal patching policies is investigated in sample topologies and their advantages are demonstrated.

Abstract-In this work, we investigate the use of epidemic routing in energy constrained Delay Tolerant Networks (DTNs). In epidemic routing, messages are relayed by intermediate nodes at contact opportunities, i.e., when pairs of nodes come within the transmission range of each other. Each node needs to decide whether to forward its message upon contact with a new node based on its own residual energy level and the age of that message. We mathematically characterize the fundamental tradeoff between energy conservation and a measure of Quality of Service as a dynamic energy-dependent optimal control problem. We prove that in the mean-field regime, the optimal dynamic forwarding decisions follow simple threshold-based structures in which the forwarding threshold for each node depends on its current remaining energy. We then characterize the nature of this dependence. Our simulations reveal that the optimal dynamic policy significantly outperforms heuristics.

We analyze optimal strategies for the allocation of a finite budget that can be invested in different advertising channels over time with the objective of influencing social opinions in a network of individuals. In our analysis, we consider both exogenous influence mechanisms, such as advertising campaigns, as well as endogenous mechanisms of social influence, such as word-ofmouth and peer-pressure, which are modeled using diffusion dynamics. We show that for a broad family of objective functions, the optimal influence strategy at every time uses all channels at either their maximum rate or not at all, i.e., a bang-bang strategy. Furthermore, we prove that the number of switches between these extremes is bounded above by a term that is typically much smaller than the number of agents. This means that the optimal influence strategy is to exert maximum effort in waves for every channel, and then cease effort and let the effects propagate. We also show that, at the beginning of the campaign, the total cost-adjusted reach of an exogenous advertising channel determines its relative value. In contrast, as we approach our investment horizon (e.g., election day), the optimal strategy is to invest in channels able to target individuals instead of broad-reaching channels. We demonstrate that the optimal influence strategies are easily computable in several practical cases, and explicitly characterize the optimal controls for the case of linear objective functions in closed form. Finally, we see that, in the canonical example of designing an election campaign, identifying late-deciders is a critical component in the optimal design. I. INTRODUCTIONO PINIONS are important definers of real-world outcomes: they affect who is elected for political office [1], which policies are successful [2], and which products are bought by customers [3]. The proliferation of online media has complicated [4], sped up [5], and enhanced [6] opinion formation processes. The opinion formation process can be affected by interested parties through advertising channels, which are media by which messages are distributed to a target audience. Political campaigns and marketing departments apportion their advertising budgets between such channels (e.g., TV ads, website banner ads, billboards) in order to maximize some ultimate goal (e.g., votes, sales) [7], though the extent of the effect of these efforts is a matter of debate [1], [8]. The importance of this decision has increased in conjunction with the increasing resources devoted to these efforts: In 2017, over $1 trillion was spent on marketing globally [9], while $9.8 billion was spent on advertising in the 2016 US elections alone [10]. Thus, studying the related multi-channel resource allocation problem is both timely and significant.In particular, the mechanisms of opinion influence can be classified into two types based on its direct provenance. First, there are endogenous influence mechanisms (e.g., word-of-mouth), in which individuals process the expressed opinions of other individuals they me...

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