In a variety of settings, some payoff-relevant item spreads along a network of connected individuals. In some cases, the item will benefit those who receive it (for example, a music download, a stock tip, news about a new research funding source, etc.) while in other cases the impact may be negative (for example, viruses, both biological and electronic, financial contagion, and so on). Often, good and bad items may propagate along the same networks, so individuals must weigh the costs and benefits of being more or less connected to the network. The situation becomes more complicated (and more interesting) if individuals can also put effort into security, where security can be thought of as a screening technology that allows an individual to keep getting the benefits of network connectivity while blocking out the bad items. Drawing on the network literatures in economics, epidemiology, and applied math, we formulate a model of network security that can be used to study individual incentives to expand and secure networks and characterize properties of a symmetric equilibrium.