suppose we observe a security network composed of sensors with each sensor returning a value indicating whether the sensor is at risk (high value) or not (low value). ?a typical attack leaves a trail where the sensors return higher-than-normal values.? the goal is to detect a possible attack.? within a simplified framework, we will see that if the sensor do not return high-enough values (we will quantify that), then detection is impossible. formal abstract: consider the complete regular binary tree of depth m oriented from the root to the leaves.? to each node we associate a random variable and those variables are assumed to be independent.? under the null hypothesis, these random variables have the standard normal distribution while under the alternative, there is a path from the root to a leaf along which the nodes have the normal distribution with mean a and variance 1, and the standard normal distribution away from the path. we show that, as m increases, the hypotheses become separable if, and only if, a is larger than the square root of? 2 ln 2.? we obtain corresponding results for other graphs and other distributions.? the concept of predictability profile plays a crucial role in our analysis. joint work with emmanuel candes, hannes helgason and ofer zeitouni.