Are Large Random Graphs Always Safe to Hide?

We discuss winning possibilities of players in various variants of cops and robber game played on large random graphs, a testbed for various kinds of network queries, search problems in particular. We explore the use of logic frameworks to investigate such results; in particular, we show that whenever a winning condition for either player can be expressed as a certain kind of formula in first-order logic, that player almost always wins. In the process, we obtain more insight into the logic-game connection from the zero-one law perspective.