Network observability for source localization in graphs with unobserved edges

Localizing a source of diffusion is a crucial task in various applications such as epidemics quarantine and identification of trendsetters in social networks. We analyze the problem of selecting the minimum number of observed nodes that would lead to unambiguous source localization, i.e. achieve network observability, when both infection times of all the nodes, as well as the network structure cannot be fully observed. Under a simple propagation scenario, we model the assumption that, while the structure of local communities is well known, the connections between different communities are often unobserved. We present a necessary and sufficient condition for the minimum number of observed nodes in networks where all components have either a tree, a grid, a cycle or a complete graph structure. Additionally, we provide a sufficient condition for the selection of observed nodes when the components are of arbitrary structure. Through simulation, we illustrate the performance of the proposed bound.