How Many Sensors to Localize the Source? The Double Metric Dimension of Random Networks

We consider the problem of detecting the source of an epidemic that spreads in a network. The only information about the epidemic comes from a subset of nodes, which we call sensors, and which can reveal if and when they become infected. How many sensors do we need to guarantee that the epidemic source is correctly identified? The answer to this question is a known network property, called the double metric dimension (DMD); unfortunately, it is hard to compute. We compute tight bounds for the DMD of $\mathcal{G}(N, p)$ random networks. Interestingly, these bounds are non-monotonic functions of the edge density p: this implies in turn that the detectability of the source is influenced by the edge density p in a non-monotonic fashion in $\mathcal{G}(N,p)$ networks. We show empirically that this property applies to other topologies as well.